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authorDaniel Baumann <daniel.baumann@progress-linux.org>2024-04-28 13:14:23 +0000
committerDaniel Baumann <daniel.baumann@progress-linux.org>2024-04-28 13:14:23 +0000
commit73df946d56c74384511a194dd01dbe099584fd1a (patch)
treefd0bcea490dd81327ddfbb31e215439672c9a068 /src/math
parentInitial commit. (diff)
downloadgolang-1.16-73df946d56c74384511a194dd01dbe099584fd1a.tar.xz
golang-1.16-73df946d56c74384511a194dd01dbe099584fd1a.zip
Adding upstream version 1.16.10.upstream/1.16.10upstream
Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>
Diffstat (limited to 'src/math')
-rw-r--r--src/math/abs.go14
-rw-r--r--src/math/acos_s390x.s144
-rw-r--r--src/math/acosh.go59
-rw-r--r--src/math/acosh_s390x.s158
-rw-r--r--src/math/all_test.go3812
-rw-r--r--src/math/arith_s390x.go77
-rw-r--r--src/math/arith_s390x_test.go442
-rw-r--r--src/math/asin.go55
-rw-r--r--src/math/asin_386.s30
-rw-r--r--src/math/asin_s390x.s162
-rw-r--r--src/math/asinh.go71
-rw-r--r--src/math/asinh_s390x.s213
-rw-r--r--src/math/atan.go105
-rw-r--r--src/math/atan2.go71
-rw-r--r--src/math/atan2_386.s13
-rw-r--r--src/math/atan2_s390x.s297
-rw-r--r--src/math/atan_386.s13
-rw-r--r--src/math/atan_s390x.s128
-rw-r--r--src/math/atanh.go79
-rw-r--r--src/math/atanh_s390x.s174
-rw-r--r--src/math/big/accuracy_string.go17
-rw-r--r--src/math/big/arith.go287
-rw-r--r--src/math/big/arith_386.s244
-rw-r--r--src/math/big/arith_amd64.go11
-rw-r--r--src/math/big/arith_amd64.s525
-rw-r--r--src/math/big/arith_arm.s283
-rw-r--r--src/math/big/arith_arm64.s583
-rw-r--r--src/math/big/arith_decl.go18
-rw-r--r--src/math/big/arith_decl_pure.go53
-rw-r--r--src/math/big/arith_decl_s390x.go18
-rw-r--r--src/math/big/arith_mips64x.s38
-rw-r--r--src/math/big/arith_mipsx.s38
-rw-r--r--src/math/big/arith_ppc64x.s481
-rw-r--r--src/math/big/arith_riscv64.s46
-rw-r--r--src/math/big/arith_s390x.s795
-rw-r--r--src/math/big/arith_s390x_test.go32
-rw-r--r--src/math/big/arith_test.go673
-rw-r--r--src/math/big/arith_wasm.s35
-rw-r--r--src/math/big/bits_test.go224
-rw-r--r--src/math/big/calibrate_test.go173
-rw-r--r--src/math/big/decimal.go270
-rw-r--r--src/math/big/decimal_test.go134
-rw-r--r--src/math/big/doc.go99
-rw-r--r--src/math/big/example_rat_test.go65
-rw-r--r--src/math/big/example_test.go148
-rw-r--r--src/math/big/float.go1730
-rw-r--r--src/math/big/float_test.go1858
-rw-r--r--src/math/big/floatconv.go304
-rw-r--r--src/math/big/floatconv_test.go825
-rw-r--r--src/math/big/floatexample_test.go141
-rw-r--r--src/math/big/floatmarsh.go120
-rw-r--r--src/math/big/floatmarsh_test.go139
-rw-r--r--src/math/big/ftoa.go536
-rw-r--r--src/math/big/gcd_test.go64
-rw-r--r--src/math/big/hilbert_test.go160
-rw-r--r--src/math/big/int.go1218
-rw-r--r--src/math/big/int_test.go1896
-rw-r--r--src/math/big/intconv.go257
-rw-r--r--src/math/big/intconv_test.go431
-rw-r--r--src/math/big/intmarsh.go80
-rw-r--r--src/math/big/intmarsh_test.go121
-rw-r--r--src/math/big/link_test.go63
-rw-r--r--src/math/big/nat.go1569
-rw-r--r--src/math/big/nat_test.go816
-rw-r--r--src/math/big/natconv.go512
-rw-r--r--src/math/big/natconv_test.go463
-rw-r--r--src/math/big/prime.go320
-rw-r--r--src/math/big/prime_test.go222
-rw-r--r--src/math/big/rat.go544
-rw-r--r--src/math/big/rat_test.go728
-rw-r--r--src/math/big/ratconv.go375
-rw-r--r--src/math/big/ratconv_test.go616
-rw-r--r--src/math/big/ratmarsh.go75
-rw-r--r--src/math/big/ratmarsh_test.go125
-rw-r--r--src/math/big/roundingmode_string.go16
-rw-r--r--src/math/big/sqrt.go128
-rw-r--r--src/math/big/sqrt_test.go126
-rw-r--r--src/math/bits.go62
-rw-r--r--src/math/bits/bits.go588
-rw-r--r--src/math/bits/bits_errors.go15
-rw-r--r--src/math/bits/bits_errors_bootstrap.go22
-rw-r--r--src/math/bits/bits_tables.go83
-rw-r--r--src/math/bits/bits_test.go1347
-rw-r--r--src/math/bits/example_test.go210
-rw-r--r--src/math/bits/export_test.go7
-rw-r--r--src/math/bits/make_examples.go112
-rw-r--r--src/math/bits/make_tables.go92
-rw-r--r--src/math/cbrt.go79
-rw-r--r--src/math/cbrt_s390x.s156
-rw-r--r--src/math/cmplx/abs.go13
-rw-r--r--src/math/cmplx/asin.go221
-rw-r--r--src/math/cmplx/cmath_test.go1589
-rw-r--r--src/math/cmplx/conj.go8
-rw-r--r--src/math/cmplx/example_test.go28
-rw-r--r--src/math/cmplx/exp.go72
-rw-r--r--src/math/cmplx/huge_test.go22
-rw-r--r--src/math/cmplx/isinf.go21
-rw-r--r--src/math/cmplx/isnan.go25
-rw-r--r--src/math/cmplx/log.go65
-rw-r--r--src/math/cmplx/phase.go11
-rw-r--r--src/math/cmplx/polar.go12
-rw-r--r--src/math/cmplx/pow.go81
-rw-r--r--src/math/cmplx/rect.go13
-rw-r--r--src/math/cmplx/sin.go184
-rw-r--r--src/math/cmplx/sqrt.go107
-rw-r--r--src/math/cmplx/tan.go297
-rw-r--r--src/math/const.go52
-rw-r--r--src/math/copysign.go12
-rw-r--r--src/math/cosh_s390x.s211
-rw-r--r--src/math/dim.go81
-rw-r--r--src/math/dim_amd64.s98
-rw-r--r--src/math/dim_arm64.s49
-rw-r--r--src/math/dim_riscv64.s70
-rw-r--r--src/math/dim_s390x.s96
-rw-r--r--src/math/erf.go339
-rw-r--r--src/math/erf_s390x.s293
-rw-r--r--src/math/erfc_s390x.s527
-rw-r--r--src/math/erfinv.go127
-rw-r--r--src/math/example_test.go240
-rw-r--r--src/math/exp.go191
-rw-r--r--src/math/exp2_386.s40
-rw-r--r--src/math/exp_amd64.s159
-rw-r--r--src/math/exp_arm64.s182
-rw-r--r--src/math/exp_asm.go11
-rw-r--r--src/math/exp_s390x.s177
-rw-r--r--src/math/expm1.go237
-rw-r--r--src/math/expm1_386.s57
-rw-r--r--src/math/expm1_s390x.s194
-rw-r--r--src/math/export_s390x_test.go31
-rw-r--r--src/math/export_test.go14
-rw-r--r--src/math/floor.go131
-rw-r--r--src/math/floor_386.s46
-rw-r--r--src/math/floor_amd64.s76
-rw-r--r--src/math/floor_arm64.s26
-rw-r--r--src/math/floor_ppc64x.s25
-rw-r--r--src/math/floor_s390x.s26
-rw-r--r--src/math/floor_wasm.s26
-rw-r--r--src/math/fma.go170
-rw-r--r--src/math/frexp.go33
-rw-r--r--src/math/frexp_386.s25
-rw-r--r--src/math/gamma.go221
-rw-r--r--src/math/huge_test.go115
-rw-r--r--src/math/hypot.go38
-rw-r--r--src/math/hypot_386.s59
-rw-r--r--src/math/hypot_amd64.s52
-rw-r--r--src/math/j0.go427
-rw-r--r--src/math/j1.go422
-rw-r--r--src/math/jn.go304
-rw-r--r--src/math/ldexp.go45
-rw-r--r--src/math/ldexp_386.s14
-rw-r--r--src/math/lgamma.go365
-rw-r--r--src/math/log.go123
-rw-r--r--src/math/log10.go27
-rw-r--r--src/math/log10_386.s21
-rw-r--r--src/math/log10_s390x.s156
-rw-r--r--src/math/log1p.go197
-rw-r--r--src/math/log1p_386.s27
-rw-r--r--src/math/log1p_s390x.s180
-rw-r--r--src/math/log_386.s13
-rw-r--r--src/math/log_amd64.s112
-rw-r--r--src/math/log_s390x.s168
-rw-r--r--src/math/logb.go50
-rw-r--r--src/math/mod.go46
-rw-r--r--src/math/mod_386.s17
-rw-r--r--src/math/modf.go37
-rw-r--r--src/math/modf_386.s34
-rw-r--r--src/math/modf_arm64.s18
-rw-r--r--src/math/modf_ppc64x.s17
-rw-r--r--src/math/nextafter.go49
-rw-r--r--src/math/pow.go151
-rw-r--r--src/math/pow10.go46
-rw-r--r--src/math/pow_s390x.s634
-rw-r--r--src/math/rand/example_test.go157
-rw-r--r--src/math/rand/exp.go222
-rw-r--r--src/math/rand/gen_cooked.go89
-rw-r--r--src/math/rand/normal.go157
-rw-r--r--src/math/rand/race_test.go48
-rw-r--r--src/math/rand/rand.go421
-rw-r--r--src/math/rand/rand_test.go681
-rw-r--r--src/math/rand/regress_test.go404
-rw-r--r--src/math/rand/rng.go252
-rw-r--r--src/math/rand/zipf.go77
-rw-r--r--src/math/remainder.go89
-rw-r--r--src/math/remainder_386.s17
-rw-r--r--src/math/signbit.go10
-rw-r--r--src/math/sin.go232
-rw-r--r--src/math/sin_s390x.s356
-rw-r--r--src/math/sincos.go72
-rw-r--r--src/math/sinh.go81
-rw-r--r--src/math/sinh_s390x.s251
-rw-r--r--src/math/sqrt.go144
-rw-r--r--src/math/sqrt_386.s12
-rw-r--r--src/math/sqrt_amd64.s12
-rw-r--r--src/math/sqrt_arm.s20
-rw-r--r--src/math/sqrt_arm64.s12
-rw-r--r--src/math/sqrt_mipsx.s18
-rw-r--r--src/math/sqrt_ppc64x.s14
-rw-r--r--src/math/sqrt_riscv64.s14
-rw-r--r--src/math/sqrt_s390x.s12
-rw-r--r--src/math/sqrt_wasm.s12
-rw-r--r--src/math/stubs_386.s53
-rw-r--r--src/math/stubs_amd64.s86
-rw-r--r--src/math/stubs_arm.s110
-rw-r--r--src/math/stubs_arm64.s88
-rw-r--r--src/math/stubs_mips64x.s115
-rw-r--r--src/math/stubs_mipsx.s113
-rw-r--r--src/math/stubs_ppc64x.s101
-rw-r--r--src/math/stubs_riscv64.s104
-rw-r--r--src/math/stubs_s390x.s492
-rw-r--r--src/math/stubs_wasm.s101
-rw-r--r--src/math/tan.go134
-rw-r--r--src/math/tan_s390x.s110
-rw-r--r--src/math/tanh.go99
-rw-r--r--src/math/tanh_s390x.s169
-rw-r--r--src/math/trig_reduce.go100
-rw-r--r--src/math/unsafe.go29
216 files changed, 47674 insertions, 0 deletions
diff --git a/src/math/abs.go b/src/math/abs.go
new file mode 100644
index 0000000..df83add
--- /dev/null
+++ b/src/math/abs.go
@@ -0,0 +1,14 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Abs returns the absolute value of x.
+//
+// Special cases are:
+// Abs(±Inf) = +Inf
+// Abs(NaN) = NaN
+func Abs(x float64) float64 {
+ return Float64frombits(Float64bits(x) &^ (1 << 63))
+}
diff --git a/src/math/acos_s390x.s b/src/math/acos_s390x.s
new file mode 100644
index 0000000..d2288b8
--- /dev/null
+++ b/src/math/acos_s390x.s
@@ -0,0 +1,144 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial coefficients and other constants
+DATA ·acosrodataL13<> + 0(SB)/8, $0.314159265358979323E+01 //pi
+DATA ·acosrodataL13<> + 8(SB)/8, $-0.0
+DATA ·acosrodataL13<> + 16(SB)/8, $0x7ff8000000000000 //Nan
+DATA ·acosrodataL13<> + 24(SB)/8, $-1.0
+DATA ·acosrodataL13<> + 32(SB)/8, $1.0
+DATA ·acosrodataL13<> + 40(SB)/8, $0.166666666666651626E+00
+DATA ·acosrodataL13<> + 48(SB)/8, $0.750000000042621169E-01
+DATA ·acosrodataL13<> + 56(SB)/8, $0.446428567178116477E-01
+DATA ·acosrodataL13<> + 64(SB)/8, $0.303819660378071894E-01
+DATA ·acosrodataL13<> + 72(SB)/8, $0.223715011892010405E-01
+DATA ·acosrodataL13<> + 80(SB)/8, $0.173659424522364952E-01
+DATA ·acosrodataL13<> + 88(SB)/8, $0.137810186504372266E-01
+DATA ·acosrodataL13<> + 96(SB)/8, $0.134066870961173521E-01
+DATA ·acosrodataL13<> + 104(SB)/8, $-.412335502831898721E-02
+DATA ·acosrodataL13<> + 112(SB)/8, $0.867383739532082719E-01
+DATA ·acosrodataL13<> + 120(SB)/8, $-.328765950607171649E+00
+DATA ·acosrodataL13<> + 128(SB)/8, $0.110401073869414626E+01
+DATA ·acosrodataL13<> + 136(SB)/8, $-.270694366992537307E+01
+DATA ·acosrodataL13<> + 144(SB)/8, $0.500196500770928669E+01
+DATA ·acosrodataL13<> + 152(SB)/8, $-.665866959108585165E+01
+DATA ·acosrodataL13<> + 160(SB)/8, $-.344895269334086578E+01
+DATA ·acosrodataL13<> + 168(SB)/8, $0.927437952918301659E+00
+DATA ·acosrodataL13<> + 176(SB)/8, $0.610487478874645653E+01
+DATA ·acosrodataL13<> + 184(SB)/8, $0.157079632679489656e+01
+DATA ·acosrodataL13<> + 192(SB)/8, $0.0
+GLOBL ·acosrodataL13<> + 0(SB), RODATA, $200
+
+// Acos returns the arccosine, in radians, of the argument.
+//
+// Special case is:
+// Acos(x) = NaN if x < -1 or x > 1
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·acosAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·acosrodataL13<>+0(SB), R9
+ LGDR F0, R12
+ FMOVD F0, F10
+ SRAD $32, R12
+ WORD $0xC0293FE6 //iilf %r2,1072079005
+ BYTE $0xA0
+ BYTE $0x9D
+ WORD $0xB917001C //llgtr %r1,%r12
+ CMPW R1,R2
+ BGT L2
+ FMOVD 192(R9), F8
+ FMADD F0, F0, F8
+ FMOVD 184(R9), F1
+L3:
+ WFMDB V8, V8, V2
+ FMOVD 176(R9), F6
+ FMOVD 168(R9), F0
+ FMOVD 160(R9), F4
+ WFMADB V2, V0, V6, V0
+ FMOVD 152(R9), F6
+ WFMADB V2, V4, V6, V4
+ FMOVD 144(R9), F6
+ WFMADB V2, V0, V6, V0
+ FMOVD 136(R9), F6
+ WFMADB V2, V4, V6, V4
+ FMOVD 128(R9), F6
+ WFMADB V2, V0, V6, V0
+ FMOVD 120(R9), F6
+ WFMADB V2, V4, V6, V4
+ FMOVD 112(R9), F6
+ WFMADB V2, V0, V6, V0
+ FMOVD 104(R9), F6
+ WFMADB V2, V4, V6, V4
+ FMOVD 96(R9), F6
+ WFMADB V2, V0, V6, V0
+ FMOVD 88(R9), F6
+ WFMADB V2, V4, V6, V4
+ FMOVD 80(R9), F6
+ WFMADB V2, V0, V6, V0
+ FMOVD 72(R9), F6
+ WFMADB V2, V4, V6, V4
+ FMOVD 64(R9), F6
+ WFMADB V2, V0, V6, V0
+ FMOVD 56(R9), F6
+ WFMADB V2, V4, V6, V4
+ FMOVD 48(R9), F6
+ WFMADB V2, V0, V6, V0
+ FMOVD 40(R9), F6
+ WFMADB V2, V4, V6, V2
+ FMOVD 192(R9), F4
+ WFMADB V8, V0, V2, V0
+ WFMADB V10, V8, V4, V8
+ FMADD F0, F8, F10
+ WFSDB V10, V1, V10
+L1:
+ FMOVD F10, ret+8(FP)
+ RET
+
+L2:
+ WORD $0xC0293FEF //iilf %r2,1072693247
+ BYTE $0xFF
+ BYTE $0xFF
+ CMPW R1, R2
+ BLE L12
+L4:
+ WORD $0xED009020 //cdb %f0,.L34-.L13(%r9)
+ BYTE $0x00
+ BYTE $0x19
+ BEQ L8
+ WORD $0xED009018 //cdb %f0,.L35-.L13(%r9)
+ BYTE $0x00
+ BYTE $0x19
+ BEQ L9
+ WFCEDBS V10, V10, V0
+ BVS L1
+ FMOVD 16(R9), F10
+ BR L1
+L12:
+ FMOVD 24(R9), F0
+ FMADD F10, F10, F0
+ WORD $0xB3130080 //lcdbr %f8,%f0
+ WORD $0xED009008 //cdb %f0,.L37-.L13(%r9)
+ BYTE $0x00
+ BYTE $0x19
+ FSQRT F8, F10
+L5:
+ MOVW R12, R4
+ CMPBLE R4, $0, L7
+ WORD $0xB31300AA //lcdbr %f10,%f10
+ FMOVD $0, F1
+ BR L3
+L9:
+ FMOVD 0(R9), F10
+ BR L1
+L8:
+ FMOVD $0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+L7:
+ FMOVD 0(R9), F1
+ BR L3
diff --git a/src/math/acosh.go b/src/math/acosh.go
new file mode 100644
index 0000000..41ca871
--- /dev/null
+++ b/src/math/acosh.go
@@ -0,0 +1,59 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_acosh.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// __ieee754_acosh(x)
+// Method :
+// Based on
+// acosh(x) = log [ x + sqrt(x*x-1) ]
+// we have
+// acosh(x) := log(x)+ln2, if x is large; else
+// acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
+// acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
+//
+// Special cases:
+// acosh(x) is NaN with signal if x<1.
+// acosh(NaN) is NaN without signal.
+//
+
+// Acosh returns the inverse hyperbolic cosine of x.
+//
+// Special cases are:
+// Acosh(+Inf) = +Inf
+// Acosh(x) = NaN if x < 1
+// Acosh(NaN) = NaN
+func Acosh(x float64) float64
+
+func acosh(x float64) float64 {
+ const Large = 1 << 28 // 2**28
+ // first case is special case
+ switch {
+ case x < 1 || IsNaN(x):
+ return NaN()
+ case x == 1:
+ return 0
+ case x >= Large:
+ return Log(x) + Ln2 // x > 2**28
+ case x > 2:
+ return Log(2*x - 1/(x+Sqrt(x*x-1))) // 2**28 > x > 2
+ }
+ t := x - 1
+ return Log1p(t + Sqrt(2*t+t*t)) // 2 >= x > 1
+}
diff --git a/src/math/acosh_s390x.s b/src/math/acosh_s390x.s
new file mode 100644
index 0000000..9294c48
--- /dev/null
+++ b/src/math/acosh_s390x.s
@@ -0,0 +1,158 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial coefficients and other constants
+DATA ·acoshrodataL11<> + 0(SB)/8, $-1.0
+DATA ·acoshrodataL11<> + 8(SB)/8, $.41375273347623353626
+DATA ·acoshrodataL11<> + 16(SB)/8, $.51487302528619766235E+04
+DATA ·acoshrodataL11<> + 24(SB)/8, $-1.67526912689208984375
+DATA ·acoshrodataL11<> + 32(SB)/8, $0.181818181818181826E+00
+DATA ·acoshrodataL11<> + 40(SB)/8, $-.165289256198351540E-01
+DATA ·acoshrodataL11<> + 48(SB)/8, $0.200350613573012186E-02
+DATA ·acoshrodataL11<> + 56(SB)/8, $-.273205381970859341E-03
+DATA ·acoshrodataL11<> + 64(SB)/8, $0.397389654305194527E-04
+DATA ·acoshrodataL11<> + 72(SB)/8, $0.938370938292558173E-06
+DATA ·acoshrodataL11<> + 80(SB)/8, $-.602107458843052029E-05
+DATA ·acoshrodataL11<> + 88(SB)/8, $0.212881813645679599E-07
+DATA ·acoshrodataL11<> + 96(SB)/8, $-.148682720127920854E-06
+DATA ·acoshrodataL11<> + 104(SB)/8, $-5.5
+DATA ·acoshrodataL11<> + 112(SB)/8, $0x7ff8000000000000 //Nan
+GLOBL ·acoshrodataL11<> + 0(SB), RODATA, $120
+
+// Table of log correction terms
+DATA ·acoshtab2068<> + 0(SB)/8, $0.585235384085551248E-01
+DATA ·acoshtab2068<> + 8(SB)/8, $0.412206153771168640E-01
+DATA ·acoshtab2068<> + 16(SB)/8, $0.273839003221648339E-01
+DATA ·acoshtab2068<> + 24(SB)/8, $0.166383778368856480E-01
+DATA ·acoshtab2068<> + 32(SB)/8, $0.866678223433169637E-02
+DATA ·acoshtab2068<> + 40(SB)/8, $0.319831684989627514E-02
+DATA ·acoshtab2068<> + 48(SB)/8, $0.0
+DATA ·acoshtab2068<> + 56(SB)/8, $-.113006378583725549E-02
+DATA ·acoshtab2068<> + 64(SB)/8, $-.367979419636602491E-03
+DATA ·acoshtab2068<> + 72(SB)/8, $0.213172484510484979E-02
+DATA ·acoshtab2068<> + 80(SB)/8, $0.623271047682013536E-02
+DATA ·acoshtab2068<> + 88(SB)/8, $0.118140812789696885E-01
+DATA ·acoshtab2068<> + 96(SB)/8, $0.187681358930914206E-01
+DATA ·acoshtab2068<> + 104(SB)/8, $0.269985148668178992E-01
+DATA ·acoshtab2068<> + 112(SB)/8, $0.364186619761331328E-01
+DATA ·acoshtab2068<> + 120(SB)/8, $0.469505379381388441E-01
+GLOBL ·acoshtab2068<> + 0(SB), RODATA, $128
+
+// Acosh returns the inverse hyperbolic cosine of the argument.
+//
+// Special cases are:
+// Acosh(+Inf) = +Inf
+// Acosh(x) = NaN if x < 1
+// Acosh(NaN) = NaN
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·acoshAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·acoshrodataL11<>+0(SB), R9
+ LGDR F0, R1
+ WORD $0xC0295FEF //iilf %r2,1609564159
+ BYTE $0xFF
+ BYTE $0xFF
+ SRAD $32, R1
+ CMPW R1, R2
+ BGT L2
+ WORD $0xC0293FEF //iilf %r2,1072693247
+ BYTE $0xFF
+ BYTE $0xFF
+ CMPW R1, R2
+ BGT L10
+L3:
+ WFCEDBS V0, V0, V2
+ BVS L1
+ FMOVD 112(R9), F0
+L1:
+ FMOVD F0, ret+8(FP)
+ RET
+L2:
+ WORD $0xC0297FEF //iilf %r2,2146435071
+ BYTE $0xFF
+ BYTE $0xFF
+ MOVW R1, R6
+ MOVW R2, R7
+ CMPBGT R6, R7, L1
+ FMOVD F0, F8
+ FMOVD $0, F0
+ WFADB V0, V8, V0
+ WORD $0xC0398006 //iilf %r3,2147909631
+ BYTE $0x7F
+ BYTE $0xFF
+ LGDR F0, R5
+ SRAD $32, R5
+ MOVH $0x0, R1
+ SUBW R5, R3
+ FMOVD $0, F10
+ RISBGZ $32, $47, $0, R3, R4
+ RISBGZ $57, $60, $51, R3, R3
+ BYTE $0x18 //lr %r2,%r4
+ BYTE $0x24
+ RISBGN $0, $31, $32, R4, R1
+ SUBW $0x100000, R2
+ SRAW $8, R2, R2
+ ORW $0x45000000, R2
+L5:
+ LDGR R1, F0
+ FMOVD 104(R9), F2
+ FMADD F8, F0, F2
+ FMOVD 96(R9), F4
+ WFMADB V10, V0, V2, V0
+ FMOVD 88(R9), F6
+ FMOVD 80(R9), F2
+ WFMADB V0, V6, V4, V6
+ FMOVD 72(R9), F1
+ WFMDB V0, V0, V4
+ WFMADB V0, V1, V2, V1
+ FMOVD 64(R9), F2
+ WFMADB V6, V4, V1, V6
+ FMOVD 56(R9), F1
+ RISBGZ $57, $60, $0, R3, R3
+ WFMADB V0, V2, V1, V2
+ FMOVD 48(R9), F1
+ WFMADB V4, V6, V2, V6
+ FMOVD 40(R9), F2
+ WFMADB V0, V1, V2, V1
+ VLVGF $0, R2, V2
+ WFMADB V4, V6, V1, V4
+ LDEBR F2, F2
+ FMOVD 32(R9), F6
+ WFMADB V0, V4, V6, V4
+ FMOVD 24(R9), F1
+ FMOVD 16(R9), F6
+ MOVD $·acoshtab2068<>+0(SB), R1
+ WFMADB V2, V1, V6, V2
+ FMOVD 0(R3)(R1*1), F3
+ WFMADB V0, V4, V3, V0
+ FMOVD 8(R9), F4
+ FMADD F4, F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+L10:
+ FMOVD F0, F8
+ FMOVD 0(R9), F0
+ FMADD F8, F8, F0
+ LTDBR F0, F0
+ FSQRT F0, F10
+L4:
+ WFADB V10, V8, V0
+ WORD $0xC0398006 //iilf %r3,2147909631
+ BYTE $0x7F
+ BYTE $0xFF
+ LGDR F0, R5
+ SRAD $32, R5
+ MOVH $0x0, R1
+ SUBW R5, R3
+ SRAW $8, R3, R2
+ RISBGZ $32, $47, $0, R3, R4
+ ANDW $0xFFFFFF00, R2
+ RISBGZ $57, $60, $51, R3, R3
+ ORW $0x45000000, R2
+ RISBGN $0, $31, $32, R4, R1
+ BR L5
diff --git a/src/math/all_test.go b/src/math/all_test.go
new file mode 100644
index 0000000..3aae037
--- /dev/null
+++ b/src/math/all_test.go
@@ -0,0 +1,3812 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math_test
+
+import (
+ "fmt"
+ . "math"
+ "testing"
+ "unsafe"
+)
+
+var vf = []float64{
+ 4.9790119248836735e+00,
+ 7.7388724745781045e+00,
+ -2.7688005719200159e-01,
+ -5.0106036182710749e+00,
+ 9.6362937071984173e+00,
+ 2.9263772392439646e+00,
+ 5.2290834314593066e+00,
+ 2.7279399104360102e+00,
+ 1.8253080916808550e+00,
+ -8.6859247685756013e+00,
+}
+
+// The expected results below were computed by the high precision calculators
+// at https://keisan.casio.com/. More exact input values (array vf[], above)
+// were obtained by printing them with "%.26f". The answers were calculated
+// to 26 digits (by using the "Digit number" drop-down control of each
+// calculator).
+var acos = []float64{
+ 1.0496193546107222142571536e+00,
+ 6.8584012813664425171660692e-01,
+ 1.5984878714577160325521819e+00,
+ 2.0956199361475859327461799e+00,
+ 2.7053008467824138592616927e-01,
+ 1.2738121680361776018155625e+00,
+ 1.0205369421140629186287407e+00,
+ 1.2945003481781246062157835e+00,
+ 1.3872364345374451433846657e+00,
+ 2.6231510803970463967294145e+00,
+}
+var acosh = []float64{
+ 2.4743347004159012494457618e+00,
+ 2.8576385344292769649802701e+00,
+ 7.2796961502981066190593175e-01,
+ 2.4796794418831451156471977e+00,
+ 3.0552020742306061857212962e+00,
+ 2.044238592688586588942468e+00,
+ 2.5158701513104513595766636e+00,
+ 1.99050839282411638174299e+00,
+ 1.6988625798424034227205445e+00,
+ 2.9611454842470387925531875e+00,
+}
+var asin = []float64{
+ 5.2117697218417440497416805e-01,
+ 8.8495619865825236751471477e-01,
+ -02.769154466281941332086016e-02,
+ -5.2482360935268931351485822e-01,
+ 1.3002662421166552333051524e+00,
+ 2.9698415875871901741575922e-01,
+ 5.5025938468083370060258102e-01,
+ 2.7629597861677201301553823e-01,
+ 1.83559892257451475846656e-01,
+ -1.0523547536021497774980928e+00,
+}
+var asinh = []float64{
+ 2.3083139124923523427628243e+00,
+ 2.743551594301593620039021e+00,
+ -2.7345908534880091229413487e-01,
+ -2.3145157644718338650499085e+00,
+ 2.9613652154015058521951083e+00,
+ 1.7949041616585821933067568e+00,
+ 2.3564032905983506405561554e+00,
+ 1.7287118790768438878045346e+00,
+ 1.3626658083714826013073193e+00,
+ -2.8581483626513914445234004e+00,
+}
+var atan = []float64{
+ 1.372590262129621651920085e+00,
+ 1.442290609645298083020664e+00,
+ -2.7011324359471758245192595e-01,
+ -1.3738077684543379452781531e+00,
+ 1.4673921193587666049154681e+00,
+ 1.2415173565870168649117764e+00,
+ 1.3818396865615168979966498e+00,
+ 1.2194305844639670701091426e+00,
+ 1.0696031952318783760193244e+00,
+ -1.4561721938838084990898679e+00,
+}
+var atanh = []float64{
+ 5.4651163712251938116878204e-01,
+ 1.0299474112843111224914709e+00,
+ -2.7695084420740135145234906e-02,
+ -5.5072096119207195480202529e-01,
+ 1.9943940993171843235906642e+00,
+ 3.01448604578089708203017e-01,
+ 5.8033427206942188834370595e-01,
+ 2.7987997499441511013958297e-01,
+ 1.8459947964298794318714228e-01,
+ -1.3273186910532645867272502e+00,
+}
+var atan2 = []float64{
+ 1.1088291730037004444527075e+00,
+ 9.1218183188715804018797795e-01,
+ 1.5984772603216203736068915e+00,
+ 2.0352918654092086637227327e+00,
+ 8.0391819139044720267356014e-01,
+ 1.2861075249894661588866752e+00,
+ 1.0889904479131695712182587e+00,
+ 1.3044821793397925293797357e+00,
+ 1.3902530903455392306872261e+00,
+ 2.2859857424479142655411058e+00,
+}
+var cbrt = []float64{
+ 1.7075799841925094446722675e+00,
+ 1.9779982212970353936691498e+00,
+ -6.5177429017779910853339447e-01,
+ -1.7111838886544019873338113e+00,
+ 2.1279920909827937423960472e+00,
+ 1.4303536770460741452312367e+00,
+ 1.7357021059106154902341052e+00,
+ 1.3972633462554328350552916e+00,
+ 1.2221149580905388454977636e+00,
+ -2.0556003730500069110343596e+00,
+}
+var ceil = []float64{
+ 5.0000000000000000e+00,
+ 8.0000000000000000e+00,
+ Copysign(0, -1),
+ -5.0000000000000000e+00,
+ 1.0000000000000000e+01,
+ 3.0000000000000000e+00,
+ 6.0000000000000000e+00,
+ 3.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ -8.0000000000000000e+00,
+}
+var copysign = []float64{
+ -4.9790119248836735e+00,
+ -7.7388724745781045e+00,
+ -2.7688005719200159e-01,
+ -5.0106036182710749e+00,
+ -9.6362937071984173e+00,
+ -2.9263772392439646e+00,
+ -5.2290834314593066e+00,
+ -2.7279399104360102e+00,
+ -1.8253080916808550e+00,
+ -8.6859247685756013e+00,
+}
+var cos = []float64{
+ 2.634752140995199110787593e-01,
+ 1.148551260848219865642039e-01,
+ 9.6191297325640768154550453e-01,
+ 2.938141150061714816890637e-01,
+ -9.777138189897924126294461e-01,
+ -9.7693041344303219127199518e-01,
+ 4.940088096948647263961162e-01,
+ -9.1565869021018925545016502e-01,
+ -2.517729313893103197176091e-01,
+ -7.39241351595676573201918e-01,
+}
+
+// Results for 100000 * Pi + vf[i]
+var cosLarge = []float64{
+ 2.634752141185559426744e-01,
+ 1.14855126055543100712e-01,
+ 9.61912973266488928113e-01,
+ 2.9381411499556122552e-01,
+ -9.777138189880161924641e-01,
+ -9.76930413445147608049e-01,
+ 4.940088097314976789841e-01,
+ -9.15658690217517835002e-01,
+ -2.51772931436786954751e-01,
+ -7.3924135157173099849e-01,
+}
+
+var cosh = []float64{
+ 7.2668796942212842775517446e+01,
+ 1.1479413465659254502011135e+03,
+ 1.0385767908766418550935495e+00,
+ 7.5000957789658051428857788e+01,
+ 7.655246669605357888468613e+03,
+ 9.3567491758321272072888257e+00,
+ 9.331351599270605471131735e+01,
+ 7.6833430994624643209296404e+00,
+ 3.1829371625150718153881164e+00,
+ 2.9595059261916188501640911e+03,
+}
+var erf = []float64{
+ 5.1865354817738701906913566e-01,
+ 7.2623875834137295116929844e-01,
+ -3.123458688281309990629839e-02,
+ -5.2143121110253302920437013e-01,
+ 8.2704742671312902508629582e-01,
+ 3.2101767558376376743993945e-01,
+ 5.403990312223245516066252e-01,
+ 3.0034702916738588551174831e-01,
+ 2.0369924417882241241559589e-01,
+ -7.8069386968009226729944677e-01,
+}
+var erfc = []float64{
+ 4.8134645182261298093086434e-01,
+ 2.7376124165862704883070156e-01,
+ 1.0312345868828130999062984e+00,
+ 1.5214312111025330292043701e+00,
+ 1.7295257328687097491370418e-01,
+ 6.7898232441623623256006055e-01,
+ 4.596009687776754483933748e-01,
+ 6.9965297083261411448825169e-01,
+ 7.9630075582117758758440411e-01,
+ 1.7806938696800922672994468e+00,
+}
+var erfinv = []float64{
+ 4.746037673358033586786350696e-01,
+ 8.559054432692110956388764172e-01,
+ -2.45427830571707336251331946e-02,
+ -4.78116683518973366268905506e-01,
+ 1.479804430319470983648120853e+00,
+ 2.654485787128896161882650211e-01,
+ 5.027444534221520197823192493e-01,
+ 2.466703532707627818954585670e-01,
+ 1.632011465103005426240343116e-01,
+ -1.06672334642196900710000389e+00,
+}
+var exp = []float64{
+ 1.4533071302642137507696589e+02,
+ 2.2958822575694449002537581e+03,
+ 7.5814542574851666582042306e-01,
+ 6.6668778421791005061482264e-03,
+ 1.5310493273896033740861206e+04,
+ 1.8659907517999328638667732e+01,
+ 1.8662167355098714543942057e+02,
+ 1.5301332413189378961665788e+01,
+ 6.2047063430646876349125085e+00,
+ 1.6894712385826521111610438e-04,
+}
+var expm1 = []float64{
+ 5.105047796122957327384770212e-02,
+ 8.046199708567344080562675439e-02,
+ -2.764970978891639815187418703e-03,
+ -4.8871434888875355394330300273e-02,
+ 1.0115864277221467777117227494e-01,
+ 2.969616407795910726014621657e-02,
+ 5.368214487944892300914037972e-02,
+ 2.765488851131274068067445335e-02,
+ 1.842068661871398836913874273e-02,
+ -8.3193870863553801814961137573e-02,
+}
+var expm1Large = []float64{
+ 4.2031418113550844e+21,
+ 4.0690789717473863e+33,
+ -0.9372627915981363e+00,
+ -1.0,
+ 7.077694784145933e+41,
+ 5.117936223839153e+12,
+ 5.124137759001189e+22,
+ 7.03546003972584e+11,
+ 8.456921800389698e+07,
+ -1.0,
+}
+var exp2 = []float64{
+ 3.1537839463286288034313104e+01,
+ 2.1361549283756232296144849e+02,
+ 8.2537402562185562902577219e-01,
+ 3.1021158628740294833424229e-02,
+ 7.9581744110252191462569661e+02,
+ 7.6019905892596359262696423e+00,
+ 3.7506882048388096973183084e+01,
+ 6.6250893439173561733216375e+00,
+ 3.5438267900243941544605339e+00,
+ 2.4281533133513300984289196e-03,
+}
+var fabs = []float64{
+ 4.9790119248836735e+00,
+ 7.7388724745781045e+00,
+ 2.7688005719200159e-01,
+ 5.0106036182710749e+00,
+ 9.6362937071984173e+00,
+ 2.9263772392439646e+00,
+ 5.2290834314593066e+00,
+ 2.7279399104360102e+00,
+ 1.8253080916808550e+00,
+ 8.6859247685756013e+00,
+}
+var fdim = []float64{
+ 4.9790119248836735e+00,
+ 7.7388724745781045e+00,
+ 0.0000000000000000e+00,
+ 0.0000000000000000e+00,
+ 9.6362937071984173e+00,
+ 2.9263772392439646e+00,
+ 5.2290834314593066e+00,
+ 2.7279399104360102e+00,
+ 1.8253080916808550e+00,
+ 0.0000000000000000e+00,
+}
+var floor = []float64{
+ 4.0000000000000000e+00,
+ 7.0000000000000000e+00,
+ -1.0000000000000000e+00,
+ -6.0000000000000000e+00,
+ 9.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 5.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 1.0000000000000000e+00,
+ -9.0000000000000000e+00,
+}
+var fmod = []float64{
+ 4.197615023265299782906368e-02,
+ 2.261127525421895434476482e+00,
+ 3.231794108794261433104108e-02,
+ 4.989396381728925078391512e+00,
+ 3.637062928015826201999516e-01,
+ 1.220868282268106064236690e+00,
+ 4.770916568540693347699744e+00,
+ 1.816180268691969246219742e+00,
+ 8.734595415957246977711748e-01,
+ 1.314075231424398637614104e+00,
+}
+
+type fi struct {
+ f float64
+ i int
+}
+
+var frexp = []fi{
+ {6.2237649061045918750e-01, 3},
+ {9.6735905932226306250e-01, 3},
+ {-5.5376011438400318000e-01, -1},
+ {-6.2632545228388436250e-01, 3},
+ {6.02268356699901081250e-01, 4},
+ {7.3159430981099115000e-01, 2},
+ {6.5363542893241332500e-01, 3},
+ {6.8198497760900255000e-01, 2},
+ {9.1265404584042750000e-01, 1},
+ {-5.4287029803597508250e-01, 4},
+}
+var gamma = []float64{
+ 2.3254348370739963835386613898e+01,
+ 2.991153837155317076427529816e+03,
+ -4.561154336726758060575129109e+00,
+ 7.719403468842639065959210984e-01,
+ 1.6111876618855418534325755566e+05,
+ 1.8706575145216421164173224946e+00,
+ 3.4082787447257502836734201635e+01,
+ 1.579733951448952054898583387e+00,
+ 9.3834586598354592860187267089e-01,
+ -2.093995902923148389186189429e-05,
+}
+var j0 = []float64{
+ -1.8444682230601672018219338e-01,
+ 2.27353668906331975435892e-01,
+ 9.809259936157051116270273e-01,
+ -1.741170131426226587841181e-01,
+ -2.1389448451144143352039069e-01,
+ -2.340905848928038763337414e-01,
+ -1.0029099691890912094586326e-01,
+ -1.5466726714884328135358907e-01,
+ 3.252650187653420388714693e-01,
+ -8.72218484409407250005360235e-03,
+}
+var j1 = []float64{
+ -3.251526395295203422162967e-01,
+ 1.893581711430515718062564e-01,
+ -1.3711761352467242914491514e-01,
+ 3.287486536269617297529617e-01,
+ 1.3133899188830978473849215e-01,
+ 3.660243417832986825301766e-01,
+ -3.4436769271848174665420672e-01,
+ 4.329481396640773768835036e-01,
+ 5.8181350531954794639333955e-01,
+ -2.7030574577733036112996607e-01,
+}
+var j2 = []float64{
+ 5.3837518920137802565192769e-02,
+ -1.7841678003393207281244667e-01,
+ 9.521746934916464142495821e-03,
+ 4.28958355470987397983072e-02,
+ 2.4115371837854494725492872e-01,
+ 4.842458532394520316844449e-01,
+ -3.142145220618633390125946e-02,
+ 4.720849184745124761189957e-01,
+ 3.122312022520957042957497e-01,
+ 7.096213118930231185707277e-02,
+}
+var jM3 = []float64{
+ -3.684042080996403091021151e-01,
+ 2.8157665936340887268092661e-01,
+ 4.401005480841948348343589e-04,
+ 3.629926999056814081597135e-01,
+ 3.123672198825455192489266e-02,
+ -2.958805510589623607540455e-01,
+ -3.2033177696533233403289416e-01,
+ -2.592737332129663376736604e-01,
+ -1.0241334641061485092351251e-01,
+ -2.3762660886100206491674503e-01,
+}
+var lgamma = []fi{
+ {3.146492141244545774319734e+00, 1},
+ {8.003414490659126375852113e+00, 1},
+ {1.517575735509779707488106e+00, -1},
+ {-2.588480028182145853558748e-01, 1},
+ {1.1989897050205555002007985e+01, 1},
+ {6.262899811091257519386906e-01, 1},
+ {3.5287924899091566764846037e+00, 1},
+ {4.5725644770161182299423372e-01, 1},
+ {-6.363667087767961257654854e-02, 1},
+ {-1.077385130910300066425564e+01, -1},
+}
+var log = []float64{
+ 1.605231462693062999102599e+00,
+ 2.0462560018708770653153909e+00,
+ -1.2841708730962657801275038e+00,
+ 1.6115563905281545116286206e+00,
+ 2.2655365644872016636317461e+00,
+ 1.0737652208918379856272735e+00,
+ 1.6542360106073546632707956e+00,
+ 1.0035467127723465801264487e+00,
+ 6.0174879014578057187016475e-01,
+ 2.161703872847352815363655e+00,
+}
+var logb = []float64{
+ 2.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ -2.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 3.0000000000000000e+00,
+ 1.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 1.0000000000000000e+00,
+ 0.0000000000000000e+00,
+ 3.0000000000000000e+00,
+}
+var log10 = []float64{
+ 6.9714316642508290997617083e-01,
+ 8.886776901739320576279124e-01,
+ -5.5770832400658929815908236e-01,
+ 6.998900476822994346229723e-01,
+ 9.8391002850684232013281033e-01,
+ 4.6633031029295153334285302e-01,
+ 7.1842557117242328821552533e-01,
+ 4.3583479968917773161304553e-01,
+ 2.6133617905227038228626834e-01,
+ 9.3881606348649405716214241e-01,
+}
+var log1p = []float64{
+ 4.8590257759797794104158205e-02,
+ 7.4540265965225865330849141e-02,
+ -2.7726407903942672823234024e-03,
+ -5.1404917651627649094953380e-02,
+ 9.1998280672258624681335010e-02,
+ 2.8843762576593352865894824e-02,
+ 5.0969534581863707268992645e-02,
+ 2.6913947602193238458458594e-02,
+ 1.8088493239630770262045333e-02,
+ -9.0865245631588989681559268e-02,
+}
+var log2 = []float64{
+ 2.3158594707062190618898251e+00,
+ 2.9521233862883917703341018e+00,
+ -1.8526669502700329984917062e+00,
+ 2.3249844127278861543568029e+00,
+ 3.268478366538305087466309e+00,
+ 1.5491157592596970278166492e+00,
+ 2.3865580889631732407886495e+00,
+ 1.447811865817085365540347e+00,
+ 8.6813999540425116282815557e-01,
+ 3.118679457227342224364709e+00,
+}
+var modf = [][2]float64{
+ {4.0000000000000000e+00, 9.7901192488367350108546816e-01},
+ {7.0000000000000000e+00, 7.3887247457810456552351752e-01},
+ {Copysign(0, -1), -2.7688005719200159404635997e-01},
+ {-5.0000000000000000e+00, -1.060361827107492160848778e-02},
+ {9.0000000000000000e+00, 6.3629370719841737980004837e-01},
+ {2.0000000000000000e+00, 9.2637723924396464525443662e-01},
+ {5.0000000000000000e+00, 2.2908343145930665230025625e-01},
+ {2.0000000000000000e+00, 7.2793991043601025126008608e-01},
+ {1.0000000000000000e+00, 8.2530809168085506044576505e-01},
+ {-8.0000000000000000e+00, -6.8592476857560136238589621e-01},
+}
+var nextafter32 = []float32{
+ 4.979012489318848e+00,
+ 7.738873004913330e+00,
+ -2.768800258636475e-01,
+ -5.010602951049805e+00,
+ 9.636294364929199e+00,
+ 2.926377534866333e+00,
+ 5.229084014892578e+00,
+ 2.727940082550049e+00,
+ 1.825308203697205e+00,
+ -8.685923576354980e+00,
+}
+var nextafter64 = []float64{
+ 4.97901192488367438926388786e+00,
+ 7.73887247457810545370193722e+00,
+ -2.7688005719200153853520874e-01,
+ -5.01060361827107403343006808e+00,
+ 9.63629370719841915615688777e+00,
+ 2.92637723924396508934364647e+00,
+ 5.22908343145930754047867595e+00,
+ 2.72793991043601069534929593e+00,
+ 1.82530809168085528249036997e+00,
+ -8.68592476857559958602905681e+00,
+}
+var pow = []float64{
+ 9.5282232631648411840742957e+04,
+ 5.4811599352999901232411871e+07,
+ 5.2859121715894396531132279e-01,
+ 9.7587991957286474464259698e-06,
+ 4.328064329346044846740467e+09,
+ 8.4406761805034547437659092e+02,
+ 1.6946633276191194947742146e+05,
+ 5.3449040147551939075312879e+02,
+ 6.688182138451414936380374e+01,
+ 2.0609869004248742886827439e-09,
+}
+var remainder = []float64{
+ 4.197615023265299782906368e-02,
+ 2.261127525421895434476482e+00,
+ 3.231794108794261433104108e-02,
+ -2.120723654214984321697556e-02,
+ 3.637062928015826201999516e-01,
+ 1.220868282268106064236690e+00,
+ -4.581668629186133046005125e-01,
+ -9.117596417440410050403443e-01,
+ 8.734595415957246977711748e-01,
+ 1.314075231424398637614104e+00,
+}
+var round = []float64{
+ 5,
+ 8,
+ Copysign(0, -1),
+ -5,
+ 10,
+ 3,
+ 5,
+ 3,
+ 2,
+ -9,
+}
+var signbit = []bool{
+ false,
+ false,
+ true,
+ true,
+ false,
+ false,
+ false,
+ false,
+ false,
+ true,
+}
+var sin = []float64{
+ -9.6466616586009283766724726e-01,
+ 9.9338225271646545763467022e-01,
+ -2.7335587039794393342449301e-01,
+ 9.5586257685042792878173752e-01,
+ -2.099421066779969164496634e-01,
+ 2.135578780799860532750616e-01,
+ -8.694568971167362743327708e-01,
+ 4.019566681155577786649878e-01,
+ 9.6778633541687993721617774e-01,
+ -6.734405869050344734943028e-01,
+}
+
+// Results for 100000 * Pi + vf[i]
+var sinLarge = []float64{
+ -9.646661658548936063912e-01,
+ 9.933822527198506903752e-01,
+ -2.7335587036246899796e-01,
+ 9.55862576853689321268e-01,
+ -2.099421066862688873691e-01,
+ 2.13557878070308981163e-01,
+ -8.694568970959221300497e-01,
+ 4.01956668098863248917e-01,
+ 9.67786335404528727927e-01,
+ -6.7344058693131973066e-01,
+}
+var sinh = []float64{
+ 7.2661916084208532301448439e+01,
+ 1.1479409110035194500526446e+03,
+ -2.8043136512812518927312641e-01,
+ -7.499429091181587232835164e+01,
+ 7.6552466042906758523925934e+03,
+ 9.3031583421672014313789064e+00,
+ 9.330815755828109072810322e+01,
+ 7.6179893137269146407361477e+00,
+ 3.021769180549615819524392e+00,
+ -2.95950575724449499189888e+03,
+}
+var sqrt = []float64{
+ 2.2313699659365484748756904e+00,
+ 2.7818829009464263511285458e+00,
+ 5.2619393496314796848143251e-01,
+ 2.2384377628763938724244104e+00,
+ 3.1042380236055381099288487e+00,
+ 1.7106657298385224403917771e+00,
+ 2.286718922705479046148059e+00,
+ 1.6516476350711159636222979e+00,
+ 1.3510396336454586262419247e+00,
+ 2.9471892997524949215723329e+00,
+}
+var tan = []float64{
+ -3.661316565040227801781974e+00,
+ 8.64900232648597589369854e+00,
+ -2.8417941955033612725238097e-01,
+ 3.253290185974728640827156e+00,
+ 2.147275640380293804770778e-01,
+ -2.18600910711067004921551e-01,
+ -1.760002817872367935518928e+00,
+ -4.389808914752818126249079e-01,
+ -3.843885560201130679995041e+00,
+ 9.10988793377685105753416e-01,
+}
+
+// Results for 100000 * Pi + vf[i]
+var tanLarge = []float64{
+ -3.66131656475596512705e+00,
+ 8.6490023287202547927e+00,
+ -2.841794195104782406e-01,
+ 3.2532901861033120983e+00,
+ 2.14727564046880001365e-01,
+ -2.18600910700688062874e-01,
+ -1.760002817699722747043e+00,
+ -4.38980891453536115952e-01,
+ -3.84388555942723509071e+00,
+ 9.1098879344275101051e-01,
+}
+var tanh = []float64{
+ 9.9990531206936338549262119e-01,
+ 9.9999962057085294197613294e-01,
+ -2.7001505097318677233756845e-01,
+ -9.9991110943061718603541401e-01,
+ 9.9999999146798465745022007e-01,
+ 9.9427249436125236705001048e-01,
+ 9.9994257600983138572705076e-01,
+ 9.9149409509772875982054701e-01,
+ 9.4936501296239685514466577e-01,
+ -9.9999994291374030946055701e-01,
+}
+var trunc = []float64{
+ 4.0000000000000000e+00,
+ 7.0000000000000000e+00,
+ Copysign(0, -1),
+ -5.0000000000000000e+00,
+ 9.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 5.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 1.0000000000000000e+00,
+ -8.0000000000000000e+00,
+}
+var y0 = []float64{
+ -3.053399153780788357534855e-01,
+ 1.7437227649515231515503649e-01,
+ -8.6221781263678836910392572e-01,
+ -3.100664880987498407872839e-01,
+ 1.422200649300982280645377e-01,
+ 4.000004067997901144239363e-01,
+ -3.3340749753099352392332536e-01,
+ 4.5399790746668954555205502e-01,
+ 4.8290004112497761007536522e-01,
+ 2.7036697826604756229601611e-01,
+}
+var y1 = []float64{
+ 0.15494213737457922210218611,
+ -0.2165955142081145245075746,
+ -2.4644949631241895201032829,
+ 0.1442740489541836405154505,
+ 0.2215379960518984777080163,
+ 0.3038800915160754150565448,
+ 0.0691107642452362383808547,
+ 0.2380116417809914424860165,
+ -0.20849492979459761009678934,
+ 0.0242503179793232308250804,
+}
+var y2 = []float64{
+ 0.3675780219390303613394936,
+ -0.23034826393250119879267257,
+ -16.939677983817727205631397,
+ 0.367653980523052152867791,
+ -0.0962401471767804440353136,
+ -0.1923169356184851105200523,
+ 0.35984072054267882391843766,
+ -0.2794987252299739821654982,
+ -0.7113490692587462579757954,
+ -0.2647831587821263302087457,
+}
+var yM3 = []float64{
+ -0.14035984421094849100895341,
+ -0.097535139617792072703973,
+ 242.25775994555580176377379,
+ -0.1492267014802818619511046,
+ 0.26148702629155918694500469,
+ 0.56675383593895176530394248,
+ -0.206150264009006981070575,
+ 0.64784284687568332737963658,
+ 1.3503631555901938037008443,
+ 0.1461869756579956803341844,
+}
+
+// arguments and expected results for special cases
+var vfacosSC = []float64{
+ -Pi,
+ 1,
+ Pi,
+ NaN(),
+}
+var acosSC = []float64{
+ NaN(),
+ 0,
+ NaN(),
+ NaN(),
+}
+
+var vfacoshSC = []float64{
+ Inf(-1),
+ 0.5,
+ 1,
+ Inf(1),
+ NaN(),
+}
+var acoshSC = []float64{
+ NaN(),
+ NaN(),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfasinSC = []float64{
+ -Pi,
+ Copysign(0, -1),
+ 0,
+ Pi,
+ NaN(),
+}
+var asinSC = []float64{
+ NaN(),
+ Copysign(0, -1),
+ 0,
+ NaN(),
+ NaN(),
+}
+
+var vfasinhSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var asinhSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfatanSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var atanSC = []float64{
+ -Pi / 2,
+ Copysign(0, -1),
+ 0,
+ Pi / 2,
+ NaN(),
+}
+
+var vfatanhSC = []float64{
+ Inf(-1),
+ -Pi,
+ -1,
+ Copysign(0, -1),
+ 0,
+ 1,
+ Pi,
+ Inf(1),
+ NaN(),
+}
+var atanhSC = []float64{
+ NaN(),
+ NaN(),
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+ NaN(),
+ NaN(),
+}
+var vfatan2SC = [][2]float64{
+ {Inf(-1), Inf(-1)},
+ {Inf(-1), -Pi},
+ {Inf(-1), 0},
+ {Inf(-1), +Pi},
+ {Inf(-1), Inf(1)},
+ {Inf(-1), NaN()},
+ {-Pi, Inf(-1)},
+ {-Pi, 0},
+ {-Pi, Inf(1)},
+ {-Pi, NaN()},
+ {Copysign(0, -1), Inf(-1)},
+ {Copysign(0, -1), -Pi},
+ {Copysign(0, -1), Copysign(0, -1)},
+ {Copysign(0, -1), 0},
+ {Copysign(0, -1), +Pi},
+ {Copysign(0, -1), Inf(1)},
+ {Copysign(0, -1), NaN()},
+ {0, Inf(-1)},
+ {0, -Pi},
+ {0, Copysign(0, -1)},
+ {0, 0},
+ {0, +Pi},
+ {0, Inf(1)},
+ {0, NaN()},
+ {+Pi, Inf(-1)},
+ {+Pi, 0},
+ {+Pi, Inf(1)},
+ {1.0, Inf(1)},
+ {-1.0, Inf(1)},
+ {+Pi, NaN()},
+ {Inf(1), Inf(-1)},
+ {Inf(1), -Pi},
+ {Inf(1), 0},
+ {Inf(1), +Pi},
+ {Inf(1), Inf(1)},
+ {Inf(1), NaN()},
+ {NaN(), NaN()},
+}
+var atan2SC = []float64{
+ -3 * Pi / 4, // atan2(-Inf, -Inf)
+ -Pi / 2, // atan2(-Inf, -Pi)
+ -Pi / 2, // atan2(-Inf, +0)
+ -Pi / 2, // atan2(-Inf, +Pi)
+ -Pi / 4, // atan2(-Inf, +Inf)
+ NaN(), // atan2(-Inf, NaN)
+ -Pi, // atan2(-Pi, -Inf)
+ -Pi / 2, // atan2(-Pi, +0)
+ Copysign(0, -1), // atan2(-Pi, Inf)
+ NaN(), // atan2(-Pi, NaN)
+ -Pi, // atan2(-0, -Inf)
+ -Pi, // atan2(-0, -Pi)
+ -Pi, // atan2(-0, -0)
+ Copysign(0, -1), // atan2(-0, +0)
+ Copysign(0, -1), // atan2(-0, +Pi)
+ Copysign(0, -1), // atan2(-0, +Inf)
+ NaN(), // atan2(-0, NaN)
+ Pi, // atan2(+0, -Inf)
+ Pi, // atan2(+0, -Pi)
+ Pi, // atan2(+0, -0)
+ 0, // atan2(+0, +0)
+ 0, // atan2(+0, +Pi)
+ 0, // atan2(+0, +Inf)
+ NaN(), // atan2(+0, NaN)
+ Pi, // atan2(+Pi, -Inf)
+ Pi / 2, // atan2(+Pi, +0)
+ 0, // atan2(+Pi, +Inf)
+ 0, // atan2(+1, +Inf)
+ Copysign(0, -1), // atan2(-1, +Inf)
+ NaN(), // atan2(+Pi, NaN)
+ 3 * Pi / 4, // atan2(+Inf, -Inf)
+ Pi / 2, // atan2(+Inf, -Pi)
+ Pi / 2, // atan2(+Inf, +0)
+ Pi / 2, // atan2(+Inf, +Pi)
+ Pi / 4, // atan2(+Inf, +Inf)
+ NaN(), // atan2(+Inf, NaN)
+ NaN(), // atan2(NaN, NaN)
+}
+
+var vfcbrtSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var cbrtSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfceilSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var ceilSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfcopysignSC = []float64{
+ Inf(-1),
+ Inf(1),
+ NaN(),
+}
+var copysignSC = []float64{
+ Inf(-1),
+ Inf(-1),
+ NaN(),
+}
+
+var vfcosSC = []float64{
+ Inf(-1),
+ Inf(1),
+ NaN(),
+}
+var cosSC = []float64{
+ NaN(),
+ NaN(),
+ NaN(),
+}
+
+var vfcoshSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var coshSC = []float64{
+ Inf(1),
+ 1,
+ 1,
+ Inf(1),
+ NaN(),
+}
+
+var vferfSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+ -1000,
+ 1000,
+}
+var erfSC = []float64{
+ -1,
+ Copysign(0, -1),
+ 0,
+ 1,
+ NaN(),
+ -1,
+ 1,
+}
+
+var vferfcSC = []float64{
+ Inf(-1),
+ Inf(1),
+ NaN(),
+ -1000,
+ 1000,
+}
+var erfcSC = []float64{
+ 2,
+ 0,
+ NaN(),
+ 2,
+ 0,
+}
+
+var vferfinvSC = []float64{
+ 1,
+ -1,
+ 0,
+ Inf(-1),
+ Inf(1),
+ NaN(),
+}
+var erfinvSC = []float64{
+ Inf(+1),
+ Inf(-1),
+ 0,
+ NaN(),
+ NaN(),
+ NaN(),
+}
+
+var vferfcinvSC = []float64{
+ 0,
+ 2,
+ 1,
+ Inf(1),
+ Inf(-1),
+ NaN(),
+}
+var erfcinvSC = []float64{
+ Inf(+1),
+ Inf(-1),
+ 0,
+ NaN(),
+ NaN(),
+ NaN(),
+}
+
+var vfexpSC = []float64{
+ Inf(-1),
+ -2000,
+ 2000,
+ Inf(1),
+ NaN(),
+ // smallest float64 that overflows Exp(x)
+ 7.097827128933841e+02,
+ // Issue 18912
+ 1.48852223e+09,
+ 1.4885222e+09,
+ 1,
+ // near zero
+ 3.725290298461915e-09,
+ // denormal
+ -740,
+}
+var expSC = []float64{
+ 0,
+ 0,
+ Inf(1),
+ Inf(1),
+ NaN(),
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ 2.718281828459045,
+ 1.0000000037252903,
+ 4.2e-322,
+}
+
+var vfexp2SC = []float64{
+ Inf(-1),
+ -2000,
+ 2000,
+ Inf(1),
+ NaN(),
+ // smallest float64 that overflows Exp2(x)
+ 1024,
+ // near underflow
+ -1.07399999999999e+03,
+ // near zero
+ 3.725290298461915e-09,
+}
+var exp2SC = []float64{
+ 0,
+ 0,
+ Inf(1),
+ Inf(1),
+ NaN(),
+ Inf(1),
+ 5e-324,
+ 1.0000000025821745,
+}
+
+var vfexpm1SC = []float64{
+ Inf(-1),
+ -710,
+ Copysign(0, -1),
+ 0,
+ 710,
+ Inf(1),
+ NaN(),
+}
+var expm1SC = []float64{
+ -1,
+ -1,
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ Inf(1),
+ NaN(),
+}
+
+var vffabsSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var fabsSC = []float64{
+ Inf(1),
+ 0,
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vffdimSC = [][2]float64{
+ {Inf(-1), Inf(-1)},
+ {Inf(-1), Inf(1)},
+ {Inf(-1), NaN()},
+ {Copysign(0, -1), Copysign(0, -1)},
+ {Copysign(0, -1), 0},
+ {0, Copysign(0, -1)},
+ {0, 0},
+ {Inf(1), Inf(-1)},
+ {Inf(1), Inf(1)},
+ {Inf(1), NaN()},
+ {NaN(), Inf(-1)},
+ {NaN(), Copysign(0, -1)},
+ {NaN(), 0},
+ {NaN(), Inf(1)},
+ {NaN(), NaN()},
+}
+var nan = Float64frombits(0xFFF8000000000000) // SSE2 DIVSD 0/0
+var vffdim2SC = [][2]float64{
+ {Inf(-1), Inf(-1)},
+ {Inf(-1), Inf(1)},
+ {Inf(-1), nan},
+ {Copysign(0, -1), Copysign(0, -1)},
+ {Copysign(0, -1), 0},
+ {0, Copysign(0, -1)},
+ {0, 0},
+ {Inf(1), Inf(-1)},
+ {Inf(1), Inf(1)},
+ {Inf(1), nan},
+ {nan, Inf(-1)},
+ {nan, Copysign(0, -1)},
+ {nan, 0},
+ {nan, Inf(1)},
+ {nan, nan},
+}
+var fdimSC = []float64{
+ NaN(),
+ 0,
+ NaN(),
+ 0,
+ 0,
+ 0,
+ 0,
+ Inf(1),
+ NaN(),
+ NaN(),
+ NaN(),
+ NaN(),
+ NaN(),
+ NaN(),
+ NaN(),
+}
+var fmaxSC = []float64{
+ Inf(-1),
+ Inf(1),
+ NaN(),
+ Copysign(0, -1),
+ 0,
+ 0,
+ 0,
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ NaN(),
+ NaN(),
+ NaN(),
+ Inf(1),
+ NaN(),
+}
+var fminSC = []float64{
+ Inf(-1),
+ Inf(-1),
+ Inf(-1),
+ Copysign(0, -1),
+ Copysign(0, -1),
+ Copysign(0, -1),
+ 0,
+ Inf(-1),
+ Inf(1),
+ NaN(),
+ Inf(-1),
+ NaN(),
+ NaN(),
+ NaN(),
+ NaN(),
+}
+
+var vffmodSC = [][2]float64{
+ {Inf(-1), Inf(-1)},
+ {Inf(-1), -Pi},
+ {Inf(-1), 0},
+ {Inf(-1), Pi},
+ {Inf(-1), Inf(1)},
+ {Inf(-1), NaN()},
+ {-Pi, Inf(-1)},
+ {-Pi, 0},
+ {-Pi, Inf(1)},
+ {-Pi, NaN()},
+ {Copysign(0, -1), Inf(-1)},
+ {Copysign(0, -1), 0},
+ {Copysign(0, -1), Inf(1)},
+ {Copysign(0, -1), NaN()},
+ {0, Inf(-1)},
+ {0, 0},
+ {0, Inf(1)},
+ {0, NaN()},
+ {Pi, Inf(-1)},
+ {Pi, 0},
+ {Pi, Inf(1)},
+ {Pi, NaN()},
+ {Inf(1), Inf(-1)},
+ {Inf(1), -Pi},
+ {Inf(1), 0},
+ {Inf(1), Pi},
+ {Inf(1), Inf(1)},
+ {Inf(1), NaN()},
+ {NaN(), Inf(-1)},
+ {NaN(), -Pi},
+ {NaN(), 0},
+ {NaN(), Pi},
+ {NaN(), Inf(1)},
+ {NaN(), NaN()},
+}
+var fmodSC = []float64{
+ NaN(), // fmod(-Inf, -Inf)
+ NaN(), // fmod(-Inf, -Pi)
+ NaN(), // fmod(-Inf, 0)
+ NaN(), // fmod(-Inf, Pi)
+ NaN(), // fmod(-Inf, +Inf)
+ NaN(), // fmod(-Inf, NaN)
+ -Pi, // fmod(-Pi, -Inf)
+ NaN(), // fmod(-Pi, 0)
+ -Pi, // fmod(-Pi, +Inf)
+ NaN(), // fmod(-Pi, NaN)
+ Copysign(0, -1), // fmod(-0, -Inf)
+ NaN(), // fmod(-0, 0)
+ Copysign(0, -1), // fmod(-0, Inf)
+ NaN(), // fmod(-0, NaN)
+ 0, // fmod(0, -Inf)
+ NaN(), // fmod(0, 0)
+ 0, // fmod(0, +Inf)
+ NaN(), // fmod(0, NaN)
+ Pi, // fmod(Pi, -Inf)
+ NaN(), // fmod(Pi, 0)
+ Pi, // fmod(Pi, +Inf)
+ NaN(), // fmod(Pi, NaN)
+ NaN(), // fmod(+Inf, -Inf)
+ NaN(), // fmod(+Inf, -Pi)
+ NaN(), // fmod(+Inf, 0)
+ NaN(), // fmod(+Inf, Pi)
+ NaN(), // fmod(+Inf, +Inf)
+ NaN(), // fmod(+Inf, NaN)
+ NaN(), // fmod(NaN, -Inf)
+ NaN(), // fmod(NaN, -Pi)
+ NaN(), // fmod(NaN, 0)
+ NaN(), // fmod(NaN, Pi)
+ NaN(), // fmod(NaN, +Inf)
+ NaN(), // fmod(NaN, NaN)
+}
+
+var vffrexpSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var frexpSC = []fi{
+ {Inf(-1), 0},
+ {Copysign(0, -1), 0},
+ {0, 0},
+ {Inf(1), 0},
+ {NaN(), 0},
+}
+
+var vfgamma = [][2]float64{
+ {Inf(1), Inf(1)},
+ {Inf(-1), NaN()},
+ {0, Inf(1)},
+ {Copysign(0, -1), Inf(-1)},
+ {NaN(), NaN()},
+ {-1, NaN()},
+ {-2, NaN()},
+ {-3, NaN()},
+ {-1e16, NaN()},
+ {-1e300, NaN()},
+ {1.7e308, Inf(1)},
+
+ // Test inputs inspired by Python test suite.
+ // Outputs computed at high precision by PARI/GP.
+ // If recomputing table entries, be careful to use
+ // high-precision (%.1000g) formatting of the float64 inputs.
+ // For example, -2.0000000000000004 is the float64 with exact value
+ // -2.00000000000000044408920985626161695, and
+ // gamma(-2.0000000000000004) = -1249999999999999.5386078562728167651513, while
+ // gamma(-2.00000000000000044408920985626161695) = -1125899906826907.2044875028130093136826.
+ // Thus the table lists -1.1258999068426235e+15 as the answer.
+ {0.5, 1.772453850905516},
+ {1.5, 0.886226925452758},
+ {2.5, 1.329340388179137},
+ {3.5, 3.3233509704478426},
+ {-0.5, -3.544907701811032},
+ {-1.5, 2.363271801207355},
+ {-2.5, -0.9453087204829419},
+ {-3.5, 0.2700882058522691},
+ {0.1, 9.51350769866873},
+ {0.01, 99.4325851191506},
+ {1e-08, 9.999999942278434e+07},
+ {1e-16, 1e+16},
+ {0.001, 999.4237724845955},
+ {1e-16, 1e+16},
+ {1e-308, 1e+308},
+ {5.6e-309, 1.7857142857142864e+308},
+ {5.5e-309, Inf(1)},
+ {1e-309, Inf(1)},
+ {1e-323, Inf(1)},
+ {5e-324, Inf(1)},
+ {-0.1, -10.686287021193193},
+ {-0.01, -100.58719796441078},
+ {-1e-08, -1.0000000057721567e+08},
+ {-1e-16, -1e+16},
+ {-0.001, -1000.5782056293586},
+ {-1e-16, -1e+16},
+ {-1e-308, -1e+308},
+ {-5.6e-309, -1.7857142857142864e+308},
+ {-5.5e-309, Inf(-1)},
+ {-1e-309, Inf(-1)},
+ {-1e-323, Inf(-1)},
+ {-5e-324, Inf(-1)},
+ {-0.9999999999999999, -9.007199254740992e+15},
+ {-1.0000000000000002, 4.5035996273704955e+15},
+ {-1.9999999999999998, 2.2517998136852485e+15},
+ {-2.0000000000000004, -1.1258999068426235e+15},
+ {-100.00000000000001, -7.540083334883109e-145},
+ {-99.99999999999999, 7.540083334884096e-145},
+ {17, 2.0922789888e+13},
+ {171, 7.257415615307999e+306},
+ {171.6, 1.5858969096672565e+308},
+ {171.624, 1.7942117599248104e+308},
+ {171.625, Inf(1)},
+ {172, Inf(1)},
+ {2000, Inf(1)},
+ {-100.5, -3.3536908198076787e-159},
+ {-160.5, -5.255546447007829e-286},
+ {-170.5, -3.3127395215386074e-308},
+ {-171.5, 1.9316265431712e-310},
+ {-176.5, -1.196e-321},
+ {-177.5, 5e-324},
+ {-178.5, Copysign(0, -1)},
+ {-179.5, 0},
+ {-201.0001, 0},
+ {-202.9999, Copysign(0, -1)},
+ {-1000.5, Copysign(0, -1)},
+ {-1.0000000003e+09, Copysign(0, -1)},
+ {-4.5035996273704955e+15, 0},
+ {-63.349078729022985, 4.177797167776188e-88},
+ {-127.45117632943295, 1.183111089623681e-214},
+}
+
+var vfhypotSC = [][2]float64{
+ {Inf(-1), Inf(-1)},
+ {Inf(-1), 0},
+ {Inf(-1), Inf(1)},
+ {Inf(-1), NaN()},
+ {Copysign(0, -1), Copysign(0, -1)},
+ {Copysign(0, -1), 0},
+ {0, Copysign(0, -1)},
+ {0, 0}, // +0, +0
+ {0, Inf(-1)},
+ {0, Inf(1)},
+ {0, NaN()},
+ {Inf(1), Inf(-1)},
+ {Inf(1), 0},
+ {Inf(1), Inf(1)},
+ {Inf(1), NaN()},
+ {NaN(), Inf(-1)},
+ {NaN(), 0},
+ {NaN(), Inf(1)},
+ {NaN(), NaN()},
+}
+var hypotSC = []float64{
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ 0,
+ 0,
+ 0,
+ 0,
+ Inf(1),
+ Inf(1),
+ NaN(),
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ NaN(),
+ Inf(1),
+ NaN(),
+}
+
+var ilogbSC = []int{
+ MaxInt32,
+ MinInt32,
+ MaxInt32,
+ MaxInt32,
+}
+
+var vfj0SC = []float64{
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var j0SC = []float64{
+ 0,
+ 1,
+ 0,
+ NaN(),
+}
+var j1SC = []float64{
+ 0,
+ 0,
+ 0,
+ NaN(),
+}
+var j2SC = []float64{
+ 0,
+ 0,
+ 0,
+ NaN(),
+}
+var jM3SC = []float64{
+ 0,
+ 0,
+ 0,
+ NaN(),
+}
+
+var vfldexpSC = []fi{
+ {0, 0},
+ {0, -1075},
+ {0, 1024},
+ {Copysign(0, -1), 0},
+ {Copysign(0, -1), -1075},
+ {Copysign(0, -1), 1024},
+ {Inf(1), 0},
+ {Inf(1), -1024},
+ {Inf(-1), 0},
+ {Inf(-1), -1024},
+ {NaN(), -1024},
+ {10, int(1) << (uint64(unsafe.Sizeof(0)-1) * 8)},
+ {10, -(int(1) << (uint64(unsafe.Sizeof(0)-1) * 8))},
+}
+var ldexpSC = []float64{
+ 0,
+ 0,
+ 0,
+ Copysign(0, -1),
+ Copysign(0, -1),
+ Copysign(0, -1),
+ Inf(1),
+ Inf(1),
+ Inf(-1),
+ Inf(-1),
+ NaN(),
+ Inf(1),
+ 0,
+}
+
+var vflgammaSC = []float64{
+ Inf(-1),
+ -3,
+ 0,
+ 1,
+ 2,
+ Inf(1),
+ NaN(),
+}
+var lgammaSC = []fi{
+ {Inf(-1), 1},
+ {Inf(1), 1},
+ {Inf(1), 1},
+ {0, 1},
+ {0, 1},
+ {Inf(1), 1},
+ {NaN(), 1},
+}
+
+var vflogSC = []float64{
+ Inf(-1),
+ -Pi,
+ Copysign(0, -1),
+ 0,
+ 1,
+ Inf(1),
+ NaN(),
+}
+var logSC = []float64{
+ NaN(),
+ NaN(),
+ Inf(-1),
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vflogbSC = []float64{
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var logbSC = []float64{
+ Inf(1),
+ Inf(-1),
+ Inf(1),
+ NaN(),
+}
+
+var vflog1pSC = []float64{
+ Inf(-1),
+ -Pi,
+ -1,
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+ 4503599627370496.5, // Issue #29488
+}
+var log1pSC = []float64{
+ NaN(),
+ NaN(),
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+ 36.04365338911715, // Issue #29488
+}
+
+var vfmodfSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ Inf(1),
+ NaN(),
+}
+var modfSC = [][2]float64{
+ {Inf(-1), NaN()}, // [2]float64{Copysign(0, -1), Inf(-1)},
+ {Copysign(0, -1), Copysign(0, -1)},
+ {Inf(1), NaN()}, // [2]float64{0, Inf(1)},
+ {NaN(), NaN()},
+}
+
+var vfnextafter32SC = [][2]float32{
+ {0, 0},
+ {0, float32(Copysign(0, -1))},
+ {0, -1},
+ {0, float32(NaN())},
+ {float32(Copysign(0, -1)), 1},
+ {float32(Copysign(0, -1)), 0},
+ {float32(Copysign(0, -1)), float32(Copysign(0, -1))},
+ {float32(Copysign(0, -1)), -1},
+ {float32(NaN()), 0},
+ {float32(NaN()), float32(NaN())},
+}
+var nextafter32SC = []float32{
+ 0,
+ 0,
+ -1.401298464e-45, // Float32frombits(0x80000001)
+ float32(NaN()),
+ 1.401298464e-45, // Float32frombits(0x00000001)
+ float32(Copysign(0, -1)),
+ float32(Copysign(0, -1)),
+ -1.401298464e-45, // Float32frombits(0x80000001)
+ float32(NaN()),
+ float32(NaN()),
+}
+
+var vfnextafter64SC = [][2]float64{
+ {0, 0},
+ {0, Copysign(0, -1)},
+ {0, -1},
+ {0, NaN()},
+ {Copysign(0, -1), 1},
+ {Copysign(0, -1), 0},
+ {Copysign(0, -1), Copysign(0, -1)},
+ {Copysign(0, -1), -1},
+ {NaN(), 0},
+ {NaN(), NaN()},
+}
+var nextafter64SC = []float64{
+ 0,
+ 0,
+ -4.9406564584124654418e-324, // Float64frombits(0x8000000000000001)
+ NaN(),
+ 4.9406564584124654418e-324, // Float64frombits(0x0000000000000001)
+ Copysign(0, -1),
+ Copysign(0, -1),
+ -4.9406564584124654418e-324, // Float64frombits(0x8000000000000001)
+ NaN(),
+ NaN(),
+}
+
+var vfpowSC = [][2]float64{
+ {Inf(-1), -Pi},
+ {Inf(-1), -3},
+ {Inf(-1), Copysign(0, -1)},
+ {Inf(-1), 0},
+ {Inf(-1), 1},
+ {Inf(-1), 3},
+ {Inf(-1), Pi},
+ {Inf(-1), 0.5},
+ {Inf(-1), NaN()},
+
+ {-Pi, Inf(-1)},
+ {-Pi, -Pi},
+ {-Pi, Copysign(0, -1)},
+ {-Pi, 0},
+ {-Pi, 1},
+ {-Pi, Pi},
+ {-Pi, Inf(1)},
+ {-Pi, NaN()},
+
+ {-1, Inf(-1)},
+ {-1, Inf(1)},
+ {-1, NaN()},
+ {-1 / 2, Inf(-1)},
+ {-1 / 2, Inf(1)},
+ {Copysign(0, -1), Inf(-1)},
+ {Copysign(0, -1), -Pi},
+ {Copysign(0, -1), -0.5},
+ {Copysign(0, -1), -3},
+ {Copysign(0, -1), 3},
+ {Copysign(0, -1), Pi},
+ {Copysign(0, -1), 0.5},
+ {Copysign(0, -1), Inf(1)},
+
+ {0, Inf(-1)},
+ {0, -Pi},
+ {0, -3},
+ {0, Copysign(0, -1)},
+ {0, 0},
+ {0, 3},
+ {0, Pi},
+ {0, Inf(1)},
+ {0, NaN()},
+
+ {1 / 2, Inf(-1)},
+ {1 / 2, Inf(1)},
+ {1, Inf(-1)},
+ {1, Inf(1)},
+ {1, NaN()},
+
+ {Pi, Inf(-1)},
+ {Pi, Copysign(0, -1)},
+ {Pi, 0},
+ {Pi, 1},
+ {Pi, Inf(1)},
+ {Pi, NaN()},
+ {Inf(1), -Pi},
+ {Inf(1), Copysign(0, -1)},
+ {Inf(1), 0},
+ {Inf(1), 1},
+ {Inf(1), Pi},
+ {Inf(1), NaN()},
+ {NaN(), -Pi},
+ {NaN(), Copysign(0, -1)},
+ {NaN(), 0},
+ {NaN(), 1},
+ {NaN(), Pi},
+ {NaN(), NaN()},
+
+ // Issue #7394 overflow checks
+ {2, float64(1 << 32)},
+ {2, -float64(1 << 32)},
+ {-2, float64(1<<32 + 1)},
+ {1 / 2, float64(1 << 45)},
+ {1 / 2, -float64(1 << 45)},
+ {Nextafter(1, 2), float64(1 << 63)},
+ {Nextafter(1, -2), float64(1 << 63)},
+ {Nextafter(-1, 2), float64(1 << 63)},
+ {Nextafter(-1, -2), float64(1 << 63)},
+}
+var powSC = []float64{
+ 0, // pow(-Inf, -Pi)
+ Copysign(0, -1), // pow(-Inf, -3)
+ 1, // pow(-Inf, -0)
+ 1, // pow(-Inf, +0)
+ Inf(-1), // pow(-Inf, 1)
+ Inf(-1), // pow(-Inf, 3)
+ Inf(1), // pow(-Inf, Pi)
+ Inf(1), // pow(-Inf, 0.5)
+ NaN(), // pow(-Inf, NaN)
+ 0, // pow(-Pi, -Inf)
+ NaN(), // pow(-Pi, -Pi)
+ 1, // pow(-Pi, -0)
+ 1, // pow(-Pi, +0)
+ -Pi, // pow(-Pi, 1)
+ NaN(), // pow(-Pi, Pi)
+ Inf(1), // pow(-Pi, +Inf)
+ NaN(), // pow(-Pi, NaN)
+ 1, // pow(-1, -Inf) IEEE 754-2008
+ 1, // pow(-1, +Inf) IEEE 754-2008
+ NaN(), // pow(-1, NaN)
+ Inf(1), // pow(-1/2, -Inf)
+ 0, // pow(-1/2, +Inf)
+ Inf(1), // pow(-0, -Inf)
+ Inf(1), // pow(-0, -Pi)
+ Inf(1), // pow(-0, -0.5)
+ Inf(-1), // pow(-0, -3) IEEE 754-2008
+ Copysign(0, -1), // pow(-0, 3) IEEE 754-2008
+ 0, // pow(-0, +Pi)
+ 0, // pow(-0, 0.5)
+ 0, // pow(-0, +Inf)
+ Inf(1), // pow(+0, -Inf)
+ Inf(1), // pow(+0, -Pi)
+ Inf(1), // pow(+0, -3)
+ 1, // pow(+0, -0)
+ 1, // pow(+0, +0)
+ 0, // pow(+0, 3)
+ 0, // pow(+0, +Pi)
+ 0, // pow(+0, +Inf)
+ NaN(), // pow(+0, NaN)
+ Inf(1), // pow(1/2, -Inf)
+ 0, // pow(1/2, +Inf)
+ 1, // pow(1, -Inf) IEEE 754-2008
+ 1, // pow(1, +Inf) IEEE 754-2008
+ 1, // pow(1, NaN) IEEE 754-2008
+ 0, // pow(+Pi, -Inf)
+ 1, // pow(+Pi, -0)
+ 1, // pow(+Pi, +0)
+ Pi, // pow(+Pi, 1)
+ Inf(1), // pow(+Pi, +Inf)
+ NaN(), // pow(+Pi, NaN)
+ 0, // pow(+Inf, -Pi)
+ 1, // pow(+Inf, -0)
+ 1, // pow(+Inf, +0)
+ Inf(1), // pow(+Inf, 1)
+ Inf(1), // pow(+Inf, Pi)
+ NaN(), // pow(+Inf, NaN)
+ NaN(), // pow(NaN, -Pi)
+ 1, // pow(NaN, -0)
+ 1, // pow(NaN, +0)
+ NaN(), // pow(NaN, 1)
+ NaN(), // pow(NaN, +Pi)
+ NaN(), // pow(NaN, NaN)
+
+ // Issue #7394 overflow checks
+ Inf(1), // pow(2, float64(1 << 32))
+ 0, // pow(2, -float64(1 << 32))
+ Inf(-1), // pow(-2, float64(1<<32 + 1))
+ 0, // pow(1/2, float64(1 << 45))
+ Inf(1), // pow(1/2, -float64(1 << 45))
+ Inf(1), // pow(Nextafter(1, 2), float64(1 << 63))
+ 0, // pow(Nextafter(1, -2), float64(1 << 63))
+ 0, // pow(Nextafter(-1, 2), float64(1 << 63))
+ Inf(1), // pow(Nextafter(-1, -2), float64(1 << 63))
+}
+
+var vfpow10SC = []int{
+ MinInt32,
+ -324,
+ -323,
+ -50,
+ -22,
+ -1,
+ 0,
+ 1,
+ 22,
+ 50,
+ 100,
+ 200,
+ 308,
+ 309,
+ MaxInt32,
+}
+
+var pow10SC = []float64{
+ 0, // pow10(MinInt32)
+ 0, // pow10(-324)
+ 1.0e-323, // pow10(-323)
+ 1.0e-50, // pow10(-50)
+ 1.0e-22, // pow10(-22)
+ 1.0e-1, // pow10(-1)
+ 1.0e0, // pow10(0)
+ 1.0e1, // pow10(1)
+ 1.0e22, // pow10(22)
+ 1.0e50, // pow10(50)
+ 1.0e100, // pow10(100)
+ 1.0e200, // pow10(200)
+ 1.0e308, // pow10(308)
+ Inf(1), // pow10(309)
+ Inf(1), // pow10(MaxInt32)
+}
+
+var vfroundSC = [][2]float64{
+ {0, 0},
+ {1.390671161567e-309, 0}, // denormal
+ {0.49999999999999994, 0}, // 0.5-epsilon
+ {0.5, 1},
+ {0.5000000000000001, 1}, // 0.5+epsilon
+ {-1.5, -2},
+ {-2.5, -3},
+ {NaN(), NaN()},
+ {Inf(1), Inf(1)},
+ {2251799813685249.5, 2251799813685250}, // 1 bit fraction
+ {2251799813685250.5, 2251799813685251},
+ {4503599627370495.5, 4503599627370496}, // 1 bit fraction, rounding to 0 bit fraction
+ {4503599627370497, 4503599627370497}, // large integer
+}
+var vfroundEvenSC = [][2]float64{
+ {0, 0},
+ {1.390671161567e-309, 0}, // denormal
+ {0.49999999999999994, 0}, // 0.5-epsilon
+ {0.5, 0},
+ {0.5000000000000001, 1}, // 0.5+epsilon
+ {-1.5, -2},
+ {-2.5, -2},
+ {NaN(), NaN()},
+ {Inf(1), Inf(1)},
+ {2251799813685249.5, 2251799813685250}, // 1 bit fraction
+ {2251799813685250.5, 2251799813685250},
+ {4503599627370495.5, 4503599627370496}, // 1 bit fraction, rounding to 0 bit fraction
+ {4503599627370497, 4503599627370497}, // large integer
+}
+
+var vfsignbitSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var signbitSC = []bool{
+ true,
+ true,
+ false,
+ false,
+ false,
+}
+
+var vfsinSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var sinSC = []float64{
+ NaN(),
+ Copysign(0, -1),
+ 0,
+ NaN(),
+ NaN(),
+}
+
+var vfsinhSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var sinhSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfsqrtSC = []float64{
+ Inf(-1),
+ -Pi,
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+ Float64frombits(2), // subnormal; see https://golang.org/issue/13013
+}
+var sqrtSC = []float64{
+ NaN(),
+ NaN(),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+ 3.1434555694052576e-162,
+}
+
+var vftanhSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var tanhSC = []float64{
+ -1,
+ Copysign(0, -1),
+ 0,
+ 1,
+ NaN(),
+}
+
+var vfy0SC = []float64{
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+ -1,
+}
+var y0SC = []float64{
+ NaN(),
+ Inf(-1),
+ 0,
+ NaN(),
+ NaN(),
+}
+var y1SC = []float64{
+ NaN(),
+ Inf(-1),
+ 0,
+ NaN(),
+ NaN(),
+}
+var y2SC = []float64{
+ NaN(),
+ Inf(-1),
+ 0,
+ NaN(),
+ NaN(),
+}
+var yM3SC = []float64{
+ NaN(),
+ Inf(1),
+ 0,
+ NaN(),
+ NaN(),
+}
+
+// arguments and expected results for boundary cases
+const (
+ SmallestNormalFloat64 = 2.2250738585072014e-308 // 2**-1022
+ LargestSubnormalFloat64 = SmallestNormalFloat64 - SmallestNonzeroFloat64
+)
+
+var vffrexpBC = []float64{
+ SmallestNormalFloat64,
+ LargestSubnormalFloat64,
+ SmallestNonzeroFloat64,
+ MaxFloat64,
+ -SmallestNormalFloat64,
+ -LargestSubnormalFloat64,
+ -SmallestNonzeroFloat64,
+ -MaxFloat64,
+}
+var frexpBC = []fi{
+ {0.5, -1021},
+ {0.99999999999999978, -1022},
+ {0.5, -1073},
+ {0.99999999999999989, 1024},
+ {-0.5, -1021},
+ {-0.99999999999999978, -1022},
+ {-0.5, -1073},
+ {-0.99999999999999989, 1024},
+}
+
+var vfldexpBC = []fi{
+ {SmallestNormalFloat64, -52},
+ {LargestSubnormalFloat64, -51},
+ {SmallestNonzeroFloat64, 1074},
+ {MaxFloat64, -(1023 + 1074)},
+ {1, -1075},
+ {-1, -1075},
+ {1, 1024},
+ {-1, 1024},
+ {1.0000000000000002, -1075},
+ {1, -1075},
+}
+var ldexpBC = []float64{
+ SmallestNonzeroFloat64,
+ 1e-323, // 2**-1073
+ 1,
+ 1e-323, // 2**-1073
+ 0,
+ Copysign(0, -1),
+ Inf(1),
+ Inf(-1),
+ SmallestNonzeroFloat64,
+ 0,
+}
+
+var logbBC = []float64{
+ -1022,
+ -1023,
+ -1074,
+ 1023,
+ -1022,
+ -1023,
+ -1074,
+ 1023,
+}
+
+// Test cases were generated with Berkeley TestFloat-3e/testfloat_gen.
+// http://www.jhauser.us/arithmetic/TestFloat.html.
+// The default rounding mode is selected (nearest/even), and exception flags are ignored.
+var fmaC = []struct{ x, y, z, want float64 }{
+ // Large exponent spread
+ {-3.999999999999087, -1.1123914289620494e-16, -7.999877929687506, -7.999877929687505},
+ {-262112.0000004768, -0.06251525855623184, 1.1102230248837136e-16, 16385.99945072085},
+ {-6.462348523533467e-27, -2.3763644720331857e-211, 4.000000000931324, 4.000000000931324},
+
+ // Effective addition
+ {-2.0000000037252907, 6.7904383376e-313, -3.3951933161e-313, -1.697607001654e-312},
+ {-0.12499999999999999, 512.007568359375, -1.4193627164960366e-16, -64.00094604492188},
+ {-2.7550648847397148e-39, -3.4028301595800694e+38, 0.9960937495343386, 1.9335955376735676},
+ {5.723369164769208e+24, 3.8149300927159385e-06, 1.84489958778182e+19, 4.028324913621874e+19},
+ {-0.4843749999990904, -3.6893487872543293e+19, 9.223653786709391e+18, 2.7093936974938993e+19},
+ {-3.8146972665201165e-06, 4.2949672959999385e+09, -2.2204460489938386e-16, -16384.000003844263},
+ {6.98156394130982e-309, -1.1072962560000002e+09, -4.4414561548793455e-308, -7.73065965765153e-300},
+
+ // Effective subtraction
+ {5e-324, 4.5, -2e-323, 0},
+ {5e-324, 7, -3.5e-323, 0},
+ {5e-324, 0.5000000000000001, -5e-324, Copysign(0, -1)},
+ {-2.1240680525e-314, -1.233647078189316e+308, -0.25781249999954525, -0.25780987964919844},
+ {8.579992955364441e-308, 0.6037391876780558, -4.4501307410480706e-308, 7.29947236107098e-309},
+ {-4.450143471986689e-308, -0.9960937499927239, -4.450419332475649e-308, -1.7659233458788e-310},
+ {1.4932076393918112, -2.2248022430460833e-308, 4.449875571054211e-308, 1.127783865601762e-308},
+
+ // Overflow
+ {-2.288020632214759e+38, -8.98846570988901e+307, 1.7696041796300924e+308, Inf(0)},
+ {1.4888652783208255e+308, -9.007199254742012e+15, -6.807282911929205e+38, Inf(-1)},
+ {9.142703268902826e+192, -1.3504889569802838e+296, -1.9082200803806996e-89, Inf(-1)},
+
+ // Finite x and y, but non-finite z.
+ {31.99218749627471, -1.7976930544991702e+308, Inf(0), Inf(0)},
+ {-1.7976931281784667e+308, -2.0009765625002265, Inf(-1), Inf(-1)},
+
+ // Special
+ {0, 0, 0, 0},
+ {-1.1754226043408471e-38, NaN(), Inf(0), NaN()},
+ {0, 0, 2.22507385643494e-308, 2.22507385643494e-308},
+ {-8.65697792e+09, NaN(), -7.516192799999999e+09, NaN()},
+ {-0.00012207403779029757, 3.221225471996093e+09, NaN(), NaN()},
+ {Inf(-1), 0.1252441407414153, -1.387184532981584e-76, Inf(-1)},
+ {Inf(0), 1.525878907671432e-05, -9.214364835452549e+18, Inf(0)},
+
+ // Random
+ {0.1777916152213626, -32.000015266239636, -2.2204459148334633e-16, -5.689334401293007},
+ {-2.0816681711722314e-16, -0.4997558592585846, -0.9465627129124969, -0.9465627129124968},
+ {-1.9999997615814211, 1.8518819259933516e+19, 16.874999999999996, -3.703763410463646e+19},
+ {-0.12499994039717421, 32767.99999976135, -2.0752587082923246e+19, -2.075258708292325e+19},
+ {7.705600568510257e-34, -1.801432979000528e+16, -0.17224197722973714, -0.17224197722973716},
+ {3.8988133103758913e-308, -0.9848632812499999, 3.893879244098556e-308, 5.40811742605814e-310},
+ {-0.012651981190687427, 6.911985574912436e+38, 6.669240527007144e+18, -8.745031148409496e+36},
+ {4.612811918325842e+18, 1.4901161193847641e-08, 2.6077032311277997e-08, 6.873625395187494e+10},
+ {-9.094947033611148e-13, 4.450691014249257e-308, 2.086006742350485e-308, 2.086006742346437e-308},
+ {-7.751454006381804e-05, 5.588653777189071e-308, -2.2207280111272877e-308, -2.2211612130544025e-308},
+}
+
+func tolerance(a, b, e float64) bool {
+ // Multiplying by e here can underflow denormal values to zero.
+ // Check a==b so that at least if a and b are small and identical
+ // we say they match.
+ if a == b {
+ return true
+ }
+ d := a - b
+ if d < 0 {
+ d = -d
+ }
+
+ // note: b is correct (expected) value, a is actual value.
+ // make error tolerance a fraction of b, not a.
+ if b != 0 {
+ e = e * b
+ if e < 0 {
+ e = -e
+ }
+ }
+ return d < e
+}
+func close(a, b float64) bool { return tolerance(a, b, 1e-14) }
+func veryclose(a, b float64) bool { return tolerance(a, b, 4e-16) }
+func soclose(a, b, e float64) bool { return tolerance(a, b, e) }
+func alike(a, b float64) bool {
+ switch {
+ case IsNaN(a) && IsNaN(b):
+ return true
+ case a == b:
+ return Signbit(a) == Signbit(b)
+ }
+ return false
+}
+
+func TestNaN(t *testing.T) {
+ f64 := NaN()
+ if f64 == f64 {
+ t.Fatalf("NaN() returns %g, expected NaN", f64)
+ }
+ f32 := float32(f64)
+ if f32 == f32 {
+ t.Fatalf("float32(NaN()) is %g, expected NaN", f32)
+ }
+}
+
+func TestAcos(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Acos(a); !close(acos[i], f) {
+ t.Errorf("Acos(%g) = %g, want %g", a, f, acos[i])
+ }
+ }
+ for i := 0; i < len(vfacosSC); i++ {
+ if f := Acos(vfacosSC[i]); !alike(acosSC[i], f) {
+ t.Errorf("Acos(%g) = %g, want %g", vfacosSC[i], f, acosSC[i])
+ }
+ }
+}
+
+func TestAcosh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := 1 + Abs(vf[i])
+ if f := Acosh(a); !veryclose(acosh[i], f) {
+ t.Errorf("Acosh(%g) = %g, want %g", a, f, acosh[i])
+ }
+ }
+ for i := 0; i < len(vfacoshSC); i++ {
+ if f := Acosh(vfacoshSC[i]); !alike(acoshSC[i], f) {
+ t.Errorf("Acosh(%g) = %g, want %g", vfacoshSC[i], f, acoshSC[i])
+ }
+ }
+}
+
+func TestAsin(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Asin(a); !veryclose(asin[i], f) {
+ t.Errorf("Asin(%g) = %g, want %g", a, f, asin[i])
+ }
+ }
+ for i := 0; i < len(vfasinSC); i++ {
+ if f := Asin(vfasinSC[i]); !alike(asinSC[i], f) {
+ t.Errorf("Asin(%g) = %g, want %g", vfasinSC[i], f, asinSC[i])
+ }
+ }
+}
+
+func TestAsinh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Asinh(vf[i]); !veryclose(asinh[i], f) {
+ t.Errorf("Asinh(%g) = %g, want %g", vf[i], f, asinh[i])
+ }
+ }
+ for i := 0; i < len(vfasinhSC); i++ {
+ if f := Asinh(vfasinhSC[i]); !alike(asinhSC[i], f) {
+ t.Errorf("Asinh(%g) = %g, want %g", vfasinhSC[i], f, asinhSC[i])
+ }
+ }
+}
+
+func TestAtan(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Atan(vf[i]); !veryclose(atan[i], f) {
+ t.Errorf("Atan(%g) = %g, want %g", vf[i], f, atan[i])
+ }
+ }
+ for i := 0; i < len(vfatanSC); i++ {
+ if f := Atan(vfatanSC[i]); !alike(atanSC[i], f) {
+ t.Errorf("Atan(%g) = %g, want %g", vfatanSC[i], f, atanSC[i])
+ }
+ }
+}
+
+func TestAtanh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Atanh(a); !veryclose(atanh[i], f) {
+ t.Errorf("Atanh(%g) = %g, want %g", a, f, atanh[i])
+ }
+ }
+ for i := 0; i < len(vfatanhSC); i++ {
+ if f := Atanh(vfatanhSC[i]); !alike(atanhSC[i], f) {
+ t.Errorf("Atanh(%g) = %g, want %g", vfatanhSC[i], f, atanhSC[i])
+ }
+ }
+}
+
+func TestAtan2(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Atan2(10, vf[i]); !veryclose(atan2[i], f) {
+ t.Errorf("Atan2(10, %g) = %g, want %g", vf[i], f, atan2[i])
+ }
+ }
+ for i := 0; i < len(vfatan2SC); i++ {
+ if f := Atan2(vfatan2SC[i][0], vfatan2SC[i][1]); !alike(atan2SC[i], f) {
+ t.Errorf("Atan2(%g, %g) = %g, want %g", vfatan2SC[i][0], vfatan2SC[i][1], f, atan2SC[i])
+ }
+ }
+}
+
+func TestCbrt(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Cbrt(vf[i]); !veryclose(cbrt[i], f) {
+ t.Errorf("Cbrt(%g) = %g, want %g", vf[i], f, cbrt[i])
+ }
+ }
+ for i := 0; i < len(vfcbrtSC); i++ {
+ if f := Cbrt(vfcbrtSC[i]); !alike(cbrtSC[i], f) {
+ t.Errorf("Cbrt(%g) = %g, want %g", vfcbrtSC[i], f, cbrtSC[i])
+ }
+ }
+}
+
+func TestCeil(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Ceil(vf[i]); !alike(ceil[i], f) {
+ t.Errorf("Ceil(%g) = %g, want %g", vf[i], f, ceil[i])
+ }
+ }
+ for i := 0; i < len(vfceilSC); i++ {
+ if f := Ceil(vfceilSC[i]); !alike(ceilSC[i], f) {
+ t.Errorf("Ceil(%g) = %g, want %g", vfceilSC[i], f, ceilSC[i])
+ }
+ }
+}
+
+func TestCopysign(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Copysign(vf[i], -1); copysign[i] != f {
+ t.Errorf("Copysign(%g, -1) = %g, want %g", vf[i], f, copysign[i])
+ }
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := Copysign(vf[i], 1); -copysign[i] != f {
+ t.Errorf("Copysign(%g, 1) = %g, want %g", vf[i], f, -copysign[i])
+ }
+ }
+ for i := 0; i < len(vfcopysignSC); i++ {
+ if f := Copysign(vfcopysignSC[i], -1); !alike(copysignSC[i], f) {
+ t.Errorf("Copysign(%g, -1) = %g, want %g", vfcopysignSC[i], f, copysignSC[i])
+ }
+ }
+}
+
+func TestCos(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Cos(vf[i]); !veryclose(cos[i], f) {
+ t.Errorf("Cos(%g) = %g, want %g", vf[i], f, cos[i])
+ }
+ }
+ for i := 0; i < len(vfcosSC); i++ {
+ if f := Cos(vfcosSC[i]); !alike(cosSC[i], f) {
+ t.Errorf("Cos(%g) = %g, want %g", vfcosSC[i], f, cosSC[i])
+ }
+ }
+}
+
+func TestCosh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Cosh(vf[i]); !close(cosh[i], f) {
+ t.Errorf("Cosh(%g) = %g, want %g", vf[i], f, cosh[i])
+ }
+ }
+ for i := 0; i < len(vfcoshSC); i++ {
+ if f := Cosh(vfcoshSC[i]); !alike(coshSC[i], f) {
+ t.Errorf("Cosh(%g) = %g, want %g", vfcoshSC[i], f, coshSC[i])
+ }
+ }
+}
+
+func TestErf(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Erf(a); !veryclose(erf[i], f) {
+ t.Errorf("Erf(%g) = %g, want %g", a, f, erf[i])
+ }
+ }
+ for i := 0; i < len(vferfSC); i++ {
+ if f := Erf(vferfSC[i]); !alike(erfSC[i], f) {
+ t.Errorf("Erf(%g) = %g, want %g", vferfSC[i], f, erfSC[i])
+ }
+ }
+}
+
+func TestErfc(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Erfc(a); !veryclose(erfc[i], f) {
+ t.Errorf("Erfc(%g) = %g, want %g", a, f, erfc[i])
+ }
+ }
+ for i := 0; i < len(vferfcSC); i++ {
+ if f := Erfc(vferfcSC[i]); !alike(erfcSC[i], f) {
+ t.Errorf("Erfc(%g) = %g, want %g", vferfcSC[i], f, erfcSC[i])
+ }
+ }
+}
+
+func TestErfinv(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Erfinv(a); !veryclose(erfinv[i], f) {
+ t.Errorf("Erfinv(%g) = %g, want %g", a, f, erfinv[i])
+ }
+ }
+ for i := 0; i < len(vferfinvSC); i++ {
+ if f := Erfinv(vferfinvSC[i]); !alike(erfinvSC[i], f) {
+ t.Errorf("Erfinv(%g) = %g, want %g", vferfinvSC[i], f, erfinvSC[i])
+ }
+ }
+ for x := -0.9; x <= 0.90; x += 1e-2 {
+ if f := Erf(Erfinv(x)); !close(x, f) {
+ t.Errorf("Erf(Erfinv(%g)) = %g, want %g", x, f, x)
+ }
+ }
+ for x := -0.9; x <= 0.90; x += 1e-2 {
+ if f := Erfinv(Erf(x)); !close(x, f) {
+ t.Errorf("Erfinv(Erf(%g)) = %g, want %g", x, f, x)
+ }
+ }
+}
+
+func TestErfcinv(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := 1.0 - (vf[i] / 10)
+ if f := Erfcinv(a); !veryclose(erfinv[i], f) {
+ t.Errorf("Erfcinv(%g) = %g, want %g", a, f, erfinv[i])
+ }
+ }
+ for i := 0; i < len(vferfcinvSC); i++ {
+ if f := Erfcinv(vferfcinvSC[i]); !alike(erfcinvSC[i], f) {
+ t.Errorf("Erfcinv(%g) = %g, want %g", vferfcinvSC[i], f, erfcinvSC[i])
+ }
+ }
+ for x := 0.1; x <= 1.9; x += 1e-2 {
+ if f := Erfc(Erfcinv(x)); !close(x, f) {
+ t.Errorf("Erfc(Erfcinv(%g)) = %g, want %g", x, f, x)
+ }
+ }
+ for x := 0.1; x <= 1.9; x += 1e-2 {
+ if f := Erfcinv(Erfc(x)); !close(x, f) {
+ t.Errorf("Erfcinv(Erfc(%g)) = %g, want %g", x, f, x)
+ }
+ }
+}
+
+func TestExp(t *testing.T) {
+ testExp(t, Exp, "Exp")
+ testExp(t, ExpGo, "ExpGo")
+}
+
+func testExp(t *testing.T, Exp func(float64) float64, name string) {
+ for i := 0; i < len(vf); i++ {
+ if f := Exp(vf[i]); !veryclose(exp[i], f) {
+ t.Errorf("%s(%g) = %g, want %g", name, vf[i], f, exp[i])
+ }
+ }
+ for i := 0; i < len(vfexpSC); i++ {
+ if f := Exp(vfexpSC[i]); !alike(expSC[i], f) {
+ t.Errorf("%s(%g) = %g, want %g", name, vfexpSC[i], f, expSC[i])
+ }
+ }
+}
+
+func TestExpm1(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 100
+ if f := Expm1(a); !veryclose(expm1[i], f) {
+ t.Errorf("Expm1(%g) = %g, want %g", a, f, expm1[i])
+ }
+ }
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] * 10
+ if f := Expm1(a); !close(expm1Large[i], f) {
+ t.Errorf("Expm1(%g) = %g, want %g", a, f, expm1Large[i])
+ }
+ }
+ for i := 0; i < len(vfexpm1SC); i++ {
+ if f := Expm1(vfexpm1SC[i]); !alike(expm1SC[i], f) {
+ t.Errorf("Expm1(%g) = %g, want %g", vfexpm1SC[i], f, expm1SC[i])
+ }
+ }
+}
+
+func TestExp2(t *testing.T) {
+ testExp2(t, Exp2, "Exp2")
+ testExp2(t, Exp2Go, "Exp2Go")
+}
+
+func testExp2(t *testing.T, Exp2 func(float64) float64, name string) {
+ for i := 0; i < len(vf); i++ {
+ if f := Exp2(vf[i]); !close(exp2[i], f) {
+ t.Errorf("%s(%g) = %g, want %g", name, vf[i], f, exp2[i])
+ }
+ }
+ for i := 0; i < len(vfexp2SC); i++ {
+ if f := Exp2(vfexp2SC[i]); !alike(exp2SC[i], f) {
+ t.Errorf("%s(%g) = %g, want %g", name, vfexp2SC[i], f, exp2SC[i])
+ }
+ }
+ for n := -1074; n < 1024; n++ {
+ f := Exp2(float64(n))
+ vf := Ldexp(1, n)
+ if f != vf {
+ t.Errorf("%s(%d) = %g, want %g", name, n, f, vf)
+ }
+ }
+}
+
+func TestAbs(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Abs(vf[i]); fabs[i] != f {
+ t.Errorf("Abs(%g) = %g, want %g", vf[i], f, fabs[i])
+ }
+ }
+ for i := 0; i < len(vffabsSC); i++ {
+ if f := Abs(vffabsSC[i]); !alike(fabsSC[i], f) {
+ t.Errorf("Abs(%g) = %g, want %g", vffabsSC[i], f, fabsSC[i])
+ }
+ }
+}
+
+func TestDim(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Dim(vf[i], 0); fdim[i] != f {
+ t.Errorf("Dim(%g, %g) = %g, want %g", vf[i], 0.0, f, fdim[i])
+ }
+ }
+ for i := 0; i < len(vffdimSC); i++ {
+ if f := Dim(vffdimSC[i][0], vffdimSC[i][1]); !alike(fdimSC[i], f) {
+ t.Errorf("Dim(%g, %g) = %g, want %g", vffdimSC[i][0], vffdimSC[i][1], f, fdimSC[i])
+ }
+ }
+ for i := 0; i < len(vffdim2SC); i++ {
+ if f := Dim(vffdim2SC[i][0], vffdim2SC[i][1]); !alike(fdimSC[i], f) {
+ t.Errorf("Dim(%g, %g) = %g, want %g", vffdim2SC[i][0], vffdim2SC[i][1], f, fdimSC[i])
+ }
+ }
+}
+
+func TestFloor(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Floor(vf[i]); !alike(floor[i], f) {
+ t.Errorf("Floor(%g) = %g, want %g", vf[i], f, floor[i])
+ }
+ }
+ for i := 0; i < len(vfceilSC); i++ {
+ if f := Floor(vfceilSC[i]); !alike(ceilSC[i], f) {
+ t.Errorf("Floor(%g) = %g, want %g", vfceilSC[i], f, ceilSC[i])
+ }
+ }
+}
+
+func TestMax(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Max(vf[i], ceil[i]); ceil[i] != f {
+ t.Errorf("Max(%g, %g) = %g, want %g", vf[i], ceil[i], f, ceil[i])
+ }
+ }
+ for i := 0; i < len(vffdimSC); i++ {
+ if f := Max(vffdimSC[i][0], vffdimSC[i][1]); !alike(fmaxSC[i], f) {
+ t.Errorf("Max(%g, %g) = %g, want %g", vffdimSC[i][0], vffdimSC[i][1], f, fmaxSC[i])
+ }
+ }
+ for i := 0; i < len(vffdim2SC); i++ {
+ if f := Max(vffdim2SC[i][0], vffdim2SC[i][1]); !alike(fmaxSC[i], f) {
+ t.Errorf("Max(%g, %g) = %g, want %g", vffdim2SC[i][0], vffdim2SC[i][1], f, fmaxSC[i])
+ }
+ }
+}
+
+func TestMin(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Min(vf[i], floor[i]); floor[i] != f {
+ t.Errorf("Min(%g, %g) = %g, want %g", vf[i], floor[i], f, floor[i])
+ }
+ }
+ for i := 0; i < len(vffdimSC); i++ {
+ if f := Min(vffdimSC[i][0], vffdimSC[i][1]); !alike(fminSC[i], f) {
+ t.Errorf("Min(%g, %g) = %g, want %g", vffdimSC[i][0], vffdimSC[i][1], f, fminSC[i])
+ }
+ }
+ for i := 0; i < len(vffdim2SC); i++ {
+ if f := Min(vffdim2SC[i][0], vffdim2SC[i][1]); !alike(fminSC[i], f) {
+ t.Errorf("Min(%g, %g) = %g, want %g", vffdim2SC[i][0], vffdim2SC[i][1], f, fminSC[i])
+ }
+ }
+}
+
+func TestMod(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Mod(10, vf[i]); fmod[i] != f {
+ t.Errorf("Mod(10, %g) = %g, want %g", vf[i], f, fmod[i])
+ }
+ }
+ for i := 0; i < len(vffmodSC); i++ {
+ if f := Mod(vffmodSC[i][0], vffmodSC[i][1]); !alike(fmodSC[i], f) {
+ t.Errorf("Mod(%g, %g) = %g, want %g", vffmodSC[i][0], vffmodSC[i][1], f, fmodSC[i])
+ }
+ }
+ // verify precision of result for extreme inputs
+ if f := Mod(5.9790119248836734e+200, 1.1258465975523544); 0.6447968302508578 != f {
+ t.Errorf("Remainder(5.9790119248836734e+200, 1.1258465975523544) = %g, want 0.6447968302508578", f)
+ }
+}
+
+func TestFrexp(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f, j := Frexp(vf[i]); !veryclose(frexp[i].f, f) || frexp[i].i != j {
+ t.Errorf("Frexp(%g) = %g, %d, want %g, %d", vf[i], f, j, frexp[i].f, frexp[i].i)
+ }
+ }
+ for i := 0; i < len(vffrexpSC); i++ {
+ if f, j := Frexp(vffrexpSC[i]); !alike(frexpSC[i].f, f) || frexpSC[i].i != j {
+ t.Errorf("Frexp(%g) = %g, %d, want %g, %d", vffrexpSC[i], f, j, frexpSC[i].f, frexpSC[i].i)
+ }
+ }
+ for i := 0; i < len(vffrexpBC); i++ {
+ if f, j := Frexp(vffrexpBC[i]); !alike(frexpBC[i].f, f) || frexpBC[i].i != j {
+ t.Errorf("Frexp(%g) = %g, %d, want %g, %d", vffrexpBC[i], f, j, frexpBC[i].f, frexpBC[i].i)
+ }
+ }
+}
+
+func TestGamma(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Gamma(vf[i]); !close(gamma[i], f) {
+ t.Errorf("Gamma(%g) = %g, want %g", vf[i], f, gamma[i])
+ }
+ }
+ for _, g := range vfgamma {
+ f := Gamma(g[0])
+ var ok bool
+ if IsNaN(g[1]) || IsInf(g[1], 0) || g[1] == 0 || f == 0 {
+ ok = alike(g[1], f)
+ } else if g[0] > -50 && g[0] <= 171 {
+ ok = veryclose(g[1], f)
+ } else {
+ ok = close(g[1], f)
+ }
+ if !ok {
+ t.Errorf("Gamma(%g) = %g, want %g", g[0], f, g[1])
+ }
+ }
+}
+
+func TestHypot(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(1e200 * tanh[i] * Sqrt(2))
+ if f := Hypot(1e200*tanh[i], 1e200*tanh[i]); !veryclose(a, f) {
+ t.Errorf("Hypot(%g, %g) = %g, want %g", 1e200*tanh[i], 1e200*tanh[i], f, a)
+ }
+ }
+ for i := 0; i < len(vfhypotSC); i++ {
+ if f := Hypot(vfhypotSC[i][0], vfhypotSC[i][1]); !alike(hypotSC[i], f) {
+ t.Errorf("Hypot(%g, %g) = %g, want %g", vfhypotSC[i][0], vfhypotSC[i][1], f, hypotSC[i])
+ }
+ }
+}
+
+func TestHypotGo(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(1e200 * tanh[i] * Sqrt(2))
+ if f := HypotGo(1e200*tanh[i], 1e200*tanh[i]); !veryclose(a, f) {
+ t.Errorf("HypotGo(%g, %g) = %g, want %g", 1e200*tanh[i], 1e200*tanh[i], f, a)
+ }
+ }
+ for i := 0; i < len(vfhypotSC); i++ {
+ if f := HypotGo(vfhypotSC[i][0], vfhypotSC[i][1]); !alike(hypotSC[i], f) {
+ t.Errorf("HypotGo(%g, %g) = %g, want %g", vfhypotSC[i][0], vfhypotSC[i][1], f, hypotSC[i])
+ }
+ }
+}
+
+func TestIlogb(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := frexp[i].i - 1 // adjust because fr in the interval [½, 1)
+ if e := Ilogb(vf[i]); a != e {
+ t.Errorf("Ilogb(%g) = %d, want %d", vf[i], e, a)
+ }
+ }
+ for i := 0; i < len(vflogbSC); i++ {
+ if e := Ilogb(vflogbSC[i]); ilogbSC[i] != e {
+ t.Errorf("Ilogb(%g) = %d, want %d", vflogbSC[i], e, ilogbSC[i])
+ }
+ }
+ for i := 0; i < len(vffrexpBC); i++ {
+ if e := Ilogb(vffrexpBC[i]); int(logbBC[i]) != e {
+ t.Errorf("Ilogb(%g) = %d, want %d", vffrexpBC[i], e, int(logbBC[i]))
+ }
+ }
+}
+
+func TestJ0(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := J0(vf[i]); !soclose(j0[i], f, 4e-14) {
+ t.Errorf("J0(%g) = %g, want %g", vf[i], f, j0[i])
+ }
+ }
+ for i := 0; i < len(vfj0SC); i++ {
+ if f := J0(vfj0SC[i]); !alike(j0SC[i], f) {
+ t.Errorf("J0(%g) = %g, want %g", vfj0SC[i], f, j0SC[i])
+ }
+ }
+}
+
+func TestJ1(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := J1(vf[i]); !close(j1[i], f) {
+ t.Errorf("J1(%g) = %g, want %g", vf[i], f, j1[i])
+ }
+ }
+ for i := 0; i < len(vfj0SC); i++ {
+ if f := J1(vfj0SC[i]); !alike(j1SC[i], f) {
+ t.Errorf("J1(%g) = %g, want %g", vfj0SC[i], f, j1SC[i])
+ }
+ }
+}
+
+func TestJn(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Jn(2, vf[i]); !close(j2[i], f) {
+ t.Errorf("Jn(2, %g) = %g, want %g", vf[i], f, j2[i])
+ }
+ if f := Jn(-3, vf[i]); !close(jM3[i], f) {
+ t.Errorf("Jn(-3, %g) = %g, want %g", vf[i], f, jM3[i])
+ }
+ }
+ for i := 0; i < len(vfj0SC); i++ {
+ if f := Jn(2, vfj0SC[i]); !alike(j2SC[i], f) {
+ t.Errorf("Jn(2, %g) = %g, want %g", vfj0SC[i], f, j2SC[i])
+ }
+ if f := Jn(-3, vfj0SC[i]); !alike(jM3SC[i], f) {
+ t.Errorf("Jn(-3, %g) = %g, want %g", vfj0SC[i], f, jM3SC[i])
+ }
+ }
+}
+
+func TestLdexp(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Ldexp(frexp[i].f, frexp[i].i); !veryclose(vf[i], f) {
+ t.Errorf("Ldexp(%g, %d) = %g, want %g", frexp[i].f, frexp[i].i, f, vf[i])
+ }
+ }
+ for i := 0; i < len(vffrexpSC); i++ {
+ if f := Ldexp(frexpSC[i].f, frexpSC[i].i); !alike(vffrexpSC[i], f) {
+ t.Errorf("Ldexp(%g, %d) = %g, want %g", frexpSC[i].f, frexpSC[i].i, f, vffrexpSC[i])
+ }
+ }
+ for i := 0; i < len(vfldexpSC); i++ {
+ if f := Ldexp(vfldexpSC[i].f, vfldexpSC[i].i); !alike(ldexpSC[i], f) {
+ t.Errorf("Ldexp(%g, %d) = %g, want %g", vfldexpSC[i].f, vfldexpSC[i].i, f, ldexpSC[i])
+ }
+ }
+ for i := 0; i < len(vffrexpBC); i++ {
+ if f := Ldexp(frexpBC[i].f, frexpBC[i].i); !alike(vffrexpBC[i], f) {
+ t.Errorf("Ldexp(%g, %d) = %g, want %g", frexpBC[i].f, frexpBC[i].i, f, vffrexpBC[i])
+ }
+ }
+ for i := 0; i < len(vfldexpBC); i++ {
+ if f := Ldexp(vfldexpBC[i].f, vfldexpBC[i].i); !alike(ldexpBC[i], f) {
+ t.Errorf("Ldexp(%g, %d) = %g, want %g", vfldexpBC[i].f, vfldexpBC[i].i, f, ldexpBC[i])
+ }
+ }
+}
+
+func TestLgamma(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f, s := Lgamma(vf[i]); !close(lgamma[i].f, f) || lgamma[i].i != s {
+ t.Errorf("Lgamma(%g) = %g, %d, want %g, %d", vf[i], f, s, lgamma[i].f, lgamma[i].i)
+ }
+ }
+ for i := 0; i < len(vflgammaSC); i++ {
+ if f, s := Lgamma(vflgammaSC[i]); !alike(lgammaSC[i].f, f) || lgammaSC[i].i != s {
+ t.Errorf("Lgamma(%g) = %g, %d, want %g, %d", vflgammaSC[i], f, s, lgammaSC[i].f, lgammaSC[i].i)
+ }
+ }
+}
+
+func TestLog(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Log(a); log[i] != f {
+ t.Errorf("Log(%g) = %g, want %g", a, f, log[i])
+ }
+ }
+ if f := Log(10); f != Ln10 {
+ t.Errorf("Log(%g) = %g, want %g", 10.0, f, Ln10)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := Log(vflogSC[i]); !alike(logSC[i], f) {
+ t.Errorf("Log(%g) = %g, want %g", vflogSC[i], f, logSC[i])
+ }
+ }
+}
+
+func TestLogb(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Logb(vf[i]); logb[i] != f {
+ t.Errorf("Logb(%g) = %g, want %g", vf[i], f, logb[i])
+ }
+ }
+ for i := 0; i < len(vflogbSC); i++ {
+ if f := Logb(vflogbSC[i]); !alike(logbSC[i], f) {
+ t.Errorf("Logb(%g) = %g, want %g", vflogbSC[i], f, logbSC[i])
+ }
+ }
+ for i := 0; i < len(vffrexpBC); i++ {
+ if f := Logb(vffrexpBC[i]); !alike(logbBC[i], f) {
+ t.Errorf("Logb(%g) = %g, want %g", vffrexpBC[i], f, logbBC[i])
+ }
+ }
+}
+
+func TestLog10(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Log10(a); !veryclose(log10[i], f) {
+ t.Errorf("Log10(%g) = %g, want %g", a, f, log10[i])
+ }
+ }
+ if f := Log10(E); f != Log10E {
+ t.Errorf("Log10(%g) = %g, want %g", E, f, Log10E)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := Log10(vflogSC[i]); !alike(logSC[i], f) {
+ t.Errorf("Log10(%g) = %g, want %g", vflogSC[i], f, logSC[i])
+ }
+ }
+}
+
+func TestLog1p(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 100
+ if f := Log1p(a); !veryclose(log1p[i], f) {
+ t.Errorf("Log1p(%g) = %g, want %g", a, f, log1p[i])
+ }
+ }
+ a := 9.0
+ if f := Log1p(a); f != Ln10 {
+ t.Errorf("Log1p(%g) = %g, want %g", a, f, Ln10)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := Log1p(vflog1pSC[i]); !alike(log1pSC[i], f) {
+ t.Errorf("Log1p(%g) = %g, want %g", vflog1pSC[i], f, log1pSC[i])
+ }
+ }
+}
+
+func TestLog2(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Log2(a); !veryclose(log2[i], f) {
+ t.Errorf("Log2(%g) = %g, want %g", a, f, log2[i])
+ }
+ }
+ if f := Log2(E); f != Log2E {
+ t.Errorf("Log2(%g) = %g, want %g", E, f, Log2E)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := Log2(vflogSC[i]); !alike(logSC[i], f) {
+ t.Errorf("Log2(%g) = %g, want %g", vflogSC[i], f, logSC[i])
+ }
+ }
+ for i := -1074; i <= 1023; i++ {
+ f := Ldexp(1, i)
+ l := Log2(f)
+ if l != float64(i) {
+ t.Errorf("Log2(2**%d) = %g, want %d", i, l, i)
+ }
+ }
+}
+
+func TestModf(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f, g := Modf(vf[i]); !veryclose(modf[i][0], f) || !veryclose(modf[i][1], g) {
+ t.Errorf("Modf(%g) = %g, %g, want %g, %g", vf[i], f, g, modf[i][0], modf[i][1])
+ }
+ }
+ for i := 0; i < len(vfmodfSC); i++ {
+ if f, g := Modf(vfmodfSC[i]); !alike(modfSC[i][0], f) || !alike(modfSC[i][1], g) {
+ t.Errorf("Modf(%g) = %g, %g, want %g, %g", vfmodfSC[i], f, g, modfSC[i][0], modfSC[i][1])
+ }
+ }
+}
+
+func TestNextafter32(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ vfi := float32(vf[i])
+ if f := Nextafter32(vfi, 10); nextafter32[i] != f {
+ t.Errorf("Nextafter32(%g, %g) = %g want %g", vfi, 10.0, f, nextafter32[i])
+ }
+ }
+ for i := 0; i < len(vfnextafter32SC); i++ {
+ if f := Nextafter32(vfnextafter32SC[i][0], vfnextafter32SC[i][1]); !alike(float64(nextafter32SC[i]), float64(f)) {
+ t.Errorf("Nextafter32(%g, %g) = %g want %g", vfnextafter32SC[i][0], vfnextafter32SC[i][1], f, nextafter32SC[i])
+ }
+ }
+}
+
+func TestNextafter64(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Nextafter(vf[i], 10); nextafter64[i] != f {
+ t.Errorf("Nextafter64(%g, %g) = %g want %g", vf[i], 10.0, f, nextafter64[i])
+ }
+ }
+ for i := 0; i < len(vfnextafter64SC); i++ {
+ if f := Nextafter(vfnextafter64SC[i][0], vfnextafter64SC[i][1]); !alike(nextafter64SC[i], f) {
+ t.Errorf("Nextafter64(%g, %g) = %g want %g", vfnextafter64SC[i][0], vfnextafter64SC[i][1], f, nextafter64SC[i])
+ }
+ }
+}
+
+func TestPow(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Pow(10, vf[i]); !close(pow[i], f) {
+ t.Errorf("Pow(10, %g) = %g, want %g", vf[i], f, pow[i])
+ }
+ }
+ for i := 0; i < len(vfpowSC); i++ {
+ if f := Pow(vfpowSC[i][0], vfpowSC[i][1]); !alike(powSC[i], f) {
+ t.Errorf("Pow(%g, %g) = %g, want %g", vfpowSC[i][0], vfpowSC[i][1], f, powSC[i])
+ }
+ }
+}
+
+func TestPow10(t *testing.T) {
+ for i := 0; i < len(vfpow10SC); i++ {
+ if f := Pow10(vfpow10SC[i]); !alike(pow10SC[i], f) {
+ t.Errorf("Pow10(%d) = %g, want %g", vfpow10SC[i], f, pow10SC[i])
+ }
+ }
+}
+
+func TestRemainder(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Remainder(10, vf[i]); remainder[i] != f {
+ t.Errorf("Remainder(10, %g) = %g, want %g", vf[i], f, remainder[i])
+ }
+ }
+ for i := 0; i < len(vffmodSC); i++ {
+ if f := Remainder(vffmodSC[i][0], vffmodSC[i][1]); !alike(fmodSC[i], f) {
+ t.Errorf("Remainder(%g, %g) = %g, want %g", vffmodSC[i][0], vffmodSC[i][1], f, fmodSC[i])
+ }
+ }
+ // verify precision of result for extreme inputs
+ if f := Remainder(5.9790119248836734e+200, 1.1258465975523544); -0.4810497673014966 != f {
+ t.Errorf("Remainder(5.9790119248836734e+200, 1.1258465975523544) = %g, want -0.4810497673014966", f)
+ }
+ // verify that sign is correct when r == 0.
+ test := func(x, y float64) {
+ if r := Remainder(x, y); r == 0 && Signbit(r) != Signbit(x) {
+ t.Errorf("Remainder(x=%f, y=%f) = %f, sign of (zero) result should agree with sign of x", x, y, r)
+ }
+ }
+ for x := 0.0; x <= 3.0; x += 1 {
+ for y := 1.0; y <= 3.0; y += 1 {
+ test(x, y)
+ test(x, -y)
+ test(-x, y)
+ test(-x, -y)
+ }
+ }
+}
+
+func TestRound(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Round(vf[i]); !alike(round[i], f) {
+ t.Errorf("Round(%g) = %g, want %g", vf[i], f, round[i])
+ }
+ }
+ for i := 0; i < len(vfroundSC); i++ {
+ if f := Round(vfroundSC[i][0]); !alike(vfroundSC[i][1], f) {
+ t.Errorf("Round(%g) = %g, want %g", vfroundSC[i][0], f, vfroundSC[i][1])
+ }
+ }
+}
+
+func TestRoundToEven(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := RoundToEven(vf[i]); !alike(round[i], f) {
+ t.Errorf("RoundToEven(%g) = %g, want %g", vf[i], f, round[i])
+ }
+ }
+ for i := 0; i < len(vfroundEvenSC); i++ {
+ if f := RoundToEven(vfroundEvenSC[i][0]); !alike(vfroundEvenSC[i][1], f) {
+ t.Errorf("RoundToEven(%g) = %g, want %g", vfroundEvenSC[i][0], f, vfroundEvenSC[i][1])
+ }
+ }
+}
+
+func TestSignbit(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Signbit(vf[i]); signbit[i] != f {
+ t.Errorf("Signbit(%g) = %t, want %t", vf[i], f, signbit[i])
+ }
+ }
+ for i := 0; i < len(vfsignbitSC); i++ {
+ if f := Signbit(vfsignbitSC[i]); signbitSC[i] != f {
+ t.Errorf("Signbit(%g) = %t, want %t", vfsignbitSC[i], f, signbitSC[i])
+ }
+ }
+}
+func TestSin(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Sin(vf[i]); !veryclose(sin[i], f) {
+ t.Errorf("Sin(%g) = %g, want %g", vf[i], f, sin[i])
+ }
+ }
+ for i := 0; i < len(vfsinSC); i++ {
+ if f := Sin(vfsinSC[i]); !alike(sinSC[i], f) {
+ t.Errorf("Sin(%g) = %g, want %g", vfsinSC[i], f, sinSC[i])
+ }
+ }
+}
+
+func TestSincos(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if s, c := Sincos(vf[i]); !veryclose(sin[i], s) || !veryclose(cos[i], c) {
+ t.Errorf("Sincos(%g) = %g, %g want %g, %g", vf[i], s, c, sin[i], cos[i])
+ }
+ }
+}
+
+func TestSinh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Sinh(vf[i]); !close(sinh[i], f) {
+ t.Errorf("Sinh(%g) = %g, want %g", vf[i], f, sinh[i])
+ }
+ }
+ for i := 0; i < len(vfsinhSC); i++ {
+ if f := Sinh(vfsinhSC[i]); !alike(sinhSC[i], f) {
+ t.Errorf("Sinh(%g) = %g, want %g", vfsinhSC[i], f, sinhSC[i])
+ }
+ }
+}
+
+func TestSqrt(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := SqrtGo(a); sqrt[i] != f {
+ t.Errorf("SqrtGo(%g) = %g, want %g", a, f, sqrt[i])
+ }
+ a = Abs(vf[i])
+ if f := Sqrt(a); sqrt[i] != f {
+ t.Errorf("Sqrt(%g) = %g, want %g", a, f, sqrt[i])
+ }
+ }
+ for i := 0; i < len(vfsqrtSC); i++ {
+ if f := SqrtGo(vfsqrtSC[i]); !alike(sqrtSC[i], f) {
+ t.Errorf("SqrtGo(%g) = %g, want %g", vfsqrtSC[i], f, sqrtSC[i])
+ }
+ if f := Sqrt(vfsqrtSC[i]); !alike(sqrtSC[i], f) {
+ t.Errorf("Sqrt(%g) = %g, want %g", vfsqrtSC[i], f, sqrtSC[i])
+ }
+ }
+}
+
+func TestTan(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Tan(vf[i]); !veryclose(tan[i], f) {
+ t.Errorf("Tan(%g) = %g, want %g", vf[i], f, tan[i])
+ }
+ }
+ // same special cases as Sin
+ for i := 0; i < len(vfsinSC); i++ {
+ if f := Tan(vfsinSC[i]); !alike(sinSC[i], f) {
+ t.Errorf("Tan(%g) = %g, want %g", vfsinSC[i], f, sinSC[i])
+ }
+ }
+}
+
+func TestTanh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Tanh(vf[i]); !veryclose(tanh[i], f) {
+ t.Errorf("Tanh(%g) = %g, want %g", vf[i], f, tanh[i])
+ }
+ }
+ for i := 0; i < len(vftanhSC); i++ {
+ if f := Tanh(vftanhSC[i]); !alike(tanhSC[i], f) {
+ t.Errorf("Tanh(%g) = %g, want %g", vftanhSC[i], f, tanhSC[i])
+ }
+ }
+}
+
+func TestTrunc(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Trunc(vf[i]); !alike(trunc[i], f) {
+ t.Errorf("Trunc(%g) = %g, want %g", vf[i], f, trunc[i])
+ }
+ }
+ for i := 0; i < len(vfceilSC); i++ {
+ if f := Trunc(vfceilSC[i]); !alike(ceilSC[i], f) {
+ t.Errorf("Trunc(%g) = %g, want %g", vfceilSC[i], f, ceilSC[i])
+ }
+ }
+}
+
+func TestY0(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Y0(a); !close(y0[i], f) {
+ t.Errorf("Y0(%g) = %g, want %g", a, f, y0[i])
+ }
+ }
+ for i := 0; i < len(vfy0SC); i++ {
+ if f := Y0(vfy0SC[i]); !alike(y0SC[i], f) {
+ t.Errorf("Y0(%g) = %g, want %g", vfy0SC[i], f, y0SC[i])
+ }
+ }
+}
+
+func TestY1(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Y1(a); !soclose(y1[i], f, 2e-14) {
+ t.Errorf("Y1(%g) = %g, want %g", a, f, y1[i])
+ }
+ }
+ for i := 0; i < len(vfy0SC); i++ {
+ if f := Y1(vfy0SC[i]); !alike(y1SC[i], f) {
+ t.Errorf("Y1(%g) = %g, want %g", vfy0SC[i], f, y1SC[i])
+ }
+ }
+}
+
+func TestYn(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Yn(2, a); !close(y2[i], f) {
+ t.Errorf("Yn(2, %g) = %g, want %g", a, f, y2[i])
+ }
+ if f := Yn(-3, a); !close(yM3[i], f) {
+ t.Errorf("Yn(-3, %g) = %g, want %g", a, f, yM3[i])
+ }
+ }
+ for i := 0; i < len(vfy0SC); i++ {
+ if f := Yn(2, vfy0SC[i]); !alike(y2SC[i], f) {
+ t.Errorf("Yn(2, %g) = %g, want %g", vfy0SC[i], f, y2SC[i])
+ }
+ if f := Yn(-3, vfy0SC[i]); !alike(yM3SC[i], f) {
+ t.Errorf("Yn(-3, %g) = %g, want %g", vfy0SC[i], f, yM3SC[i])
+ }
+ }
+ if f := Yn(0, 0); !alike(Inf(-1), f) {
+ t.Errorf("Yn(0, 0) = %g, want %g", f, Inf(-1))
+ }
+}
+
+var PortableFMA = FMA // hide call from compiler intrinsic; falls back to portable code
+
+func TestFMA(t *testing.T) {
+ for _, c := range fmaC {
+ got := FMA(c.x, c.y, c.z)
+ if !alike(got, c.want) {
+ t.Errorf("FMA(%g,%g,%g) == %g; want %g", c.x, c.y, c.z, got, c.want)
+ }
+ got = PortableFMA(c.x, c.y, c.z)
+ if !alike(got, c.want) {
+ t.Errorf("PortableFMA(%g,%g,%g) == %g; want %g", c.x, c.y, c.z, got, c.want)
+ }
+ }
+}
+
+// Check that math functions of high angle values
+// return accurate results. [Since (vf[i] + large) - large != vf[i],
+// testing for Trig(vf[i] + large) == Trig(vf[i]), where large is
+// a multiple of 2*Pi, is misleading.]
+func TestLargeCos(t *testing.T) {
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1 := cosLarge[i]
+ f2 := Cos(vf[i] + large)
+ if !close(f1, f2) {
+ t.Errorf("Cos(%g) = %g, want %g", vf[i]+large, f2, f1)
+ }
+ }
+}
+
+func TestLargeSin(t *testing.T) {
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1 := sinLarge[i]
+ f2 := Sin(vf[i] + large)
+ if !close(f1, f2) {
+ t.Errorf("Sin(%g) = %g, want %g", vf[i]+large, f2, f1)
+ }
+ }
+}
+
+func TestLargeSincos(t *testing.T) {
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1, g1 := sinLarge[i], cosLarge[i]
+ f2, g2 := Sincos(vf[i] + large)
+ if !close(f1, f2) || !close(g1, g2) {
+ t.Errorf("Sincos(%g) = %g, %g, want %g, %g", vf[i]+large, f2, g2, f1, g1)
+ }
+ }
+}
+
+func TestLargeTan(t *testing.T) {
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1 := tanLarge[i]
+ f2 := Tan(vf[i] + large)
+ if !close(f1, f2) {
+ t.Errorf("Tan(%g) = %g, want %g", vf[i]+large, f2, f1)
+ }
+ }
+}
+
+// Check that trigReduce matches the standard reduction results for input values
+// below reduceThreshold.
+func TestTrigReduce(t *testing.T) {
+ inputs := make([]float64, len(vf))
+ // all of the standard inputs
+ copy(inputs, vf)
+ // all of the large inputs
+ large := float64(100000 * Pi)
+ for _, v := range vf {
+ inputs = append(inputs, v+large)
+ }
+ // Also test some special inputs, Pi and right below the reduceThreshold
+ inputs = append(inputs, Pi, Nextafter(ReduceThreshold, 0))
+ for _, x := range inputs {
+ // reduce the value to compare
+ j, z := TrigReduce(x)
+ xred := float64(j)*(Pi/4) + z
+
+ if f, fred := Sin(x), Sin(xred); !close(f, fred) {
+ t.Errorf("Sin(trigReduce(%g)) != Sin(%g), got %g, want %g", x, x, fred, f)
+ }
+ if f, fred := Cos(x), Cos(xred); !close(f, fred) {
+ t.Errorf("Cos(trigReduce(%g)) != Cos(%g), got %g, want %g", x, x, fred, f)
+ }
+ if f, fred := Tan(x), Tan(xred); !close(f, fred) {
+ t.Errorf(" Tan(trigReduce(%g)) != Tan(%g), got %g, want %g", x, x, fred, f)
+ }
+ f, g := Sincos(x)
+ fred, gred := Sincos(xred)
+ if !close(f, fred) || !close(g, gred) {
+ t.Errorf(" Sincos(trigReduce(%g)) != Sincos(%g), got %g, %g, want %g, %g", x, x, fred, gred, f, g)
+ }
+ }
+}
+
+// Check that math constants are accepted by compiler
+// and have right value (assumes strconv.ParseFloat works).
+// https://golang.org/issue/201
+
+type floatTest struct {
+ val interface{}
+ name string
+ str string
+}
+
+var floatTests = []floatTest{
+ {float64(MaxFloat64), "MaxFloat64", "1.7976931348623157e+308"},
+ {float64(SmallestNonzeroFloat64), "SmallestNonzeroFloat64", "5e-324"},
+ {float32(MaxFloat32), "MaxFloat32", "3.4028235e+38"},
+ {float32(SmallestNonzeroFloat32), "SmallestNonzeroFloat32", "1e-45"},
+}
+
+func TestFloatMinMax(t *testing.T) {
+ for _, tt := range floatTests {
+ s := fmt.Sprint(tt.val)
+ if s != tt.str {
+ t.Errorf("Sprint(%v) = %s, want %s", tt.name, s, tt.str)
+ }
+ }
+}
+
+// Benchmarks
+
+// Global exported variables are used to store the
+// return values of functions measured in the benchmarks.
+// Storing the results in these variables prevents the compiler
+// from completely optimizing the benchmarked functions away.
+var (
+ GlobalI int
+ GlobalB bool
+ GlobalF float64
+)
+
+func BenchmarkAcos(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Acos(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkAcosh(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Acosh(1.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkAsin(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Asin(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkAsinh(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Asinh(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkAtan(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Atan(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkAtanh(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Atanh(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkAtan2(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Atan2(.5, 1)
+ }
+ GlobalF = x
+}
+
+func BenchmarkCbrt(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Cbrt(10)
+ }
+ GlobalF = x
+}
+
+func BenchmarkCeil(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Ceil(.5)
+ }
+ GlobalF = x
+}
+
+var copysignNeg = -1.0
+
+func BenchmarkCopysign(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Copysign(.5, copysignNeg)
+ }
+ GlobalF = x
+}
+
+func BenchmarkCos(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Cos(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkCosh(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Cosh(2.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkErf(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Erf(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkErfc(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Erfc(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkErfinv(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Erfinv(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkErfcinv(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Erfcinv(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkExp(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Exp(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkExpGo(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = ExpGo(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkExpm1(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Expm1(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkExp2(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Exp2(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkExp2Go(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Exp2Go(.5)
+ }
+ GlobalF = x
+}
+
+var absPos = .5
+
+func BenchmarkAbs(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Abs(absPos)
+ }
+ GlobalF = x
+
+}
+
+func BenchmarkDim(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Dim(GlobalF, x)
+ }
+ GlobalF = x
+}
+
+func BenchmarkFloor(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Floor(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkMax(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Max(10, 3)
+ }
+ GlobalF = x
+}
+
+func BenchmarkMin(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Min(10, 3)
+ }
+ GlobalF = x
+}
+
+func BenchmarkMod(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Mod(10, 3)
+ }
+ GlobalF = x
+}
+
+func BenchmarkFrexp(b *testing.B) {
+ x := 0.0
+ y := 0
+ for i := 0; i < b.N; i++ {
+ x, y = Frexp(8)
+ }
+ GlobalF = x
+ GlobalI = y
+}
+
+func BenchmarkGamma(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Gamma(2.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkHypot(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Hypot(3, 4)
+ }
+ GlobalF = x
+}
+
+func BenchmarkHypotGo(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = HypotGo(3, 4)
+ }
+ GlobalF = x
+}
+
+func BenchmarkIlogb(b *testing.B) {
+ x := 0
+ for i := 0; i < b.N; i++ {
+ x = Ilogb(.5)
+ }
+ GlobalI = x
+}
+
+func BenchmarkJ0(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = J0(2.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkJ1(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = J1(2.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkJn(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Jn(2, 2.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkLdexp(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Ldexp(.5, 2)
+ }
+ GlobalF = x
+}
+
+func BenchmarkLgamma(b *testing.B) {
+ x := 0.0
+ y := 0
+ for i := 0; i < b.N; i++ {
+ x, y = Lgamma(2.5)
+ }
+ GlobalF = x
+ GlobalI = y
+}
+
+func BenchmarkLog(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Log(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkLogb(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Logb(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkLog1p(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Log1p(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkLog10(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Log10(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkLog2(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Log2(.5)
+ }
+ GlobalF += x
+}
+
+func BenchmarkModf(b *testing.B) {
+ x := 0.0
+ y := 0.0
+ for i := 0; i < b.N; i++ {
+ x, y = Modf(1.5)
+ }
+ GlobalF += x
+ GlobalF += y
+}
+
+func BenchmarkNextafter32(b *testing.B) {
+ x := float32(0.0)
+ for i := 0; i < b.N; i++ {
+ x = Nextafter32(.5, 1)
+ }
+ GlobalF = float64(x)
+}
+
+func BenchmarkNextafter64(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Nextafter(.5, 1)
+ }
+ GlobalF = x
+}
+
+func BenchmarkPowInt(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Pow(2, 2)
+ }
+ GlobalF = x
+}
+
+func BenchmarkPowFrac(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Pow(2.5, 1.5)
+ }
+ GlobalF = x
+}
+
+var pow10pos = int(300)
+
+func BenchmarkPow10Pos(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Pow10(pow10pos)
+ }
+ GlobalF = x
+}
+
+var pow10neg = int(-300)
+
+func BenchmarkPow10Neg(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Pow10(pow10neg)
+ }
+ GlobalF = x
+}
+
+var roundNeg = float64(-2.5)
+
+func BenchmarkRound(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Round(roundNeg)
+ }
+ GlobalF = x
+}
+
+func BenchmarkRoundToEven(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = RoundToEven(roundNeg)
+ }
+ GlobalF = x
+}
+
+func BenchmarkRemainder(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Remainder(10, 3)
+ }
+ GlobalF = x
+}
+
+var signbitPos = 2.5
+
+func BenchmarkSignbit(b *testing.B) {
+ x := false
+ for i := 0; i < b.N; i++ {
+ x = Signbit(signbitPos)
+ }
+ GlobalB = x
+}
+
+func BenchmarkSin(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Sin(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkSincos(b *testing.B) {
+ x := 0.0
+ y := 0.0
+ for i := 0; i < b.N; i++ {
+ x, y = Sincos(.5)
+ }
+ GlobalF += x
+ GlobalF += y
+}
+
+func BenchmarkSinh(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Sinh(2.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkSqrtIndirect(b *testing.B) {
+ x, y := 0.0, 10.0
+ f := Sqrt
+ for i := 0; i < b.N; i++ {
+ x += f(y)
+ }
+ GlobalF = x
+}
+
+func BenchmarkSqrtLatency(b *testing.B) {
+ x := 10.0
+ for i := 0; i < b.N; i++ {
+ x = Sqrt(x)
+ }
+ GlobalF = x
+}
+
+func BenchmarkSqrtIndirectLatency(b *testing.B) {
+ x := 10.0
+ f := Sqrt
+ for i := 0; i < b.N; i++ {
+ x = f(x)
+ }
+ GlobalF = x
+}
+
+func BenchmarkSqrtGoLatency(b *testing.B) {
+ x := 10.0
+ for i := 0; i < b.N; i++ {
+ x = SqrtGo(x)
+ }
+ GlobalF = x
+}
+
+func isPrime(i int) bool {
+ // Yes, this is a dumb way to write this code,
+ // but calling Sqrt repeatedly in this way demonstrates
+ // the benefit of using a direct SQRT instruction on systems
+ // that have one, whereas the obvious loop seems not to
+ // demonstrate such a benefit.
+ for j := 2; float64(j) <= Sqrt(float64(i)); j++ {
+ if i%j == 0 {
+ return false
+ }
+ }
+ return true
+}
+
+func BenchmarkSqrtPrime(b *testing.B) {
+ x := false
+ for i := 0; i < b.N; i++ {
+ x = isPrime(100003)
+ }
+ GlobalB = x
+}
+
+func BenchmarkTan(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Tan(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkTanh(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Tanh(2.5)
+ }
+ GlobalF = x
+}
+func BenchmarkTrunc(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Trunc(.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkY0(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Y0(2.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkY1(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Y1(2.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkYn(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Yn(2, 2.5)
+ }
+ GlobalF = x
+}
+
+func BenchmarkFloat64bits(b *testing.B) {
+ y := uint64(0)
+ for i := 0; i < b.N; i++ {
+ y = Float64bits(roundNeg)
+ }
+ GlobalI = int(y)
+}
+
+var roundUint64 = uint64(5)
+
+func BenchmarkFloat64frombits(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = Float64frombits(roundUint64)
+ }
+ GlobalF = x
+}
+
+var roundFloat32 = float32(-2.5)
+
+func BenchmarkFloat32bits(b *testing.B) {
+ y := uint32(0)
+ for i := 0; i < b.N; i++ {
+ y = Float32bits(roundFloat32)
+ }
+ GlobalI = int(y)
+}
+
+var roundUint32 = uint32(5)
+
+func BenchmarkFloat32frombits(b *testing.B) {
+ x := float32(0.0)
+ for i := 0; i < b.N; i++ {
+ x = Float32frombits(roundUint32)
+ }
+ GlobalF = float64(x)
+}
+
+func BenchmarkFMA(b *testing.B) {
+ x := 0.0
+ for i := 0; i < b.N; i++ {
+ x = FMA(E, Pi, x)
+ }
+ GlobalF = x
+}
diff --git a/src/math/arith_s390x.go b/src/math/arith_s390x.go
new file mode 100644
index 0000000..90a7d4f
--- /dev/null
+++ b/src/math/arith_s390x.go
@@ -0,0 +1,77 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+import "internal/cpu"
+
+func log10TrampolineSetup(x float64) float64
+func log10Asm(x float64) float64
+
+func cosTrampolineSetup(x float64) float64
+func cosAsm(x float64) float64
+
+func coshTrampolineSetup(x float64) float64
+func coshAsm(x float64) float64
+
+func sinTrampolineSetup(x float64) float64
+func sinAsm(x float64) float64
+
+func sinhTrampolineSetup(x float64) float64
+func sinhAsm(x float64) float64
+
+func tanhTrampolineSetup(x float64) float64
+func tanhAsm(x float64) float64
+
+func log1pTrampolineSetup(x float64) float64
+func log1pAsm(x float64) float64
+
+func atanhTrampolineSetup(x float64) float64
+func atanhAsm(x float64) float64
+
+func acosTrampolineSetup(x float64) float64
+func acosAsm(x float64) float64
+
+func acoshTrampolineSetup(x float64) float64
+func acoshAsm(x float64) float64
+
+func asinTrampolineSetup(x float64) float64
+func asinAsm(x float64) float64
+
+func asinhTrampolineSetup(x float64) float64
+func asinhAsm(x float64) float64
+
+func erfTrampolineSetup(x float64) float64
+func erfAsm(x float64) float64
+
+func erfcTrampolineSetup(x float64) float64
+func erfcAsm(x float64) float64
+
+func atanTrampolineSetup(x float64) float64
+func atanAsm(x float64) float64
+
+func atan2TrampolineSetup(x, y float64) float64
+func atan2Asm(x, y float64) float64
+
+func cbrtTrampolineSetup(x float64) float64
+func cbrtAsm(x float64) float64
+
+func logTrampolineSetup(x float64) float64
+func logAsm(x float64) float64
+
+func tanTrampolineSetup(x float64) float64
+func tanAsm(x float64) float64
+
+func expTrampolineSetup(x float64) float64
+func expAsm(x float64) float64
+
+func expm1TrampolineSetup(x float64) float64
+func expm1Asm(x float64) float64
+
+func powTrampolineSetup(x, y float64) float64
+func powAsm(x, y float64) float64
+
+// hasVX reports whether the machine has the z/Architecture
+// vector facility installed and enabled.
+var hasVX = cpu.S390X.HasVX
diff --git a/src/math/arith_s390x_test.go b/src/math/arith_s390x_test.go
new file mode 100644
index 0000000..cfbc7b7
--- /dev/null
+++ b/src/math/arith_s390x_test.go
@@ -0,0 +1,442 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Tests whether the non vector routines are working, even when the tests are run on a
+// vector-capable machine.
+package math_test
+
+import (
+ . "math"
+ "testing"
+)
+
+func TestCosNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := CosNoVec(vf[i]); !veryclose(cos[i], f) {
+ t.Errorf("Cos(%g) = %g, want %g", vf[i], f, cos[i])
+ }
+ }
+ for i := 0; i < len(vfcosSC); i++ {
+ if f := CosNoVec(vfcosSC[i]); !alike(cosSC[i], f) {
+ t.Errorf("Cos(%g) = %g, want %g", vfcosSC[i], f, cosSC[i])
+ }
+ }
+}
+
+func TestCoshNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := CoshNoVec(vf[i]); !close(cosh[i], f) {
+ t.Errorf("Cosh(%g) = %g, want %g", vf[i], f, cosh[i])
+ }
+ }
+ for i := 0; i < len(vfcoshSC); i++ {
+ if f := CoshNoVec(vfcoshSC[i]); !alike(coshSC[i], f) {
+ t.Errorf("Cosh(%g) = %g, want %g", vfcoshSC[i], f, coshSC[i])
+ }
+ }
+}
+func TestSinNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := SinNoVec(vf[i]); !veryclose(sin[i], f) {
+ t.Errorf("Sin(%g) = %g, want %g", vf[i], f, sin[i])
+ }
+ }
+ for i := 0; i < len(vfsinSC); i++ {
+ if f := SinNoVec(vfsinSC[i]); !alike(sinSC[i], f) {
+ t.Errorf("Sin(%g) = %g, want %g", vfsinSC[i], f, sinSC[i])
+ }
+ }
+}
+
+func TestSinhNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := SinhNoVec(vf[i]); !close(sinh[i], f) {
+ t.Errorf("Sinh(%g) = %g, want %g", vf[i], f, sinh[i])
+ }
+ }
+ for i := 0; i < len(vfsinhSC); i++ {
+ if f := SinhNoVec(vfsinhSC[i]); !alike(sinhSC[i], f) {
+ t.Errorf("Sinh(%g) = %g, want %g", vfsinhSC[i], f, sinhSC[i])
+ }
+ }
+}
+
+// Check that math functions of high angle values
+// return accurate results. [Since (vf[i] + large) - large != vf[i],
+// testing for Trig(vf[i] + large) == Trig(vf[i]), where large is
+// a multiple of 2*Pi, is misleading.]
+func TestLargeCosNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1 := cosLarge[i]
+ f2 := CosNoVec(vf[i] + large)
+ if !close(f1, f2) {
+ t.Errorf("Cos(%g) = %g, want %g", vf[i]+large, f2, f1)
+ }
+ }
+}
+
+func TestLargeSinNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1 := sinLarge[i]
+ f2 := SinNoVec(vf[i] + large)
+ if !close(f1, f2) {
+ t.Errorf("Sin(%g) = %g, want %g", vf[i]+large, f2, f1)
+ }
+ }
+}
+
+func TestLargeTanNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1 := tanLarge[i]
+ f2 := TanNovec(vf[i] + large)
+ if !close(f1, f2) {
+ t.Errorf("Tan(%g) = %g, want %g", vf[i]+large, f2, f1)
+ }
+ }
+}
+
+func TestTanNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := TanNovec(vf[i]); !veryclose(tan[i], f) {
+ t.Errorf("Tan(%g) = %g, want %g", vf[i], f, tan[i])
+ }
+ }
+ // same special cases as Sin
+ for i := 0; i < len(vfsinSC); i++ {
+ if f := TanNovec(vfsinSC[i]); !alike(sinSC[i], f) {
+ t.Errorf("Tan(%g) = %g, want %g", vfsinSC[i], f, sinSC[i])
+ }
+ }
+}
+
+func TestTanhNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := TanhNoVec(vf[i]); !veryclose(tanh[i], f) {
+ t.Errorf("Tanh(%g) = %g, want %g", vf[i], f, tanh[i])
+ }
+ }
+ for i := 0; i < len(vftanhSC); i++ {
+ if f := TanhNoVec(vftanhSC[i]); !alike(tanhSC[i], f) {
+ t.Errorf("Tanh(%g) = %g, want %g", vftanhSC[i], f, tanhSC[i])
+ }
+ }
+
+}
+
+func TestLog10Novec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Log10NoVec(a); !veryclose(log10[i], f) {
+ t.Errorf("Log10(%g) = %g, want %g", a, f, log10[i])
+ }
+ }
+ if f := Log10NoVec(E); f != Log10E {
+ t.Errorf("Log10(%g) = %g, want %g", E, f, Log10E)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := Log10NoVec(vflogSC[i]); !alike(logSC[i], f) {
+ t.Errorf("Log10(%g) = %g, want %g", vflogSC[i], f, logSC[i])
+ }
+ }
+}
+
+func TestLog1pNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 100
+ if f := Log1pNovec(a); !veryclose(log1p[i], f) {
+ t.Errorf("Log1p(%g) = %g, want %g", a, f, log1p[i])
+ }
+ }
+ a := 9.0
+ if f := Log1pNovec(a); f != Ln10 {
+ t.Errorf("Log1p(%g) = %g, want %g", a, f, Ln10)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := Log1pNovec(vflog1pSC[i]); !alike(log1pSC[i], f) {
+ t.Errorf("Log1p(%g) = %g, want %g", vflog1pSC[i], f, log1pSC[i])
+ }
+ }
+}
+
+func TestAtanhNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := AtanhNovec(a); !veryclose(atanh[i], f) {
+ t.Errorf("Atanh(%g) = %g, want %g", a, f, atanh[i])
+ }
+ }
+ for i := 0; i < len(vfatanhSC); i++ {
+ if f := AtanhNovec(vfatanhSC[i]); !alike(atanhSC[i], f) {
+ t.Errorf("Atanh(%g) = %g, want %g", vfatanhSC[i], f, atanhSC[i])
+ }
+ }
+}
+
+func TestAcosNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := AcosNovec(a); !close(acos[i], f) {
+ t.Errorf("Acos(%g) = %g, want %g", a, f, acos[i])
+ }
+ }
+ for i := 0; i < len(vfacosSC); i++ {
+ if f := AcosNovec(vfacosSC[i]); !alike(acosSC[i], f) {
+ t.Errorf("Acos(%g) = %g, want %g", vfacosSC[i], f, acosSC[i])
+ }
+ }
+}
+
+func TestAsinNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := AsinNovec(a); !veryclose(asin[i], f) {
+ t.Errorf("Asin(%g) = %g, want %g", a, f, asin[i])
+ }
+ }
+ for i := 0; i < len(vfasinSC); i++ {
+ if f := AsinNovec(vfasinSC[i]); !alike(asinSC[i], f) {
+ t.Errorf("Asin(%g) = %g, want %g", vfasinSC[i], f, asinSC[i])
+ }
+ }
+}
+
+func TestAcoshNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ a := 1 + Abs(vf[i])
+ if f := AcoshNovec(a); !veryclose(acosh[i], f) {
+ t.Errorf("Acosh(%g) = %g, want %g", a, f, acosh[i])
+ }
+ }
+ for i := 0; i < len(vfacoshSC); i++ {
+ if f := AcoshNovec(vfacoshSC[i]); !alike(acoshSC[i], f) {
+ t.Errorf("Acosh(%g) = %g, want %g", vfacoshSC[i], f, acoshSC[i])
+ }
+ }
+}
+
+func TestAsinhNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := AsinhNovec(vf[i]); !veryclose(asinh[i], f) {
+ t.Errorf("Asinh(%g) = %g, want %g", vf[i], f, asinh[i])
+ }
+ }
+ for i := 0; i < len(vfasinhSC); i++ {
+ if f := AsinhNovec(vfasinhSC[i]); !alike(asinhSC[i], f) {
+ t.Errorf("Asinh(%g) = %g, want %g", vfasinhSC[i], f, asinhSC[i])
+ }
+ }
+}
+
+func TestErfNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := ErfNovec(a); !veryclose(erf[i], f) {
+ t.Errorf("Erf(%g) = %g, want %g", a, f, erf[i])
+ }
+ }
+ for i := 0; i < len(vferfSC); i++ {
+ if f := ErfNovec(vferfSC[i]); !alike(erfSC[i], f) {
+ t.Errorf("Erf(%g) = %g, want %g", vferfSC[i], f, erfSC[i])
+ }
+ }
+}
+
+func TestErfcNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := ErfcNovec(a); !veryclose(erfc[i], f) {
+ t.Errorf("Erfc(%g) = %g, want %g", a, f, erfc[i])
+ }
+ }
+ for i := 0; i < len(vferfcSC); i++ {
+ if f := ErfcNovec(vferfcSC[i]); !alike(erfcSC[i], f) {
+ t.Errorf("Erfc(%g) = %g, want %g", vferfcSC[i], f, erfcSC[i])
+ }
+ }
+}
+
+func TestAtanNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := AtanNovec(vf[i]); !veryclose(atan[i], f) {
+ t.Errorf("Atan(%g) = %g, want %g", vf[i], f, atan[i])
+ }
+ }
+ for i := 0; i < len(vfatanSC); i++ {
+ if f := AtanNovec(vfatanSC[i]); !alike(atanSC[i], f) {
+ t.Errorf("Atan(%g) = %g, want %g", vfatanSC[i], f, atanSC[i])
+ }
+ }
+}
+
+func TestAtan2Novec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := Atan2Novec(10, vf[i]); !veryclose(atan2[i], f) {
+ t.Errorf("Atan2(10, %g) = %g, want %g", vf[i], f, atan2[i])
+ }
+ }
+ for i := 0; i < len(vfatan2SC); i++ {
+ if f := Atan2Novec(vfatan2SC[i][0], vfatan2SC[i][1]); !alike(atan2SC[i], f) {
+ t.Errorf("Atan2(%g, %g) = %g, want %g", vfatan2SC[i][0], vfatan2SC[i][1], f, atan2SC[i])
+ }
+ }
+}
+
+func TestCbrtNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := CbrtNovec(vf[i]); !veryclose(cbrt[i], f) {
+ t.Errorf("Cbrt(%g) = %g, want %g", vf[i], f, cbrt[i])
+ }
+ }
+ for i := 0; i < len(vfcbrtSC); i++ {
+ if f := CbrtNovec(vfcbrtSC[i]); !alike(cbrtSC[i], f) {
+ t.Errorf("Cbrt(%g) = %g, want %g", vfcbrtSC[i], f, cbrtSC[i])
+ }
+ }
+}
+
+func TestLogNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := LogNovec(a); log[i] != f {
+ t.Errorf("Log(%g) = %g, want %g", a, f, log[i])
+ }
+ }
+ if f := LogNovec(10); f != Ln10 {
+ t.Errorf("Log(%g) = %g, want %g", 10.0, f, Ln10)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := LogNovec(vflogSC[i]); !alike(logSC[i], f) {
+ t.Errorf("Log(%g) = %g, want %g", vflogSC[i], f, logSC[i])
+ }
+ }
+}
+
+func TestExpNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ testExpNovec(t, Exp, "Exp")
+ testExpNovec(t, ExpGo, "ExpGo")
+}
+
+func testExpNovec(t *testing.T, Exp func(float64) float64, name string) {
+ for i := 0; i < len(vf); i++ {
+ if f := ExpNovec(vf[i]); !veryclose(exp[i], f) {
+ t.Errorf("%s(%g) = %g, want %g", name, vf[i], f, exp[i])
+ }
+ }
+ for i := 0; i < len(vfexpSC); i++ {
+ if f := ExpNovec(vfexpSC[i]); !alike(expSC[i], f) {
+ t.Errorf("%s(%g) = %g, want %g", name, vfexpSC[i], f, expSC[i])
+ }
+ }
+}
+
+func TestExpm1Novec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 100
+ if f := Expm1Novec(a); !veryclose(expm1[i], f) {
+ t.Errorf("Expm1(%g) = %g, want %g", a, f, expm1[i])
+ }
+ }
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] * 10
+ if f := Expm1Novec(a); !close(expm1Large[i], f) {
+ t.Errorf("Expm1(%g) = %g, want %g", a, f, expm1Large[i])
+ }
+ }
+ for i := 0; i < len(vfexpm1SC); i++ {
+ if f := Expm1Novec(vfexpm1SC[i]); !alike(expm1SC[i], f) {
+ t.Errorf("Expm1(%g) = %g, want %g", vfexpm1SC[i], f, expm1SC[i])
+ }
+ }
+}
+
+func TestPowNovec(t *testing.T) {
+ if !HasVX {
+ t.Skipf("no vector support")
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := PowNovec(10, vf[i]); !close(pow[i], f) {
+ t.Errorf("Pow(10, %g) = %g, want %g", vf[i], f, pow[i])
+ }
+ }
+ for i := 0; i < len(vfpowSC); i++ {
+ if f := PowNovec(vfpowSC[i][0], vfpowSC[i][1]); !alike(powSC[i], f) {
+ t.Errorf("Pow(%g, %g) = %g, want %g", vfpowSC[i][0], vfpowSC[i][1], f, powSC[i])
+ }
+ }
+}
diff --git a/src/math/asin.go b/src/math/asin.go
new file mode 100644
index 0000000..88b851e
--- /dev/null
+++ b/src/math/asin.go
@@ -0,0 +1,55 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point arcsine and arccosine.
+
+ They are implemented by computing the arctangent
+ after appropriate range reduction.
+*/
+
+// Asin returns the arcsine, in radians, of x.
+//
+// Special cases are:
+// Asin(±0) = ±0
+// Asin(x) = NaN if x < -1 or x > 1
+func Asin(x float64) float64
+
+func asin(x float64) float64 {
+ if x == 0 {
+ return x // special case
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ if x > 1 {
+ return NaN() // special case
+ }
+
+ temp := Sqrt(1 - x*x)
+ if x > 0.7 {
+ temp = Pi/2 - satan(temp/x)
+ } else {
+ temp = satan(x / temp)
+ }
+
+ if sign {
+ temp = -temp
+ }
+ return temp
+}
+
+// Acos returns the arccosine, in radians, of x.
+//
+// Special case is:
+// Acos(x) = NaN if x < -1 or x > 1
+func Acos(x float64) float64
+
+func acos(x float64) float64 {
+ return Pi/2 - Asin(x)
+}
diff --git a/src/math/asin_386.s b/src/math/asin_386.s
new file mode 100644
index 0000000..7dab390
--- /dev/null
+++ b/src/math/asin_386.s
@@ -0,0 +1,30 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Asin(x float64) float64
+TEXT ·Asin(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=sin(x)
+ FMOVD F0, F1 // F0=sin(x), F1=sin(x)
+ FMULD F0, F0 // F0=sin(x)*sin(x), F1=sin(x)
+ FLD1 // F0=1, F1=sin(x)*sin(x), F2=sin(x)
+ FSUBRDP F0, F1 // F0=1-sin(x)*sin(x) (=cos(x)*cos(x)), F1=sin(x)
+ FSQRT // F0=cos(x), F1=sin(x)
+ FPATAN // F0=arcsin(sin(x))=x
+ FMOVDP F0, ret+8(FP)
+ RET
+
+// func Acos(x float64) float64
+TEXT ·Acos(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=cos(x)
+ FMOVD F0, F1 // F0=cos(x), F1=cos(x)
+ FMULD F0, F0 // F0=cos(x)*cos(x), F1=cos(x)
+ FLD1 // F0=1, F1=cos(x)*cos(x), F2=cos(x)
+ FSUBRDP F0, F1 // F0=1-cos(x)*cos(x) (=sin(x)*sin(x)), F1=cos(x)
+ FSQRT // F0=sin(x), F1=cos(x)
+ FXCHD F0, F1 // F0=cos(x), F1=sin(x)
+ FPATAN // F0=arccos(cos(x))=x
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/asin_s390x.s b/src/math/asin_s390x.s
new file mode 100644
index 0000000..dc54d05
--- /dev/null
+++ b/src/math/asin_s390x.s
@@ -0,0 +1,162 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial coefficients and other constants
+DATA ·asinrodataL15<> + 0(SB)/8, $-1.309611320495605469
+DATA ·asinrodataL15<> + 8(SB)/8, $0x3ff921fb54442d18
+DATA ·asinrodataL15<> + 16(SB)/8, $0xbff921fb54442d18
+DATA ·asinrodataL15<> + 24(SB)/8, $1.309611320495605469
+DATA ·asinrodataL15<> + 32(SB)/8, $-0.0
+DATA ·asinrodataL15<> + 40(SB)/8, $1.199437040755305217
+DATA ·asinrodataL15<> + 48(SB)/8, $0.166666666666651626E+00
+DATA ·asinrodataL15<> + 56(SB)/8, $0.750000000042621169E-01
+DATA ·asinrodataL15<> + 64(SB)/8, $0.446428567178116477E-01
+DATA ·asinrodataL15<> + 72(SB)/8, $0.303819660378071894E-01
+DATA ·asinrodataL15<> + 80(SB)/8, $0.223715011892010405E-01
+DATA ·asinrodataL15<> + 88(SB)/8, $0.173659424522364952E-01
+DATA ·asinrodataL15<> + 96(SB)/8, $0.137810186504372266E-01
+DATA ·asinrodataL15<> + 104(SB)/8, $0.134066870961173521E-01
+DATA ·asinrodataL15<> + 112(SB)/8, $-.412335502831898721E-02
+DATA ·asinrodataL15<> + 120(SB)/8, $0.867383739532082719E-01
+DATA ·asinrodataL15<> + 128(SB)/8, $-.328765950607171649E+00
+DATA ·asinrodataL15<> + 136(SB)/8, $0.110401073869414626E+01
+DATA ·asinrodataL15<> + 144(SB)/8, $-.270694366992537307E+01
+DATA ·asinrodataL15<> + 152(SB)/8, $0.500196500770928669E+01
+DATA ·asinrodataL15<> + 160(SB)/8, $-.665866959108585165E+01
+DATA ·asinrodataL15<> + 168(SB)/8, $-.344895269334086578E+01
+DATA ·asinrodataL15<> + 176(SB)/8, $0.927437952918301659E+00
+DATA ·asinrodataL15<> + 184(SB)/8, $0.610487478874645653E+01
+DATA ·asinrodataL15<> + 192(SB)/8, $0x7ff8000000000000 //+Inf
+DATA ·asinrodataL15<> + 200(SB)/8, $-1.0
+DATA ·asinrodataL15<> + 208(SB)/8, $1.0
+DATA ·asinrodataL15<> + 216(SB)/8, $1.00000000000000000e-20
+GLOBL ·asinrodataL15<> + 0(SB), RODATA, $224
+
+// Asin returns the arcsine, in radians, of the argument.
+//
+// Special cases are:
+// Asin(±0) = ±0=
+// Asin(x) = NaN if x < -1 or x > 1
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·asinAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·asinrodataL15<>+0(SB), R9
+ LGDR F0, R7
+ FMOVD F0, F8
+ SRAD $32, R7
+ WORD $0xC0193FE6 //iilf %r1,1072079005
+ BYTE $0xA0
+ BYTE $0x9D
+ WORD $0xB91700C7 //llgtr %r12,%r7
+ MOVW R12, R8
+ MOVW R1, R6
+ CMPBGT R8, R6, L2
+ WORD $0xC0193BFF //iilf %r1,1006632959
+ BYTE $0xFF
+ BYTE $0xFF
+ MOVW R1, R6
+ CMPBGT R8, R6, L13
+L3:
+ FMOVD 216(R9), F0
+ FMADD F0, F8, F8
+L1:
+ FMOVD F8, ret+8(FP)
+ RET
+L2:
+ WORD $0xC0193FEF //iilf %r1,1072693247
+ BYTE $0xFF
+ BYTE $0xFF
+ CMPW R12, R1
+ BLE L14
+L5:
+ WORD $0xED0090D0 //cdb %f0,.L17-.L15(%r9)
+ BYTE $0x00
+ BYTE $0x19
+ BEQ L9
+ WORD $0xED0090C8 //cdb %f0,.L18-.L15(%r9)
+ BYTE $0x00
+ BYTE $0x19
+ BEQ L10
+ WFCEDBS V8, V8, V0
+ BVS L1
+ FMOVD 192(R9), F8
+ BR L1
+L13:
+ WFMDB V0, V0, V10
+L4:
+ WFMDB V10, V10, V0
+ FMOVD 184(R9), F6
+ FMOVD 176(R9), F2
+ FMOVD 168(R9), F4
+ WFMADB V0, V2, V6, V2
+ FMOVD 160(R9), F6
+ WFMADB V0, V4, V6, V4
+ FMOVD 152(R9), F6
+ WFMADB V0, V2, V6, V2
+ FMOVD 144(R9), F6
+ WFMADB V0, V4, V6, V4
+ FMOVD 136(R9), F6
+ WFMADB V0, V2, V6, V2
+ WORD $0xC0193FE6 //iilf %r1,1072079005
+ BYTE $0xA0
+ BYTE $0x9D
+ FMOVD 128(R9), F6
+ WFMADB V0, V4, V6, V4
+ FMOVD 120(R9), F6
+ WFMADB V0, V2, V6, V2
+ FMOVD 112(R9), F6
+ WFMADB V0, V4, V6, V4
+ FMOVD 104(R9), F6
+ WFMADB V0, V2, V6, V2
+ FMOVD 96(R9), F6
+ WFMADB V0, V4, V6, V4
+ FMOVD 88(R9), F6
+ WFMADB V0, V2, V6, V2
+ FMOVD 80(R9), F6
+ WFMADB V0, V4, V6, V4
+ FMOVD 72(R9), F6
+ WFMADB V0, V2, V6, V2
+ FMOVD 64(R9), F6
+ WFMADB V0, V4, V6, V4
+ FMOVD 56(R9), F6
+ WFMADB V0, V2, V6, V2
+ FMOVD 48(R9), F6
+ WFMADB V0, V4, V6, V0
+ WFMDB V8, V10, V4
+ FMADD F2, F10, F0
+ FMADD F0, F4, F8
+ CMPW R12, R1
+ BLE L1
+ FMOVD 40(R9), F0
+ FMADD F0, F1, F8
+ FMOVD F8, ret+8(FP)
+ RET
+L14:
+ FMOVD 200(R9), F0
+ FMADD F8, F8, F0
+ WORD $0xB31300A0 //lcdbr %f10,%f0
+ WORD $0xED009020 //cdb %f0,.L39-.L15(%r9)
+ BYTE $0x00
+ BYTE $0x19
+ FSQRT F10, F8
+L6:
+ MOVW R7, R6
+ CMPBLE R6, $0, L8
+ WORD $0xB3130088 //lcdbr %f8,%f8
+ FMOVD 24(R9), F1
+ BR L4
+L10:
+ FMOVD 16(R9), F8
+ BR L1
+L9:
+ FMOVD 8(R9), F8
+ FMOVD F8, ret+8(FP)
+ RET
+L8:
+ FMOVD 0(R9), F1
+ BR L4
diff --git a/src/math/asinh.go b/src/math/asinh.go
new file mode 100644
index 0000000..65ae4c6
--- /dev/null
+++ b/src/math/asinh.go
@@ -0,0 +1,71 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/s_asinh.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// asinh(x)
+// Method :
+// Based on
+// asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
+// we have
+// asinh(x) := x if 1+x*x=1,
+// := sign(x)*(log(x)+ln2)) for large |x|, else
+// := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
+// := sign(x)*log1p(|x| + x**2/(1 + sqrt(1+x**2)))
+//
+
+// Asinh returns the inverse hyperbolic sine of x.
+//
+// Special cases are:
+// Asinh(±0) = ±0
+// Asinh(±Inf) = ±Inf
+// Asinh(NaN) = NaN
+func Asinh(x float64) float64
+
+func asinh(x float64) float64 {
+ const (
+ Ln2 = 6.93147180559945286227e-01 // 0x3FE62E42FEFA39EF
+ NearZero = 1.0 / (1 << 28) // 2**-28
+ Large = 1 << 28 // 2**28
+ )
+ // special cases
+ if IsNaN(x) || IsInf(x, 0) {
+ return x
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ var temp float64
+ switch {
+ case x > Large:
+ temp = Log(x) + Ln2 // |x| > 2**28
+ case x > 2:
+ temp = Log(2*x + 1/(Sqrt(x*x+1)+x)) // 2**28 > |x| > 2.0
+ case x < NearZero:
+ temp = x // |x| < 2**-28
+ default:
+ temp = Log1p(x + x*x/(1+Sqrt(1+x*x))) // 2.0 > |x| > 2**-28
+ }
+ if sign {
+ temp = -temp
+ }
+ return temp
+}
diff --git a/src/math/asinh_s390x.s b/src/math/asinh_s390x.s
new file mode 100644
index 0000000..1bcf295
--- /dev/null
+++ b/src/math/asinh_s390x.s
@@ -0,0 +1,213 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial coefficients and other constants
+DATA ·asinhrodataL18<> + 0(SB)/8, $0.749999999977387502E-01
+DATA ·asinhrodataL18<> + 8(SB)/8, $-.166666666666657082E+00
+DATA ·asinhrodataL18<> + 16(SB)/8, $0.303819368237360639E-01
+DATA ·asinhrodataL18<> + 24(SB)/8, $-.446428569571752982E-01
+DATA ·asinhrodataL18<> + 32(SB)/8, $0.173500047922695924E-01
+DATA ·asinhrodataL18<> + 40(SB)/8, $-.223719767210027185E-01
+DATA ·asinhrodataL18<> + 48(SB)/8, $0.113655037946822130E-01
+DATA ·asinhrodataL18<> + 56(SB)/8, $0.579747490622448943E-02
+DATA ·asinhrodataL18<> + 64(SB)/8, $-.139372433914359122E-01
+DATA ·asinhrodataL18<> + 72(SB)/8, $-.218674325255800840E-02
+DATA ·asinhrodataL18<> + 80(SB)/8, $-.891074277756961157E-02
+DATA ·asinhrodataL18<> + 88(SB)/8, $.41375273347623353626
+DATA ·asinhrodataL18<> + 96(SB)/8, $.51487302528619766235E+04
+DATA ·asinhrodataL18<> + 104(SB)/8, $-1.67526912689208984375
+DATA ·asinhrodataL18<> + 112(SB)/8, $0.181818181818181826E+00
+DATA ·asinhrodataL18<> + 120(SB)/8, $-.165289256198351540E-01
+DATA ·asinhrodataL18<> + 128(SB)/8, $0.200350613573012186E-02
+DATA ·asinhrodataL18<> + 136(SB)/8, $-.273205381970859341E-03
+DATA ·asinhrodataL18<> + 144(SB)/8, $0.397389654305194527E-04
+DATA ·asinhrodataL18<> + 152(SB)/8, $0.938370938292558173E-06
+DATA ·asinhrodataL18<> + 160(SB)/8, $0.212881813645679599E-07
+DATA ·asinhrodataL18<> + 168(SB)/8, $-.602107458843052029E-05
+DATA ·asinhrodataL18<> + 176(SB)/8, $-.148682720127920854E-06
+DATA ·asinhrodataL18<> + 184(SB)/8, $-5.5
+DATA ·asinhrodataL18<> + 192(SB)/8, $1.0
+DATA ·asinhrodataL18<> + 200(SB)/8, $1.0E-20
+GLOBL ·asinhrodataL18<> + 0(SB), RODATA, $208
+
+// Table of log correction terms
+DATA ·asinhtab2080<> + 0(SB)/8, $0.585235384085551248E-01
+DATA ·asinhtab2080<> + 8(SB)/8, $0.412206153771168640E-01
+DATA ·asinhtab2080<> + 16(SB)/8, $0.273839003221648339E-01
+DATA ·asinhtab2080<> + 24(SB)/8, $0.166383778368856480E-01
+DATA ·asinhtab2080<> + 32(SB)/8, $0.866678223433169637E-02
+DATA ·asinhtab2080<> + 40(SB)/8, $0.319831684989627514E-02
+DATA ·asinhtab2080<> + 48(SB)/8, $0.0
+DATA ·asinhtab2080<> + 56(SB)/8, $-.113006378583725549E-02
+DATA ·asinhtab2080<> + 64(SB)/8, $-.367979419636602491E-03
+DATA ·asinhtab2080<> + 72(SB)/8, $0.213172484510484979E-02
+DATA ·asinhtab2080<> + 80(SB)/8, $0.623271047682013536E-02
+DATA ·asinhtab2080<> + 88(SB)/8, $0.118140812789696885E-01
+DATA ·asinhtab2080<> + 96(SB)/8, $0.187681358930914206E-01
+DATA ·asinhtab2080<> + 104(SB)/8, $0.269985148668178992E-01
+DATA ·asinhtab2080<> + 112(SB)/8, $0.364186619761331328E-01
+DATA ·asinhtab2080<> + 120(SB)/8, $0.469505379381388441E-01
+GLOBL ·asinhtab2080<> + 0(SB), RODATA, $128
+
+// Asinh returns the inverse hyperbolic sine of the argument.
+//
+// Special cases are:
+// Asinh(±0) = ±0
+// Asinh(±Inf) = ±Inf
+// Asinh(NaN) = NaN
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·asinhAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·asinhrodataL18<>+0(SB), R9
+ LGDR F0, R12
+ WORD $0xC0293FDF //iilf %r2,1071644671
+ BYTE $0xFF
+ BYTE $0xFF
+ SRAD $32, R12
+ WORD $0xB917001C //llgtr %r1,%r12
+ MOVW R1, R6
+ MOVW R2, R7
+ CMPBLE R6, R7, L2
+ WORD $0xC0295FEF //iilf %r2,1609564159
+ BYTE $0xFF
+ BYTE $0xFF
+ MOVW R2, R7
+ CMPBLE R6, R7, L14
+L3:
+ WORD $0xC0297FEF //iilf %r2,2146435071
+ BYTE $0xFF
+ BYTE $0xFF
+ CMPW R1, R2
+ BGT L1
+ LTDBR F0, F0
+ FMOVD F0, F10
+ BLTU L15
+L9:
+ FMOVD $0, F0
+ WFADB V0, V10, V0
+ WORD $0xC0398006 //iilf %r3,2147909631
+ BYTE $0x7F
+ BYTE $0xFF
+ LGDR F0, R5
+ SRAD $32, R5
+ MOVH $0x0, R2
+ SUBW R5, R3
+ FMOVD $0, F8
+ RISBGZ $32, $47, $0, R3, R4
+ BYTE $0x18 //lr %r1,%r4
+ BYTE $0x14
+ RISBGN $0, $31, $32, R4, R2
+ SUBW $0x100000, R1
+ SRAW $8, R1, R1
+ ORW $0x45000000, R1
+ BR L6
+L2:
+ MOVD $0x30000000, R2
+ CMPW R1, R2
+ BGT L16
+ FMOVD 200(R9), F2
+ FMADD F2, F0, F0
+L1:
+ FMOVD F0, ret+8(FP)
+ RET
+L14:
+ LTDBR F0, F0
+ BLTU L17
+ FMOVD F0, F10
+L4:
+ FMOVD 192(R9), F2
+ WFMADB V0, V0, V2, V0
+ LTDBR F0, F0
+ FSQRT F0, F8
+L5:
+ WFADB V8, V10, V0
+ WORD $0xC0398006 //iilf %r3,2147909631
+ BYTE $0x7F
+ BYTE $0xFF
+ LGDR F0, R5
+ SRAD $32, R5
+ MOVH $0x0, R2
+ SUBW R5, R3
+ RISBGZ $32, $47, $0, R3, R4
+ SRAW $8, R4, R1
+ RISBGN $0, $31, $32, R4, R2
+ ORW $0x45000000, R1
+L6:
+ LDGR R2, F2
+ FMOVD 184(R9), F0
+ WFMADB V8, V2, V0, V8
+ FMOVD 176(R9), F4
+ WFMADB V10, V2, V8, V2
+ FMOVD 168(R9), F0
+ FMOVD 160(R9), F6
+ FMOVD 152(R9), F1
+ WFMADB V2, V6, V4, V6
+ WFMADB V2, V1, V0, V1
+ WFMDB V2, V2, V4
+ FMOVD 144(R9), F0
+ WFMADB V6, V4, V1, V6
+ FMOVD 136(R9), F1
+ RISBGZ $57, $60, $51, R3, R3
+ WFMADB V2, V0, V1, V0
+ FMOVD 128(R9), F1
+ WFMADB V4, V6, V0, V6
+ FMOVD 120(R9), F0
+ WFMADB V2, V1, V0, V1
+ VLVGF $0, R1, V0
+ WFMADB V4, V6, V1, V4
+ LDEBR F0, F0
+ FMOVD 112(R9), F6
+ WFMADB V2, V4, V6, V4
+ MOVD $·asinhtab2080<>+0(SB), R1
+ FMOVD 104(R9), F1
+ WORD $0x68331000 //ld %f3,0(%r3,%r1)
+ FMOVD 96(R9), F6
+ WFMADB V2, V4, V3, V2
+ WFMADB V0, V1, V6, V0
+ FMOVD 88(R9), F4
+ WFMADB V0, V4, V2, V0
+ MOVD R12, R6
+ CMPBGT R6, $0, L1
+
+ WORD $0xB3130000 //lcdbr %f0,%f0
+ FMOVD F0, ret+8(FP)
+ RET
+L16:
+ WFMDB V0, V0, V1
+ FMOVD 80(R9), F6
+ WFMDB V1, V1, V4
+ FMOVD 72(R9), F2
+ WFMADB V4, V2, V6, V2
+ FMOVD 64(R9), F3
+ FMOVD 56(R9), F6
+ WFMADB V4, V2, V3, V2
+ FMOVD 48(R9), F3
+ WFMADB V4, V6, V3, V6
+ FMOVD 40(R9), F5
+ FMOVD 32(R9), F3
+ WFMADB V4, V2, V5, V2
+ WFMADB V4, V6, V3, V6
+ FMOVD 24(R9), F5
+ FMOVD 16(R9), F3
+ WFMADB V4, V2, V5, V2
+ WFMADB V4, V6, V3, V6
+ FMOVD 8(R9), F5
+ FMOVD 0(R9), F3
+ WFMADB V4, V2, V5, V2
+ WFMADB V4, V6, V3, V4
+ WFMDB V0, V1, V6
+ WFMADB V1, V4, V2, V4
+ FMADD F4, F6, F0
+ FMOVD F0, ret+8(FP)
+ RET
+L17:
+ WORD $0xB31300A0 //lcdbr %f10,%f0
+ BR L4
+L15:
+ WORD $0xB31300A0 //lcdbr %f10,%f0
+ BR L9
diff --git a/src/math/atan.go b/src/math/atan.go
new file mode 100644
index 0000000..7fcc90b
--- /dev/null
+++ b/src/math/atan.go
@@ -0,0 +1,105 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point arctangent.
+*/
+
+// The original C code, the long comment, and the constants below were
+// from http://netlib.sandia.gov/cephes/cmath/atan.c, available from
+// http://www.netlib.org/cephes/cmath.tgz.
+// The go code is a version of the original C.
+//
+// atan.c
+// Inverse circular tangent (arctangent)
+//
+// SYNOPSIS:
+// double x, y, atan();
+// y = atan( x );
+//
+// DESCRIPTION:
+// Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
+//
+// Range reduction is from three intervals into the interval from zero to 0.66.
+// The approximant uses a rational function of degree 4/5 of the form
+// x + x**3 P(x)/Q(x).
+//
+// ACCURACY:
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10, 10 50000 2.4e-17 8.3e-18
+// IEEE -10, 10 10^6 1.8e-16 5.0e-17
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// xatan evaluates a series valid in the range [0, 0.66].
+func xatan(x float64) float64 {
+ const (
+ P0 = -8.750608600031904122785e-01
+ P1 = -1.615753718733365076637e+01
+ P2 = -7.500855792314704667340e+01
+ P3 = -1.228866684490136173410e+02
+ P4 = -6.485021904942025371773e+01
+ Q0 = +2.485846490142306297962e+01
+ Q1 = +1.650270098316988542046e+02
+ Q2 = +4.328810604912902668951e+02
+ Q3 = +4.853903996359136964868e+02
+ Q4 = +1.945506571482613964425e+02
+ )
+ z := x * x
+ z = z * ((((P0*z+P1)*z+P2)*z+P3)*z + P4) / (((((z+Q0)*z+Q1)*z+Q2)*z+Q3)*z + Q4)
+ z = x*z + x
+ return z
+}
+
+// satan reduces its argument (known to be positive)
+// to the range [0, 0.66] and calls xatan.
+func satan(x float64) float64 {
+ const (
+ Morebits = 6.123233995736765886130e-17 // pi/2 = PIO2 + Morebits
+ Tan3pio8 = 2.41421356237309504880 // tan(3*pi/8)
+ )
+ if x <= 0.66 {
+ return xatan(x)
+ }
+ if x > Tan3pio8 {
+ return Pi/2 - xatan(1/x) + Morebits
+ }
+ return Pi/4 + xatan((x-1)/(x+1)) + 0.5*Morebits
+}
+
+// Atan returns the arctangent, in radians, of x.
+//
+// Special cases are:
+// Atan(±0) = ±0
+// Atan(±Inf) = ±Pi/2
+func Atan(x float64) float64
+
+func atan(x float64) float64 {
+ if x == 0 {
+ return x
+ }
+ if x > 0 {
+ return satan(x)
+ }
+ return -satan(-x)
+}
diff --git a/src/math/atan2.go b/src/math/atan2.go
new file mode 100644
index 0000000..d84b332
--- /dev/null
+++ b/src/math/atan2.go
@@ -0,0 +1,71 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Atan2 returns the arc tangent of y/x, using
+// the signs of the two to determine the quadrant
+// of the return value.
+//
+// Special cases are (in order):
+// Atan2(y, NaN) = NaN
+// Atan2(NaN, x) = NaN
+// Atan2(+0, x>=0) = +0
+// Atan2(-0, x>=0) = -0
+// Atan2(+0, x<=-0) = +Pi
+// Atan2(-0, x<=-0) = -Pi
+// Atan2(y>0, 0) = +Pi/2
+// Atan2(y<0, 0) = -Pi/2
+// Atan2(+Inf, +Inf) = +Pi/4
+// Atan2(-Inf, +Inf) = -Pi/4
+// Atan2(+Inf, -Inf) = 3Pi/4
+// Atan2(-Inf, -Inf) = -3Pi/4
+// Atan2(y, +Inf) = 0
+// Atan2(y>0, -Inf) = +Pi
+// Atan2(y<0, -Inf) = -Pi
+// Atan2(+Inf, x) = +Pi/2
+// Atan2(-Inf, x) = -Pi/2
+func Atan2(y, x float64) float64
+
+func atan2(y, x float64) float64 {
+ // special cases
+ switch {
+ case IsNaN(y) || IsNaN(x):
+ return NaN()
+ case y == 0:
+ if x >= 0 && !Signbit(x) {
+ return Copysign(0, y)
+ }
+ return Copysign(Pi, y)
+ case x == 0:
+ return Copysign(Pi/2, y)
+ case IsInf(x, 0):
+ if IsInf(x, 1) {
+ switch {
+ case IsInf(y, 0):
+ return Copysign(Pi/4, y)
+ default:
+ return Copysign(0, y)
+ }
+ }
+ switch {
+ case IsInf(y, 0):
+ return Copysign(3*Pi/4, y)
+ default:
+ return Copysign(Pi, y)
+ }
+ case IsInf(y, 0):
+ return Copysign(Pi/2, y)
+ }
+
+ // Call atan and determine the quadrant.
+ q := Atan(y / x)
+ if x < 0 {
+ if q <= 0 {
+ return q + Pi
+ }
+ return q - Pi
+ }
+ return q
+}
diff --git a/src/math/atan2_386.s b/src/math/atan2_386.s
new file mode 100644
index 0000000..90d211b
--- /dev/null
+++ b/src/math/atan2_386.s
@@ -0,0 +1,13 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Atan2(y, x float64) float64 // =atan(y/x)
+TEXT ·Atan2(SB),NOSPLIT,$0
+ FMOVD y+0(FP), F0 // F0=y
+ FMOVD x+8(FP), F0 // F0=x, F1=y
+ FPATAN // F0=atan(F1/F0)
+ FMOVDP F0, ret+16(FP)
+ RET
diff --git a/src/math/atan2_s390x.s b/src/math/atan2_s390x.s
new file mode 100644
index 0000000..587b89e
--- /dev/null
+++ b/src/math/atan2_s390x.s
@@ -0,0 +1,297 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define PosInf 0x7FF0000000000000
+#define NegInf 0xFFF0000000000000
+#define NegZero 0x8000000000000000
+#define Pi 0x400921FB54442D18
+#define NegPi 0xC00921FB54442D18
+#define Pi3Div4 0x4002D97C7F3321D2 // 3Pi/4
+#define NegPi3Div4 0xC002D97C7F3321D2 // -3Pi/4
+#define PiDiv4 0x3FE921FB54442D18 // Pi/4
+#define NegPiDiv4 0xBFE921FB54442D18 // -Pi/4
+
+// Minimax polynomial coefficients and other constants
+DATA ·atan2rodataL25<> + 0(SB)/8, $0.199999999999554423E+00
+DATA ·atan2rodataL25<> + 8(SB)/8, $-.333333333333330928E+00
+DATA ·atan2rodataL25<> + 16(SB)/8, $0.111111110136634272E+00
+DATA ·atan2rodataL25<> + 24(SB)/8, $-.142857142828026806E+00
+DATA ·atan2rodataL25<> + 32(SB)/8, $0.769228118888682505E-01
+DATA ·atan2rodataL25<> + 40(SB)/8, $0.588059263575587687E-01
+DATA ·atan2rodataL25<> + 48(SB)/8, $-.909090711945939878E-01
+DATA ·atan2rodataL25<> + 56(SB)/8, $-.666641501287528609E-01
+DATA ·atan2rodataL25<> + 64(SB)/8, $0.472329433805024762E-01
+DATA ·atan2rodataL25<> + 72(SB)/8, $-.525380587584426406E-01
+DATA ·atan2rodataL25<> + 80(SB)/8, $-.422172007412067035E-01
+DATA ·atan2rodataL25<> + 88(SB)/8, $0.366935664549587481E-01
+DATA ·atan2rodataL25<> + 96(SB)/8, $0.220852012160300086E-01
+DATA ·atan2rodataL25<> + 104(SB)/8, $-.299856214685512712E-01
+DATA ·atan2rodataL25<> + 112(SB)/8, $0.726338160757602439E-02
+DATA ·atan2rodataL25<> + 120(SB)/8, $0.134893651284712515E-04
+DATA ·atan2rodataL25<> + 128(SB)/8, $-.291935324869629616E-02
+DATA ·atan2rodataL25<> + 136(SB)/8, $-.154797890856877418E-03
+DATA ·atan2rodataL25<> + 144(SB)/8, $0.843488472994227321E-03
+DATA ·atan2rodataL25<> + 152(SB)/8, $-.139950258898989925E-01
+GLOBL ·atan2rodataL25<> + 0(SB), RODATA, $160
+
+DATA ·atan2xpi2h<> + 0(SB)/8, $0x3ff330e4e4fa7b1b
+DATA ·atan2xpi2h<> + 8(SB)/8, $0xbff330e4e4fa7b1b
+DATA ·atan2xpi2h<> + 16(SB)/8, $0x400330e4e4fa7b1b
+DATA ·atan2xpi2h<> + 24(SB)/8, $0xc00330e4e4fa7b1b
+GLOBL ·atan2xpi2h<> + 0(SB), RODATA, $32
+DATA ·atan2xpim<> + 0(SB)/8, $0x3ff4f42b00000000
+GLOBL ·atan2xpim<> + 0(SB), RODATA, $8
+
+// Atan2 returns the arc tangent of y/x, using
+// the signs of the two to determine the quadrant
+// of the return value.
+//
+// Special cases are (in order):
+// Atan2(y, NaN) = NaN
+// Atan2(NaN, x) = NaN
+// Atan2(+0, x>=0) = +0
+// Atan2(-0, x>=0) = -0
+// Atan2(+0, x<=-0) = +Pi
+// Atan2(-0, x<=-0) = -Pi
+// Atan2(y>0, 0) = +Pi/2
+// Atan2(y<0, 0) = -Pi/2
+// Atan2(+Inf, +Inf) = +Pi/4
+// Atan2(-Inf, +Inf) = -Pi/4
+// Atan2(+Inf, -Inf) = 3Pi/4
+// Atan2(-Inf, -Inf) = -3Pi/4
+// Atan2(y, +Inf) = 0
+// Atan2(y>0, -Inf) = +Pi
+// Atan2(y<0, -Inf) = -Pi
+// Atan2(+Inf, x) = +Pi/2
+// Atan2(-Inf, x) = -Pi/2
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·atan2Asm(SB), NOSPLIT, $0-24
+ // special case
+ MOVD x+0(FP), R1
+ MOVD y+8(FP), R2
+
+ // special case Atan2(NaN, y) = NaN
+ MOVD $~(1<<63), R5
+ AND R1, R5 // x = |x|
+ MOVD $PosInf, R3
+ CMPUBLT R3, R5, returnX
+
+ // special case Atan2(x, NaN) = NaN
+ MOVD $~(1<<63), R5
+ AND R2, R5
+ CMPUBLT R3, R5, returnY
+
+ MOVD $NegZero, R3
+ CMPUBEQ R3, R1, xIsNegZero
+
+ MOVD $0, R3
+ CMPUBEQ R3, R1, xIsPosZero
+
+ MOVD $PosInf, R4
+ CMPUBEQ R4, R2, yIsPosInf
+
+ MOVD $NegInf, R4
+ CMPUBEQ R4, R2, yIsNegInf
+ BR Normal
+xIsNegZero:
+ // special case Atan(-0, y>=0) = -0
+ MOVD $0, R4
+ CMPBLE R4, R2, returnX
+
+ //special case Atan2(-0, y<=-0) = -Pi
+ MOVD $NegZero, R4
+ CMPBGE R4, R2, returnNegPi
+ BR Normal
+xIsPosZero:
+ //special case Atan2(0, 0) = 0
+ MOVD $0, R4
+ CMPUBEQ R4, R2, returnX
+
+ //special case Atan2(0, y<=-0) = Pi
+ MOVD $NegZero, R4
+ CMPBGE R4, R2, returnPi
+ BR Normal
+yIsNegInf:
+ //special case Atan2(+Inf, -Inf) = 3Pi/4
+ MOVD $PosInf, R3
+ CMPUBEQ R3, R1, posInfNegInf
+
+ //special case Atan2(-Inf, -Inf) = -3Pi/4
+ MOVD $NegInf, R3
+ CMPUBEQ R3, R1, negInfNegInf
+ BR Normal
+yIsPosInf:
+ //special case Atan2(+Inf, +Inf) = Pi/4
+ MOVD $PosInf, R3
+ CMPUBEQ R3, R1, posInfPosInf
+
+ //special case Atan2(-Inf, +Inf) = -Pi/4
+ MOVD $NegInf, R3
+ CMPUBEQ R3, R1, negInfPosInf
+
+ //special case Atan2(x, +Inf) = Copysign(0, x)
+ CMPBLT R1, $0, returnNegZero
+ BR returnPosZero
+
+Normal:
+ FMOVD x+0(FP), F0
+ FMOVD y+8(FP), F2
+ MOVD $·atan2rodataL25<>+0(SB), R9
+ LGDR F0, R2
+ LGDR F2, R1
+ RISBGNZ $32, $63, $32, R2, R2
+ RISBGNZ $32, $63, $32, R1, R1
+ WORD $0xB9170032 //llgtr %r3,%r2
+ RISBGZ $63, $63, $33, R2, R5
+ WORD $0xB9170041 //llgtr %r4,%r1
+ WFLCDB V0, V20
+ MOVW R4, R6
+ MOVW R3, R7
+ CMPUBLT R6, R7, L17
+ WFDDB V2, V0, V3
+ ADDW $2, R5, R2
+ MOVW R4, R6
+ MOVW R3, R7
+ CMPUBLE R6, R7, L20
+L3:
+ WFMDB V3, V3, V4
+ VLEG $0, 152(R9), V18
+ VLEG $0, 144(R9), V16
+ FMOVD 136(R9), F1
+ FMOVD 128(R9), F5
+ FMOVD 120(R9), F6
+ WFMADB V4, V16, V5, V16
+ WFMADB V4, V6, V1, V6
+ FMOVD 112(R9), F7
+ WFMDB V4, V4, V1
+ WFMADB V4, V7, V18, V7
+ VLEG $0, 104(R9), V18
+ WFMADB V1, V6, V16, V6
+ CMPWU R4, R3
+ FMOVD 96(R9), F5
+ VLEG $0, 88(R9), V16
+ WFMADB V4, V5, V18, V5
+ VLEG $0, 80(R9), V18
+ VLEG $0, 72(R9), V22
+ WFMADB V4, V16, V18, V16
+ VLEG $0, 64(R9), V18
+ WFMADB V1, V7, V5, V7
+ WFMADB V4, V18, V22, V18
+ WFMDB V1, V1, V5
+ WFMADB V1, V16, V18, V16
+ VLEG $0, 56(R9), V18
+ WFMADB V5, V6, V7, V6
+ VLEG $0, 48(R9), V22
+ FMOVD 40(R9), F7
+ WFMADB V4, V7, V18, V7
+ VLEG $0, 32(R9), V18
+ WFMADB V5, V6, V16, V6
+ WFMADB V4, V18, V22, V18
+ VLEG $0, 24(R9), V16
+ WFMADB V1, V7, V18, V7
+ VLEG $0, 16(R9), V18
+ VLEG $0, 8(R9), V22
+ WFMADB V4, V18, V16, V18
+ VLEG $0, 0(R9), V16
+ WFMADB V5, V6, V7, V6
+ WFMADB V4, V16, V22, V16
+ FMUL F3, F4
+ WFMADB V1, V18, V16, V1
+ FMADD F6, F5, F1
+ WFMADB V4, V1, V3, V4
+ BLT L18
+ BGT L7
+ LTDBR F2, F2
+ BLTU L21
+L8:
+ LTDBR F0, F0
+ BLTU L22
+L9:
+ WFCHDBS V2, V0, V0
+ BNE L18
+L7:
+ MOVW R1, R6
+ CMPBGE R6, $0, L1
+L18:
+ RISBGZ $58, $60, $3, R2, R2
+ MOVD $·atan2xpi2h<>+0(SB), R1
+ MOVD ·atan2xpim<>+0(SB), R3
+ LDGR R3, F0
+ WORD $0xED021000 //madb %f4,%f0,0(%r2,%r1)
+ BYTE $0x40
+ BYTE $0x1E
+L1:
+ FMOVD F4, ret+16(FP)
+ RET
+
+L20:
+ LTDBR F2, F2
+ BLTU L23
+ FMOVD F2, F6
+L4:
+ LTDBR F0, F0
+ BLTU L24
+ FMOVD F0, F4
+L5:
+ WFCHDBS V6, V4, V4
+ BEQ L3
+L17:
+ WFDDB V0, V2, V4
+ BYTE $0x18 //lr %r2,%r5
+ BYTE $0x25
+ WORD $0xB3130034 //lcdbr %f3,%f4
+ BR L3
+L23:
+ WORD $0xB3130062 //lcdbr %f6,%f2
+ BR L4
+L22:
+ VLR V20, V0
+ BR L9
+L21:
+ WORD $0xB3130022 //lcdbr %f2,%f2
+ BR L8
+L24:
+ VLR V20, V4
+ BR L5
+returnX: //the result is same as the first argument
+ MOVD R1, ret+16(FP)
+ RET
+returnY: //the result is same as the second argument
+ MOVD R2, ret+16(FP)
+ RET
+returnPi:
+ MOVD $Pi, R1
+ MOVD R1, ret+16(FP)
+ RET
+returnNegPi:
+ MOVD $NegPi, R1
+ MOVD R1, ret+16(FP)
+ RET
+posInfNegInf:
+ MOVD $Pi3Div4, R1
+ MOVD R1, ret+16(FP)
+ RET
+negInfNegInf:
+ MOVD $NegPi3Div4, R1
+ MOVD R1, ret+16(FP)
+ RET
+posInfPosInf:
+ MOVD $PiDiv4, R1
+ MOVD R1, ret+16(FP)
+ RET
+negInfPosInf:
+ MOVD $NegPiDiv4, R1
+ MOVD R1, ret+16(FP)
+ RET
+returnNegZero:
+ MOVD $NegZero, R1
+ MOVD R1, ret+16(FP)
+ RET
+returnPosZero:
+ MOVD $0, ret+16(FP)
+ RET
diff --git a/src/math/atan_386.s b/src/math/atan_386.s
new file mode 100644
index 0000000..43e79b9
--- /dev/null
+++ b/src/math/atan_386.s
@@ -0,0 +1,13 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Atan(x float64) float64
+TEXT ·Atan(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FLD1 // F0=1, F1=x
+ FPATAN // F0=atan(F1/F0)
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/atan_s390x.s b/src/math/atan_s390x.s
new file mode 100644
index 0000000..3a7e59b
--- /dev/null
+++ b/src/math/atan_s390x.s
@@ -0,0 +1,128 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial coefficients and other constants
+DATA ·atanrodataL8<> + 0(SB)/8, $0.199999999999554423E+00
+DATA ·atanrodataL8<> + 8(SB)/8, $0.111111110136634272E+00
+DATA ·atanrodataL8<> + 16(SB)/8, $-.142857142828026806E+00
+DATA ·atanrodataL8<> + 24(SB)/8, $-.333333333333330928E+00
+DATA ·atanrodataL8<> + 32(SB)/8, $0.769228118888682505E-01
+DATA ·atanrodataL8<> + 40(SB)/8, $0.588059263575587687E-01
+DATA ·atanrodataL8<> + 48(SB)/8, $-.666641501287528609E-01
+DATA ·atanrodataL8<> + 56(SB)/8, $-.909090711945939878E-01
+DATA ·atanrodataL8<> + 64(SB)/8, $0.472329433805024762E-01
+DATA ·atanrodataL8<> + 72(SB)/8, $0.366935664549587481E-01
+DATA ·atanrodataL8<> + 80(SB)/8, $-.422172007412067035E-01
+DATA ·atanrodataL8<> + 88(SB)/8, $-.299856214685512712E-01
+DATA ·atanrodataL8<> + 96(SB)/8, $0.220852012160300086E-01
+DATA ·atanrodataL8<> + 104(SB)/8, $0.726338160757602439E-02
+DATA ·atanrodataL8<> + 112(SB)/8, $0.843488472994227321E-03
+DATA ·atanrodataL8<> + 120(SB)/8, $0.134893651284712515E-04
+DATA ·atanrodataL8<> + 128(SB)/8, $-.525380587584426406E-01
+DATA ·atanrodataL8<> + 136(SB)/8, $-.139950258898989925E-01
+DATA ·atanrodataL8<> + 144(SB)/8, $-.291935324869629616E-02
+DATA ·atanrodataL8<> + 152(SB)/8, $-.154797890856877418E-03
+GLOBL ·atanrodataL8<> + 0(SB), RODATA, $160
+
+DATA ·atanxpi2h<> + 0(SB)/8, $0x3ff330e4e4fa7b1b
+DATA ·atanxpi2h<> + 8(SB)/8, $0xbff330e4e4fa7b1b
+DATA ·atanxpi2h<> + 16(SB)/8, $0x400330e4e4fa7b1b
+DATA ·atanxpi2h<> + 24(SB)/4, $0xc00330e4e4fa7b1b
+GLOBL ·atanxpi2h<> + 0(SB), RODATA, $32
+DATA ·atanxpim<> + 0(SB)/8, $0x3ff4f42b00000000
+GLOBL ·atanxpim<> + 0(SB), RODATA, $8
+DATA ·atanxmone<> + 0(SB)/8, $-1.0
+GLOBL ·atanxmone<> + 0(SB), RODATA, $8
+
+// Atan returns the arctangent, in radians, of the argument.
+//
+// Special cases are:
+// Atan(±0) = ±0
+// Atan(±Inf) = ±Pi/2Pi
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·atanAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ //special case Atan(±0) = ±0
+ FMOVD $(0.0), F1
+ FCMPU F0, F1
+ BEQ atanIsZero
+
+ MOVD $·atanrodataL8<>+0(SB), R5
+ MOVH $0x3FE0, R3
+ LGDR F0, R1
+ RISBGNZ $32, $63, $32, R1, R1
+ RLL $16, R1, R2
+ ANDW $0x7FF0, R2
+ MOVW R2, R6
+ MOVW R3, R7
+ CMPUBLE R6, R7, L6
+ MOVD $·atanxmone<>+0(SB), R3
+ FMOVD 0(R3), F2
+ WFDDB V0, V2, V0
+ RISBGZ $63, $63, $33, R1, R1
+ MOVD $·atanxpi2h<>+0(SB), R3
+ MOVWZ R1, R1
+ SLD $3, R1, R1
+ WORD $0x68813000 //ld %f8,0(%r1,%r3)
+L6:
+ WFMDB V0, V0, V2
+ FMOVD 152(R5), F6
+ FMOVD 144(R5), F1
+ FMOVD 136(R5), F7
+ VLEG $0, 128(R5), V16
+ FMOVD 120(R5), F4
+ FMOVD 112(R5), F5
+ WFMADB V2, V4, V6, V4
+ WFMADB V2, V5, V1, V5
+ WFMDB V2, V2, V6
+ FMOVD 104(R5), F3
+ FMOVD 96(R5), F1
+ WFMADB V2, V3, V7, V3
+ MOVH $0x3FE0, R1
+ FMOVD 88(R5), F7
+ WFMADB V2, V1, V7, V1
+ FMOVD 80(R5), F7
+ WFMADB V6, V3, V1, V3
+ WFMADB V6, V4, V5, V4
+ WFMDB V6, V6, V1
+ FMOVD 72(R5), F5
+ WFMADB V2, V5, V7, V5
+ FMOVD 64(R5), F7
+ WFMADB V2, V7, V16, V7
+ VLEG $0, 56(R5), V16
+ WFMADB V6, V5, V7, V5
+ WFMADB V1, V4, V3, V4
+ FMOVD 48(R5), F7
+ FMOVD 40(R5), F3
+ WFMADB V2, V3, V7, V3
+ FMOVD 32(R5), F7
+ WFMADB V2, V7, V16, V7
+ VLEG $0, 24(R5), V16
+ WFMADB V1, V4, V5, V4
+ FMOVD 16(R5), F5
+ WFMADB V6, V3, V7, V3
+ FMOVD 8(R5), F7
+ WFMADB V2, V7, V5, V7
+ FMOVD 0(R5), F5
+ WFMADB V2, V5, V16, V5
+ WFMADB V1, V4, V3, V4
+ WFMADB V6, V7, V5, V6
+ FMUL F0, F2
+ FMADD F4, F1, F6
+ FMADD F6, F2, F0
+ MOVW R2, R6
+ MOVW R1, R7
+ CMPUBLE R6, R7, L1
+ MOVD $·atanxpim<>+0(SB), R1
+ WORD $0xED801000 //madb %f0,%f8,0(%r1)
+ BYTE $0x00
+ BYTE $0x1E
+L1:
+atanIsZero:
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/atanh.go b/src/math/atanh.go
new file mode 100644
index 0000000..8904bd6
--- /dev/null
+++ b/src/math/atanh.go
@@ -0,0 +1,79 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_atanh.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// __ieee754_atanh(x)
+// Method :
+// 1. Reduce x to positive by atanh(-x) = -atanh(x)
+// 2. For x>=0.5
+// 1 2x x
+// atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
+// 2 1 - x 1 - x
+//
+// For x<0.5
+// atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
+//
+// Special cases:
+// atanh(x) is NaN if |x| > 1 with signal;
+// atanh(NaN) is that NaN with no signal;
+// atanh(+-1) is +-INF with signal.
+//
+
+// Atanh returns the inverse hyperbolic tangent of x.
+//
+// Special cases are:
+// Atanh(1) = +Inf
+// Atanh(±0) = ±0
+// Atanh(-1) = -Inf
+// Atanh(x) = NaN if x < -1 or x > 1
+// Atanh(NaN) = NaN
+func Atanh(x float64) float64
+
+func atanh(x float64) float64 {
+ const NearZero = 1.0 / (1 << 28) // 2**-28
+ // special cases
+ switch {
+ case x < -1 || x > 1 || IsNaN(x):
+ return NaN()
+ case x == 1:
+ return Inf(1)
+ case x == -1:
+ return Inf(-1)
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ var temp float64
+ switch {
+ case x < NearZero:
+ temp = x
+ case x < 0.5:
+ temp = x + x
+ temp = 0.5 * Log1p(temp+temp*x/(1-x))
+ default:
+ temp = 0.5 * Log1p((x+x)/(1-x))
+ }
+ if sign {
+ temp = -temp
+ }
+ return temp
+}
diff --git a/src/math/atanh_s390x.s b/src/math/atanh_s390x.s
new file mode 100644
index 0000000..c4ec2b2
--- /dev/null
+++ b/src/math/atanh_s390x.s
@@ -0,0 +1,174 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial coefficients and other constants
+DATA ·atanhrodataL10<> + 0(SB)/8, $.41375273347623353626
+DATA ·atanhrodataL10<> + 8(SB)/8, $.51487302528619766235E+04
+DATA ·atanhrodataL10<> + 16(SB)/8, $-1.67526912689208984375
+DATA ·atanhrodataL10<> + 24(SB)/8, $0.181818181818181826E+00
+DATA ·atanhrodataL10<> + 32(SB)/8, $-.165289256198351540E-01
+DATA ·atanhrodataL10<> + 40(SB)/8, $0.200350613573012186E-02
+DATA ·atanhrodataL10<> + 48(SB)/8, $0.397389654305194527E-04
+DATA ·atanhrodataL10<> + 56(SB)/8, $-.273205381970859341E-03
+DATA ·atanhrodataL10<> + 64(SB)/8, $0.938370938292558173E-06
+DATA ·atanhrodataL10<> + 72(SB)/8, $-.148682720127920854E-06
+DATA ·atanhrodataL10<> + 80(SB)/8, $ 0.212881813645679599E-07
+DATA ·atanhrodataL10<> + 88(SB)/8, $-.602107458843052029E-05
+DATA ·atanhrodataL10<> + 96(SB)/8, $-5.5
+DATA ·atanhrodataL10<> + 104(SB)/8, $-0.5
+DATA ·atanhrodataL10<> + 112(SB)/8, $0.0
+DATA ·atanhrodataL10<> + 120(SB)/8, $0x7ff8000000000000 //Nan
+DATA ·atanhrodataL10<> + 128(SB)/8, $-1.0
+DATA ·atanhrodataL10<> + 136(SB)/8, $1.0
+DATA ·atanhrodataL10<> + 144(SB)/8, $1.0E-20
+GLOBL ·atanhrodataL10<> + 0(SB), RODATA, $152
+
+// Table of log correction terms
+DATA ·atanhtab2076<> + 0(SB)/8, $0.585235384085551248E-01
+DATA ·atanhtab2076<> + 8(SB)/8, $0.412206153771168640E-01
+DATA ·atanhtab2076<> + 16(SB)/8, $0.273839003221648339E-01
+DATA ·atanhtab2076<> + 24(SB)/8, $0.166383778368856480E-01
+DATA ·atanhtab2076<> + 32(SB)/8, $0.866678223433169637E-02
+DATA ·atanhtab2076<> + 40(SB)/8, $0.319831684989627514E-02
+DATA ·atanhtab2076<> + 48(SB)/8, $0.000000000000000000E+00
+DATA ·atanhtab2076<> + 56(SB)/8, $-.113006378583725549E-02
+DATA ·atanhtab2076<> + 64(SB)/8, $-.367979419636602491E-03
+DATA ·atanhtab2076<> + 72(SB)/8, $0.213172484510484979E-02
+DATA ·atanhtab2076<> + 80(SB)/8, $0.623271047682013536E-02
+DATA ·atanhtab2076<> + 88(SB)/8, $0.118140812789696885E-01
+DATA ·atanhtab2076<> + 96(SB)/8, $0.187681358930914206E-01
+DATA ·atanhtab2076<> + 104(SB)/8, $0.269985148668178992E-01
+DATA ·atanhtab2076<> + 112(SB)/8, $0.364186619761331328E-01
+DATA ·atanhtab2076<> + 120(SB)/8, $0.469505379381388441E-01
+GLOBL ·atanhtab2076<> + 0(SB), RODATA, $128
+
+// Table of +/- .5
+DATA ·atanhtabh2075<> + 0(SB)/8, $0.5
+DATA ·atanhtabh2075<> + 8(SB)/8, $-.5
+GLOBL ·atanhtabh2075<> + 0(SB), RODATA, $16
+
+// Atanh returns the inverse hyperbolic tangent of the argument.
+//
+// Special cases are:
+// Atanh(1) = +Inf
+// Atanh(±0) = ±0
+// Atanh(-1) = -Inf
+// Atanh(x) = NaN if x < -1 or x > 1
+// Atanh(NaN) = NaN
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·atanhAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·atanhrodataL10<>+0(SB), R5
+ LGDR F0, R1
+ WORD $0xC0393FEF //iilf %r3,1072693247
+ BYTE $0xFF
+ BYTE $0xFF
+ SRAD $32, R1
+ WORD $0xB9170021 //llgtr %r2,%r1
+ MOVW R2, R6
+ MOVW R3, R7
+ CMPBGT R6, R7, L2
+ WORD $0xC0392FFF //iilf %r3,805306367
+ BYTE $0xFF
+ BYTE $0xFF
+ MOVW R2, R6
+ MOVW R3, R7
+ CMPBGT R6, R7, L9
+L3:
+ FMOVD 144(R5), F2
+ FMADD F2, F0, F0
+L1:
+ FMOVD F0, ret+8(FP)
+ RET
+
+L2:
+ WORD $0xED005088 //cdb %f0,.L12-.L10(%r5)
+ BYTE $0x00
+ BYTE $0x19
+ BEQ L5
+ WORD $0xED005080 //cdb %f0,.L13-.L10(%r5)
+ BYTE $0x00
+ BYTE $0x19
+ BEQ L5
+ WFCEDBS V0, V0, V2
+ BVS L1
+ FMOVD 120(R5), F0
+ BR L1
+L5:
+ WORD $0xED005070 //ddb %f0,.L15-.L10(%r5)
+ BYTE $0x00
+ BYTE $0x1D
+ FMOVD F0, ret+8(FP)
+ RET
+
+L9:
+ FMOVD F0, F2
+ MOVD $·atanhtabh2075<>+0(SB), R2
+ SRW $31, R1, R1
+ FMOVD 104(R5), F4
+ MOVW R1, R1
+ SLD $3, R1, R1
+ WORD $0x68012000 //ld %f0,0(%r1,%r2)
+ WFMADB V2, V4, V0, V4
+ VLEG $0, 96(R5), V16
+ FDIV F4, F2
+ WORD $0xC0298006 //iilf %r2,2147909631
+ BYTE $0x7F
+ BYTE $0xFF
+ FMOVD 88(R5), F6
+ FMOVD 80(R5), F1
+ FMOVD 72(R5), F7
+ FMOVD 64(R5), F5
+ FMOVD F2, F4
+ WORD $0xED405088 //adb %f4,.L12-.L10(%r5)
+ BYTE $0x00
+ BYTE $0x1A
+ LGDR F4, R4
+ SRAD $32, R4
+ FMOVD F4, F3
+ WORD $0xED305088 //sdb %f3,.L12-.L10(%r5)
+ BYTE $0x00
+ BYTE $0x1B
+ SUBW R4, R2
+ WFSDB V3, V2, V3
+ RISBGZ $32, $47, $0, R2, R1
+ SLD $32, R1, R1
+ LDGR R1, F2
+ WFMADB V4, V2, V16, V4
+ SRAW $8, R2, R1
+ WFMADB V4, V5, V6, V5
+ WFMDB V4, V4, V6
+ WFMADB V4, V1, V7, V1
+ WFMADB V2, V3, V4, V2
+ WFMADB V1, V6, V5, V1
+ FMOVD 56(R5), F3
+ FMOVD 48(R5), F5
+ WFMADB V4, V5, V3, V4
+ FMOVD 40(R5), F3
+ FMADD F1, F6, F4
+ FMOVD 32(R5), F1
+ FMADD F3, F2, F1
+ ANDW $0xFFFFFF00, R1
+ WFMADB V6, V4, V1, V6
+ FMOVD 24(R5), F3
+ ORW $0x45000000, R1
+ WFMADB V2, V6, V3, V6
+ VLVGF $0, R1, V4
+ LDEBR F4, F4
+ RISBGZ $57, $60, $51, R2, R2
+ MOVD $·atanhtab2076<>+0(SB), R1
+ FMOVD 16(R5), F3
+ WORD $0x68521000 //ld %f5,0(%r2,%r1)
+ FMOVD 8(R5), F1
+ WFMADB V2, V6, V5, V2
+ WFMADB V4, V3, V1, V4
+ FMOVD 0(R5), F6
+ FMADD F6, F4, F2
+ FMUL F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/big/accuracy_string.go b/src/math/big/accuracy_string.go
new file mode 100644
index 0000000..1501ace
--- /dev/null
+++ b/src/math/big/accuracy_string.go
@@ -0,0 +1,17 @@
+// Code generated by "stringer -type=Accuracy"; DO NOT EDIT.
+
+package big
+
+import "strconv"
+
+const _Accuracy_name = "BelowExactAbove"
+
+var _Accuracy_index = [...]uint8{0, 5, 10, 15}
+
+func (i Accuracy) String() string {
+ i -= -1
+ if i < 0 || i >= Accuracy(len(_Accuracy_index)-1) {
+ return "Accuracy(" + strconv.FormatInt(int64(i+-1), 10) + ")"
+ }
+ return _Accuracy_name[_Accuracy_index[i]:_Accuracy_index[i+1]]
+}
diff --git a/src/math/big/arith.go b/src/math/big/arith.go
new file mode 100644
index 0000000..750ce8a
--- /dev/null
+++ b/src/math/big/arith.go
@@ -0,0 +1,287 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file provides Go implementations of elementary multi-precision
+// arithmetic operations on word vectors. These have the suffix _g.
+// These are needed for platforms without assembly implementations of these routines.
+// This file also contains elementary operations that can be implemented
+// sufficiently efficiently in Go.
+
+package big
+
+import "math/bits"
+
+// A Word represents a single digit of a multi-precision unsigned integer.
+type Word uint
+
+const (
+ _S = _W / 8 // word size in bytes
+
+ _W = bits.UintSize // word size in bits
+ _B = 1 << _W // digit base
+ _M = _B - 1 // digit mask
+)
+
+// Many of the loops in this file are of the form
+// for i := 0; i < len(z) && i < len(x) && i < len(y); i++
+// i < len(z) is the real condition.
+// However, checking i < len(x) && i < len(y) as well is faster than
+// having the compiler do a bounds check in the body of the loop;
+// remarkably it is even faster than hoisting the bounds check
+// out of the loop, by doing something like
+// _, _ = x[len(z)-1], y[len(z)-1]
+// There are other ways to hoist the bounds check out of the loop,
+// but the compiler's BCE isn't powerful enough for them (yet?).
+// See the discussion in CL 164966.
+
+// ----------------------------------------------------------------------------
+// Elementary operations on words
+//
+// These operations are used by the vector operations below.
+
+// z1<<_W + z0 = x*y
+func mulWW_g(x, y Word) (z1, z0 Word) {
+ hi, lo := bits.Mul(uint(x), uint(y))
+ return Word(hi), Word(lo)
+}
+
+// z1<<_W + z0 = x*y + c
+func mulAddWWW_g(x, y, c Word) (z1, z0 Word) {
+ hi, lo := bits.Mul(uint(x), uint(y))
+ var cc uint
+ lo, cc = bits.Add(lo, uint(c), 0)
+ return Word(hi + cc), Word(lo)
+}
+
+// nlz returns the number of leading zeros in x.
+// Wraps bits.LeadingZeros call for convenience.
+func nlz(x Word) uint {
+ return uint(bits.LeadingZeros(uint(x)))
+}
+
+// The resulting carry c is either 0 or 1.
+func addVV_g(z, x, y []Word) (c Word) {
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x) && i < len(y); i++ {
+ zi, cc := bits.Add(uint(x[i]), uint(y[i]), uint(c))
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+// The resulting carry c is either 0 or 1.
+func subVV_g(z, x, y []Word) (c Word) {
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x) && i < len(y); i++ {
+ zi, cc := bits.Sub(uint(x[i]), uint(y[i]), uint(c))
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+// The resulting carry c is either 0 or 1.
+func addVW_g(z, x []Word, y Word) (c Word) {
+ c = y
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ zi, cc := bits.Add(uint(x[i]), uint(c), 0)
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+// addVWlarge is addVW, but intended for large z.
+// The only difference is that we check on every iteration
+// whether we are done with carries,
+// and if so, switch to a much faster copy instead.
+// This is only a good idea for large z,
+// because the overhead of the check and the function call
+// outweigh the benefits when z is small.
+func addVWlarge(z, x []Word, y Word) (c Word) {
+ c = y
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ if c == 0 {
+ copy(z[i:], x[i:])
+ return
+ }
+ zi, cc := bits.Add(uint(x[i]), uint(c), 0)
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+func subVW_g(z, x []Word, y Word) (c Word) {
+ c = y
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ zi, cc := bits.Sub(uint(x[i]), uint(c), 0)
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+// subVWlarge is to subVW as addVWlarge is to addVW.
+func subVWlarge(z, x []Word, y Word) (c Word) {
+ c = y
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ if c == 0 {
+ copy(z[i:], x[i:])
+ return
+ }
+ zi, cc := bits.Sub(uint(x[i]), uint(c), 0)
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+func shlVU_g(z, x []Word, s uint) (c Word) {
+ if s == 0 {
+ copy(z, x)
+ return
+ }
+ if len(z) == 0 {
+ return
+ }
+ s &= _W - 1 // hint to the compiler that shifts by s don't need guard code
+ ŝ := _W - s
+ ŝ &= _W - 1 // ditto
+ c = x[len(z)-1] >> ŝ
+ for i := len(z) - 1; i > 0; i-- {
+ z[i] = x[i]<<s | x[i-1]>>ŝ
+ }
+ z[0] = x[0] << s
+ return
+}
+
+func shrVU_g(z, x []Word, s uint) (c Word) {
+ if s == 0 {
+ copy(z, x)
+ return
+ }
+ if len(z) == 0 {
+ return
+ }
+ s &= _W - 1 // hint to the compiler that shifts by s don't need guard code
+ ŝ := _W - s
+ ŝ &= _W - 1 // ditto
+ c = x[0] << ŝ
+ for i := 0; i < len(z)-1; i++ {
+ z[i] = x[i]>>s | x[i+1]<<ŝ
+ }
+ z[len(z)-1] = x[len(z)-1] >> s
+ return
+}
+
+func mulAddVWW_g(z, x []Word, y, r Word) (c Word) {
+ c = r
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ c, z[i] = mulAddWWW_g(x[i], y, c)
+ }
+ return
+}
+
+func addMulVVW_g(z, x []Word, y Word) (c Word) {
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ z1, z0 := mulAddWWW_g(x[i], y, z[i])
+ lo, cc := bits.Add(uint(z0), uint(c), 0)
+ c, z[i] = Word(cc), Word(lo)
+ c += z1
+ }
+ return
+}
+
+// q = ( x1 << _W + x0 - r)/y. m = floor(( _B^2 - 1 ) / d - _B). Requiring x1<y.
+// An approximate reciprocal with a reference to "Improved Division by Invariant Integers
+// (IEEE Transactions on Computers, 11 Jun. 2010)"
+func divWW(x1, x0, y, m Word) (q, r Word) {
+ s := nlz(y)
+ if s != 0 {
+ x1 = x1<<s | x0>>(_W-s)
+ x0 <<= s
+ y <<= s
+ }
+ d := uint(y)
+ // We know that
+ // m = ⎣(B^2-1)/d⎦-B
+ // ⎣(B^2-1)/d⎦ = m+B
+ // (B^2-1)/d = m+B+delta1 0 <= delta1 <= (d-1)/d
+ // B^2/d = m+B+delta2 0 <= delta2 <= 1
+ // The quotient we're trying to compute is
+ // quotient = ⎣(x1*B+x0)/d⎦
+ // = ⎣(x1*B*(B^2/d)+x0*(B^2/d))/B^2⎦
+ // = ⎣(x1*B*(m+B+delta2)+x0*(m+B+delta2))/B^2⎦
+ // = ⎣(x1*m+x1*B+x0)/B + x0*m/B^2 + delta2*(x1*B+x0)/B^2⎦
+ // The latter two terms of this three-term sum are between 0 and 1.
+ // So we can compute just the first term, and we will be low by at most 2.
+ t1, t0 := bits.Mul(uint(m), uint(x1))
+ _, c := bits.Add(t0, uint(x0), 0)
+ t1, _ = bits.Add(t1, uint(x1), c)
+ // The quotient is either t1, t1+1, or t1+2.
+ // We'll try t1 and adjust if needed.
+ qq := t1
+ // compute remainder r=x-d*q.
+ dq1, dq0 := bits.Mul(d, qq)
+ r0, b := bits.Sub(uint(x0), dq0, 0)
+ r1, _ := bits.Sub(uint(x1), dq1, b)
+ // The remainder we just computed is bounded above by B+d:
+ // r = x1*B + x0 - d*q.
+ // = x1*B + x0 - d*⎣(x1*m+x1*B+x0)/B⎦
+ // = x1*B + x0 - d*((x1*m+x1*B+x0)/B-alpha) 0 <= alpha < 1
+ // = x1*B + x0 - x1*d/B*m - x1*d - x0*d/B + d*alpha
+ // = x1*B + x0 - x1*d/B*⎣(B^2-1)/d-B⎦ - x1*d - x0*d/B + d*alpha
+ // = x1*B + x0 - x1*d/B*⎣(B^2-1)/d-B⎦ - x1*d - x0*d/B + d*alpha
+ // = x1*B + x0 - x1*d/B*((B^2-1)/d-B-beta) - x1*d - x0*d/B + d*alpha 0 <= beta < 1
+ // = x1*B + x0 - x1*B + x1/B + x1*d + x1*d/B*beta - x1*d - x0*d/B + d*alpha
+ // = x0 + x1/B + x1*d/B*beta - x0*d/B + d*alpha
+ // = x0*(1-d/B) + x1*(1+d*beta)/B + d*alpha
+ // < B*(1-d/B) + d*B/B + d because x0<B (and 1-d/B>0), x1<d, 1+d*beta<=B, alpha<1
+ // = B - d + d + d
+ // = B+d
+ // So r1 can only be 0 or 1. If r1 is 1, then we know q was too small.
+ // Add 1 to q and subtract d from r. That guarantees that r is <B, so
+ // we no longer need to keep track of r1.
+ if r1 != 0 {
+ qq++
+ r0 -= d
+ }
+ // If the remainder is still too large, increment q one more time.
+ if r0 >= d {
+ qq++
+ r0 -= d
+ }
+ return Word(qq), Word(r0 >> s)
+}
+
+func divWVW(z []Word, xn Word, x []Word, y Word) (r Word) {
+ r = xn
+ if len(x) == 1 {
+ qq, rr := bits.Div(uint(r), uint(x[0]), uint(y))
+ z[0] = Word(qq)
+ return Word(rr)
+ }
+ rec := reciprocalWord(y)
+ for i := len(z) - 1; i >= 0; i-- {
+ z[i], r = divWW(r, x[i], y, rec)
+ }
+ return r
+}
+
+// reciprocalWord return the reciprocal of the divisor. rec = floor(( _B^2 - 1 ) / u - _B). u = d1 << nlz(d1).
+func reciprocalWord(d1 Word) Word {
+ u := uint(d1 << nlz(d1))
+ x1 := ^u
+ x0 := uint(_M)
+ rec, _ := bits.Div(x1, x0, u) // (_B^2-1)/U-_B = (_B*(_M-C)+_M)/U
+ return Word(rec)
+}
diff --git a/src/math/big/arith_386.s b/src/math/big/arith_386.s
new file mode 100644
index 0000000..d0ea949
--- /dev/null
+++ b/src/math/big/arith_386.s
@@ -0,0 +1,244 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+ MOVL x+0(FP), AX
+ MULL y+4(FP)
+ MOVL DX, z1+8(FP)
+ MOVL AX, z0+12(FP)
+ RET
+
+
+// func addVV(z, x, y []Word) (c Word)
+TEXT ·addVV(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), CX
+ MOVL z_len+4(FP), BP
+ MOVL $0, BX // i = 0
+ MOVL $0, DX // c = 0
+ JMP E1
+
+L1: MOVL (SI)(BX*4), AX
+ ADDL DX, DX // restore CF
+ ADCL (CX)(BX*4), AX
+ SBBL DX, DX // save CF
+ MOVL AX, (DI)(BX*4)
+ ADDL $1, BX // i++
+
+E1: CMPL BX, BP // i < n
+ JL L1
+
+ NEGL DX
+ MOVL DX, c+36(FP)
+ RET
+
+
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SBBL instead of ADCL and label names)
+TEXT ·subVV(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), CX
+ MOVL z_len+4(FP), BP
+ MOVL $0, BX // i = 0
+ MOVL $0, DX // c = 0
+ JMP E2
+
+L2: MOVL (SI)(BX*4), AX
+ ADDL DX, DX // restore CF
+ SBBL (CX)(BX*4), AX
+ SBBL DX, DX // save CF
+ MOVL AX, (DI)(BX*4)
+ ADDL $1, BX // i++
+
+E2: CMPL BX, BP // i < n
+ JL L2
+
+ NEGL DX
+ MOVL DX, c+36(FP)
+ RET
+
+
+// func addVW(z, x []Word, y Word) (c Word)
+TEXT ·addVW(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), AX // c = y
+ MOVL z_len+4(FP), BP
+ MOVL $0, BX // i = 0
+ JMP E3
+
+L3: ADDL (SI)(BX*4), AX
+ MOVL AX, (DI)(BX*4)
+ SBBL AX, AX // save CF
+ NEGL AX
+ ADDL $1, BX // i++
+
+E3: CMPL BX, BP // i < n
+ JL L3
+
+ MOVL AX, c+28(FP)
+ RET
+
+
+// func subVW(z, x []Word, y Word) (c Word)
+TEXT ·subVW(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), AX // c = y
+ MOVL z_len+4(FP), BP
+ MOVL $0, BX // i = 0
+ JMP E4
+
+L4: MOVL (SI)(BX*4), DX
+ SUBL AX, DX
+ MOVL DX, (DI)(BX*4)
+ SBBL AX, AX // save CF
+ NEGL AX
+ ADDL $1, BX // i++
+
+E4: CMPL BX, BP // i < n
+ JL L4
+
+ MOVL AX, c+28(FP)
+ RET
+
+
+// func shlVU(z, x []Word, s uint) (c Word)
+TEXT ·shlVU(SB),NOSPLIT,$0
+ MOVL z_len+4(FP), BX // i = z
+ SUBL $1, BX // i--
+ JL X8b // i < 0 (n <= 0)
+
+ // n > 0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL s+24(FP), CX
+ MOVL (SI)(BX*4), AX // w1 = x[n-1]
+ MOVL $0, DX
+ SHLL CX, AX, DX // w1>>ŝ
+ MOVL DX, c+28(FP)
+
+ CMPL BX, $0
+ JLE X8a // i <= 0
+
+ // i > 0
+L8: MOVL AX, DX // w = w1
+ MOVL -4(SI)(BX*4), AX // w1 = x[i-1]
+ SHLL CX, AX, DX // w<<s | w1>>ŝ
+ MOVL DX, (DI)(BX*4) // z[i] = w<<s | w1>>ŝ
+ SUBL $1, BX // i--
+ JG L8 // i > 0
+
+ // i <= 0
+X8a: SHLL CX, AX // w1<<s
+ MOVL AX, (DI) // z[0] = w1<<s
+ RET
+
+X8b: MOVL $0, c+28(FP)
+ RET
+
+
+// func shrVU(z, x []Word, s uint) (c Word)
+TEXT ·shrVU(SB),NOSPLIT,$0
+ MOVL z_len+4(FP), BP
+ SUBL $1, BP // n--
+ JL X9b // n < 0 (n <= 0)
+
+ // n > 0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL s+24(FP), CX
+ MOVL (SI), AX // w1 = x[0]
+ MOVL $0, DX
+ SHRL CX, AX, DX // w1<<ŝ
+ MOVL DX, c+28(FP)
+
+ MOVL $0, BX // i = 0
+ JMP E9
+
+ // i < n-1
+L9: MOVL AX, DX // w = w1
+ MOVL 4(SI)(BX*4), AX // w1 = x[i+1]
+ SHRL CX, AX, DX // w>>s | w1<<ŝ
+ MOVL DX, (DI)(BX*4) // z[i] = w>>s | w1<<ŝ
+ ADDL $1, BX // i++
+
+E9: CMPL BX, BP
+ JL L9 // i < n-1
+
+ // i >= n-1
+X9a: SHRL CX, AX // w1>>s
+ MOVL AX, (DI)(BP*4) // z[n-1] = w1>>s
+ RET
+
+X9b: MOVL $0, c+28(FP)
+ RET
+
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), BP
+ MOVL r+28(FP), CX // c = r
+ MOVL z_len+4(FP), BX
+ LEAL (DI)(BX*4), DI
+ LEAL (SI)(BX*4), SI
+ NEGL BX // i = -n
+ JMP E5
+
+L5: MOVL (SI)(BX*4), AX
+ MULL BP
+ ADDL CX, AX
+ ADCL $0, DX
+ MOVL AX, (DI)(BX*4)
+ MOVL DX, CX
+ ADDL $1, BX // i++
+
+E5: CMPL BX, $0 // i < 0
+ JL L5
+
+ MOVL CX, c+32(FP)
+ RET
+
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), BP
+ MOVL z_len+4(FP), BX
+ LEAL (DI)(BX*4), DI
+ LEAL (SI)(BX*4), SI
+ NEGL BX // i = -n
+ MOVL $0, CX // c = 0
+ JMP E6
+
+L6: MOVL (SI)(BX*4), AX
+ MULL BP
+ ADDL CX, AX
+ ADCL $0, DX
+ ADDL AX, (DI)(BX*4)
+ ADCL $0, DX
+ MOVL DX, CX
+ ADDL $1, BX // i++
+
+E6: CMPL BX, $0 // i < 0
+ JL L6
+
+ MOVL CX, c+28(FP)
+ RET
+
+
+
diff --git a/src/math/big/arith_amd64.go b/src/math/big/arith_amd64.go
new file mode 100644
index 0000000..1d2d37c
--- /dev/null
+++ b/src/math/big/arith_amd64.go
@@ -0,0 +1,11 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go
+
+package big
+
+import "internal/cpu"
+
+var support_adx = cpu.X86.HasADX && cpu.X86.HasBMI2
diff --git a/src/math/big/arith_amd64.s b/src/math/big/arith_amd64.s
new file mode 100644
index 0000000..61043ca
--- /dev/null
+++ b/src/math/big/arith_amd64.s
@@ -0,0 +1,525 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+ MOVQ x+0(FP), AX
+ MULQ y+8(FP)
+ MOVQ DX, z1+16(FP)
+ MOVQ AX, z0+24(FP)
+ RET
+
+
+
+// The carry bit is saved with SBBQ Rx, Rx: if the carry was set, Rx is -1, otherwise it is 0.
+// It is restored with ADDQ Rx, Rx: if Rx was -1 the carry is set, otherwise it is cleared.
+// This is faster than using rotate instructions.
+
+// func addVV(z, x, y []Word) (c Word)
+TEXT ·addVV(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), DI
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), R9
+ MOVQ z+0(FP), R10
+
+ MOVQ $0, CX // c = 0
+ MOVQ $0, SI // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUBQ $4, DI // n -= 4
+ JL V1 // if n < 0 goto V1
+
+U1: // n >= 0
+ // regular loop body unrolled 4x
+ ADDQ CX, CX // restore CF
+ MOVQ 0(R8)(SI*8), R11
+ MOVQ 8(R8)(SI*8), R12
+ MOVQ 16(R8)(SI*8), R13
+ MOVQ 24(R8)(SI*8), R14
+ ADCQ 0(R9)(SI*8), R11
+ ADCQ 8(R9)(SI*8), R12
+ ADCQ 16(R9)(SI*8), R13
+ ADCQ 24(R9)(SI*8), R14
+ MOVQ R11, 0(R10)(SI*8)
+ MOVQ R12, 8(R10)(SI*8)
+ MOVQ R13, 16(R10)(SI*8)
+ MOVQ R14, 24(R10)(SI*8)
+ SBBQ CX, CX // save CF
+
+ ADDQ $4, SI // i += 4
+ SUBQ $4, DI // n -= 4
+ JGE U1 // if n >= 0 goto U1
+
+V1: ADDQ $4, DI // n += 4
+ JLE E1 // if n <= 0 goto E1
+
+L1: // n > 0
+ ADDQ CX, CX // restore CF
+ MOVQ 0(R8)(SI*8), R11
+ ADCQ 0(R9)(SI*8), R11
+ MOVQ R11, 0(R10)(SI*8)
+ SBBQ CX, CX // save CF
+
+ ADDQ $1, SI // i++
+ SUBQ $1, DI // n--
+ JG L1 // if n > 0 goto L1
+
+E1: NEGQ CX
+ MOVQ CX, c+72(FP) // return c
+ RET
+
+
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SBBQ instead of ADCQ and label names)
+TEXT ·subVV(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), DI
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), R9
+ MOVQ z+0(FP), R10
+
+ MOVQ $0, CX // c = 0
+ MOVQ $0, SI // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUBQ $4, DI // n -= 4
+ JL V2 // if n < 0 goto V2
+
+U2: // n >= 0
+ // regular loop body unrolled 4x
+ ADDQ CX, CX // restore CF
+ MOVQ 0(R8)(SI*8), R11
+ MOVQ 8(R8)(SI*8), R12
+ MOVQ 16(R8)(SI*8), R13
+ MOVQ 24(R8)(SI*8), R14
+ SBBQ 0(R9)(SI*8), R11
+ SBBQ 8(R9)(SI*8), R12
+ SBBQ 16(R9)(SI*8), R13
+ SBBQ 24(R9)(SI*8), R14
+ MOVQ R11, 0(R10)(SI*8)
+ MOVQ R12, 8(R10)(SI*8)
+ MOVQ R13, 16(R10)(SI*8)
+ MOVQ R14, 24(R10)(SI*8)
+ SBBQ CX, CX // save CF
+
+ ADDQ $4, SI // i += 4
+ SUBQ $4, DI // n -= 4
+ JGE U2 // if n >= 0 goto U2
+
+V2: ADDQ $4, DI // n += 4
+ JLE E2 // if n <= 0 goto E2
+
+L2: // n > 0
+ ADDQ CX, CX // restore CF
+ MOVQ 0(R8)(SI*8), R11
+ SBBQ 0(R9)(SI*8), R11
+ MOVQ R11, 0(R10)(SI*8)
+ SBBQ CX, CX // save CF
+
+ ADDQ $1, SI // i++
+ SUBQ $1, DI // n--
+ JG L2 // if n > 0 goto L2
+
+E2: NEGQ CX
+ MOVQ CX, c+72(FP) // return c
+ RET
+
+
+// func addVW(z, x []Word, y Word) (c Word)
+TEXT ·addVW(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), DI
+ CMPQ DI, $32
+ JG large
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), CX // c = y
+ MOVQ z+0(FP), R10
+
+ MOVQ $0, SI // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUBQ $4, DI // n -= 4
+ JL V3 // if n < 4 goto V3
+
+U3: // n >= 0
+ // regular loop body unrolled 4x
+ MOVQ 0(R8)(SI*8), R11
+ MOVQ 8(R8)(SI*8), R12
+ MOVQ 16(R8)(SI*8), R13
+ MOVQ 24(R8)(SI*8), R14
+ ADDQ CX, R11
+ ADCQ $0, R12
+ ADCQ $0, R13
+ ADCQ $0, R14
+ SBBQ CX, CX // save CF
+ NEGQ CX
+ MOVQ R11, 0(R10)(SI*8)
+ MOVQ R12, 8(R10)(SI*8)
+ MOVQ R13, 16(R10)(SI*8)
+ MOVQ R14, 24(R10)(SI*8)
+
+ ADDQ $4, SI // i += 4
+ SUBQ $4, DI // n -= 4
+ JGE U3 // if n >= 0 goto U3
+
+V3: ADDQ $4, DI // n += 4
+ JLE E3 // if n <= 0 goto E3
+
+L3: // n > 0
+ ADDQ 0(R8)(SI*8), CX
+ MOVQ CX, 0(R10)(SI*8)
+ SBBQ CX, CX // save CF
+ NEGQ CX
+
+ ADDQ $1, SI // i++
+ SUBQ $1, DI // n--
+ JG L3 // if n > 0 goto L3
+
+E3: MOVQ CX, c+56(FP) // return c
+ RET
+large:
+ JMP ·addVWlarge(SB)
+
+
+// func subVW(z, x []Word, y Word) (c Word)
+// (same as addVW except for SUBQ/SBBQ instead of ADDQ/ADCQ and label names)
+TEXT ·subVW(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), DI
+ CMPQ DI, $32
+ JG large
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), CX // c = y
+ MOVQ z+0(FP), R10
+
+ MOVQ $0, SI // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUBQ $4, DI // n -= 4
+ JL V4 // if n < 4 goto V4
+
+U4: // n >= 0
+ // regular loop body unrolled 4x
+ MOVQ 0(R8)(SI*8), R11
+ MOVQ 8(R8)(SI*8), R12
+ MOVQ 16(R8)(SI*8), R13
+ MOVQ 24(R8)(SI*8), R14
+ SUBQ CX, R11
+ SBBQ $0, R12
+ SBBQ $0, R13
+ SBBQ $0, R14
+ SBBQ CX, CX // save CF
+ NEGQ CX
+ MOVQ R11, 0(R10)(SI*8)
+ MOVQ R12, 8(R10)(SI*8)
+ MOVQ R13, 16(R10)(SI*8)
+ MOVQ R14, 24(R10)(SI*8)
+
+ ADDQ $4, SI // i += 4
+ SUBQ $4, DI // n -= 4
+ JGE U4 // if n >= 0 goto U4
+
+V4: ADDQ $4, DI // n += 4
+ JLE E4 // if n <= 0 goto E4
+
+L4: // n > 0
+ MOVQ 0(R8)(SI*8), R11
+ SUBQ CX, R11
+ MOVQ R11, 0(R10)(SI*8)
+ SBBQ CX, CX // save CF
+ NEGQ CX
+
+ ADDQ $1, SI // i++
+ SUBQ $1, DI // n--
+ JG L4 // if n > 0 goto L4
+
+E4: MOVQ CX, c+56(FP) // return c
+ RET
+large:
+ JMP ·subVWlarge(SB)
+
+
+// func shlVU(z, x []Word, s uint) (c Word)
+TEXT ·shlVU(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), BX // i = z
+ SUBQ $1, BX // i--
+ JL X8b // i < 0 (n <= 0)
+
+ // n > 0
+ MOVQ z+0(FP), R10
+ MOVQ x+24(FP), R8
+ MOVQ s+48(FP), CX
+ MOVQ (R8)(BX*8), AX // w1 = x[n-1]
+ MOVQ $0, DX
+ SHLQ CX, AX, DX // w1>>ŝ
+ MOVQ DX, c+56(FP)
+
+ CMPQ BX, $0
+ JLE X8a // i <= 0
+
+ // i > 0
+L8: MOVQ AX, DX // w = w1
+ MOVQ -8(R8)(BX*8), AX // w1 = x[i-1]
+ SHLQ CX, AX, DX // w<<s | w1>>ŝ
+ MOVQ DX, (R10)(BX*8) // z[i] = w<<s | w1>>ŝ
+ SUBQ $1, BX // i--
+ JG L8 // i > 0
+
+ // i <= 0
+X8a: SHLQ CX, AX // w1<<s
+ MOVQ AX, (R10) // z[0] = w1<<s
+ RET
+
+X8b: MOVQ $0, c+56(FP)
+ RET
+
+
+// func shrVU(z, x []Word, s uint) (c Word)
+TEXT ·shrVU(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), R11
+ SUBQ $1, R11 // n--
+ JL X9b // n < 0 (n <= 0)
+
+ // n > 0
+ MOVQ z+0(FP), R10
+ MOVQ x+24(FP), R8
+ MOVQ s+48(FP), CX
+ MOVQ (R8), AX // w1 = x[0]
+ MOVQ $0, DX
+ SHRQ CX, AX, DX // w1<<ŝ
+ MOVQ DX, c+56(FP)
+
+ MOVQ $0, BX // i = 0
+ JMP E9
+
+ // i < n-1
+L9: MOVQ AX, DX // w = w1
+ MOVQ 8(R8)(BX*8), AX // w1 = x[i+1]
+ SHRQ CX, AX, DX // w>>s | w1<<ŝ
+ MOVQ DX, (R10)(BX*8) // z[i] = w>>s | w1<<ŝ
+ ADDQ $1, BX // i++
+
+E9: CMPQ BX, R11
+ JL L9 // i < n-1
+
+ // i >= n-1
+X9a: SHRQ CX, AX // w1>>s
+ MOVQ AX, (R10)(R11*8) // z[n-1] = w1>>s
+ RET
+
+X9b: MOVQ $0, c+56(FP)
+ RET
+
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ MOVQ z+0(FP), R10
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), R9
+ MOVQ r+56(FP), CX // c = r
+ MOVQ z_len+8(FP), R11
+ MOVQ $0, BX // i = 0
+
+ CMPQ R11, $4
+ JL E5
+
+U5: // i+4 <= n
+ // regular loop body unrolled 4x
+ MOVQ (0*8)(R8)(BX*8), AX
+ MULQ R9
+ ADDQ CX, AX
+ ADCQ $0, DX
+ MOVQ AX, (0*8)(R10)(BX*8)
+ MOVQ DX, CX
+ MOVQ (1*8)(R8)(BX*8), AX
+ MULQ R9
+ ADDQ CX, AX
+ ADCQ $0, DX
+ MOVQ AX, (1*8)(R10)(BX*8)
+ MOVQ DX, CX
+ MOVQ (2*8)(R8)(BX*8), AX
+ MULQ R9
+ ADDQ CX, AX
+ ADCQ $0, DX
+ MOVQ AX, (2*8)(R10)(BX*8)
+ MOVQ DX, CX
+ MOVQ (3*8)(R8)(BX*8), AX
+ MULQ R9
+ ADDQ CX, AX
+ ADCQ $0, DX
+ MOVQ AX, (3*8)(R10)(BX*8)
+ MOVQ DX, CX
+ ADDQ $4, BX // i += 4
+
+ LEAQ 4(BX), DX
+ CMPQ DX, R11
+ JLE U5
+ JMP E5
+
+L5: MOVQ (R8)(BX*8), AX
+ MULQ R9
+ ADDQ CX, AX
+ ADCQ $0, DX
+ MOVQ AX, (R10)(BX*8)
+ MOVQ DX, CX
+ ADDQ $1, BX // i++
+
+E5: CMPQ BX, R11 // i < n
+ JL L5
+
+ MOVQ CX, c+64(FP)
+ RET
+
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ CMPB ·support_adx(SB), $1
+ JEQ adx
+ MOVQ z+0(FP), R10
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), R9
+ MOVQ z_len+8(FP), R11
+ MOVQ $0, BX // i = 0
+ MOVQ $0, CX // c = 0
+ MOVQ R11, R12
+ ANDQ $-2, R12
+ CMPQ R11, $2
+ JAE A6
+ JMP E6
+
+A6:
+ MOVQ (R8)(BX*8), AX
+ MULQ R9
+ ADDQ (R10)(BX*8), AX
+ ADCQ $0, DX
+ ADDQ CX, AX
+ ADCQ $0, DX
+ MOVQ DX, CX
+ MOVQ AX, (R10)(BX*8)
+
+ MOVQ (8)(R8)(BX*8), AX
+ MULQ R9
+ ADDQ (8)(R10)(BX*8), AX
+ ADCQ $0, DX
+ ADDQ CX, AX
+ ADCQ $0, DX
+ MOVQ DX, CX
+ MOVQ AX, (8)(R10)(BX*8)
+
+ ADDQ $2, BX
+ CMPQ BX, R12
+ JL A6
+ JMP E6
+
+L6: MOVQ (R8)(BX*8), AX
+ MULQ R9
+ ADDQ CX, AX
+ ADCQ $0, DX
+ ADDQ AX, (R10)(BX*8)
+ ADCQ $0, DX
+ MOVQ DX, CX
+ ADDQ $1, BX // i++
+
+E6: CMPQ BX, R11 // i < n
+ JL L6
+
+ MOVQ CX, c+56(FP)
+ RET
+
+adx:
+ MOVQ z_len+8(FP), R11
+ MOVQ z+0(FP), R10
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), DX
+ MOVQ $0, BX // i = 0
+ MOVQ $0, CX // carry
+ CMPQ R11, $8
+ JAE adx_loop_header
+ CMPQ BX, R11
+ JL adx_short
+ MOVQ CX, c+56(FP)
+ RET
+
+adx_loop_header:
+ MOVQ R11, R13
+ ANDQ $-8, R13
+adx_loop:
+ XORQ R9, R9 // unset flags
+ MULXQ (R8), SI, DI
+ ADCXQ CX,SI
+ ADOXQ (R10), SI
+ MOVQ SI,(R10)
+
+ MULXQ 8(R8), AX, CX
+ ADCXQ DI, AX
+ ADOXQ 8(R10), AX
+ MOVQ AX, 8(R10)
+
+ MULXQ 16(R8), SI, DI
+ ADCXQ CX, SI
+ ADOXQ 16(R10), SI
+ MOVQ SI, 16(R10)
+
+ MULXQ 24(R8), AX, CX
+ ADCXQ DI, AX
+ ADOXQ 24(R10), AX
+ MOVQ AX, 24(R10)
+
+ MULXQ 32(R8), SI, DI
+ ADCXQ CX, SI
+ ADOXQ 32(R10), SI
+ MOVQ SI, 32(R10)
+
+ MULXQ 40(R8), AX, CX
+ ADCXQ DI, AX
+ ADOXQ 40(R10), AX
+ MOVQ AX, 40(R10)
+
+ MULXQ 48(R8), SI, DI
+ ADCXQ CX, SI
+ ADOXQ 48(R10), SI
+ MOVQ SI, 48(R10)
+
+ MULXQ 56(R8), AX, CX
+ ADCXQ DI, AX
+ ADOXQ 56(R10), AX
+ MOVQ AX, 56(R10)
+
+ ADCXQ R9, CX
+ ADOXQ R9, CX
+
+ ADDQ $64, R8
+ ADDQ $64, R10
+ ADDQ $8, BX
+
+ CMPQ BX, R13
+ JL adx_loop
+ MOVQ z+0(FP), R10
+ MOVQ x+24(FP), R8
+ CMPQ BX, R11
+ JL adx_short
+ MOVQ CX, c+56(FP)
+ RET
+
+adx_short:
+ MULXQ (R8)(BX*8), SI, DI
+ ADDQ CX, SI
+ ADCQ $0, DI
+ ADDQ SI, (R10)(BX*8)
+ ADCQ $0, DI
+ MOVQ DI, CX
+ ADDQ $1, BX // i++
+
+ CMPQ BX, R11
+ JL adx_short
+
+ MOVQ CX, c+56(FP)
+ RET
+
+
+
diff --git a/src/math/big/arith_arm.s b/src/math/big/arith_arm.s
new file mode 100644
index 0000000..cbf7445
--- /dev/null
+++ b/src/math/big/arith_arm.s
@@ -0,0 +1,283 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// func addVV(z, x, y []Word) (c Word)
+TEXT ·addVV(SB),NOSPLIT,$0
+ ADD.S $0, R0 // clear carry flag
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R4
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ ADD R4<<2, R1, R4
+ B E1
+L1:
+ MOVW.P 4(R2), R5
+ MOVW.P 4(R3), R6
+ ADC.S R6, R5
+ MOVW.P R5, 4(R1)
+E1:
+ TEQ R1, R4
+ BNE L1
+
+ MOVW $0, R0
+ MOVW.CS $1, R0
+ MOVW R0, c+36(FP)
+ RET
+
+
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SBC instead of ADC and label names)
+TEXT ·subVV(SB),NOSPLIT,$0
+ SUB.S $0, R0 // clear borrow flag
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R4
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ ADD R4<<2, R1, R4
+ B E2
+L2:
+ MOVW.P 4(R2), R5
+ MOVW.P 4(R3), R6
+ SBC.S R6, R5
+ MOVW.P R5, 4(R1)
+E2:
+ TEQ R1, R4
+ BNE L2
+
+ MOVW $0, R0
+ MOVW.CC $1, R0
+ MOVW R0, c+36(FP)
+ RET
+
+
+// func addVW(z, x []Word, y Word) (c Word)
+TEXT ·addVW(SB),NOSPLIT,$0
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R4
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ ADD R4<<2, R1, R4
+ TEQ R1, R4
+ BNE L3a
+ MOVW R3, c+28(FP)
+ RET
+L3a:
+ MOVW.P 4(R2), R5
+ ADD.S R3, R5
+ MOVW.P R5, 4(R1)
+ B E3
+L3:
+ MOVW.P 4(R2), R5
+ ADC.S $0, R5
+ MOVW.P R5, 4(R1)
+E3:
+ TEQ R1, R4
+ BNE L3
+
+ MOVW $0, R0
+ MOVW.CS $1, R0
+ MOVW R0, c+28(FP)
+ RET
+
+
+// func subVW(z, x []Word, y Word) (c Word)
+TEXT ·subVW(SB),NOSPLIT,$0
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R4
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ ADD R4<<2, R1, R4
+ TEQ R1, R4
+ BNE L4a
+ MOVW R3, c+28(FP)
+ RET
+L4a:
+ MOVW.P 4(R2), R5
+ SUB.S R3, R5
+ MOVW.P R5, 4(R1)
+ B E4
+L4:
+ MOVW.P 4(R2), R5
+ SBC.S $0, R5
+ MOVW.P R5, 4(R1)
+E4:
+ TEQ R1, R4
+ BNE L4
+
+ MOVW $0, R0
+ MOVW.CC $1, R0
+ MOVW R0, c+28(FP)
+ RET
+
+
+// func shlVU(z, x []Word, s uint) (c Word)
+TEXT ·shlVU(SB),NOSPLIT,$0
+ MOVW z_len+4(FP), R5
+ TEQ $0, R5
+ BEQ X7
+
+ MOVW z+0(FP), R1
+ MOVW x+12(FP), R2
+ ADD R5<<2, R2, R2
+ ADD R5<<2, R1, R5
+ MOVW s+24(FP), R3
+ TEQ $0, R3 // shift 0 is special
+ BEQ Y7
+ ADD $4, R1 // stop one word early
+ MOVW $32, R4
+ SUB R3, R4
+ MOVW $0, R7
+
+ MOVW.W -4(R2), R6
+ MOVW R6<<R3, R7
+ MOVW R6>>R4, R6
+ MOVW R6, c+28(FP)
+ B E7
+
+L7:
+ MOVW.W -4(R2), R6
+ ORR R6>>R4, R7
+ MOVW.W R7, -4(R5)
+ MOVW R6<<R3, R7
+E7:
+ TEQ R1, R5
+ BNE L7
+
+ MOVW R7, -4(R5)
+ RET
+
+Y7: // copy loop, because shift 0 == shift 32
+ MOVW.W -4(R2), R6
+ MOVW.W R6, -4(R5)
+ TEQ R1, R5
+ BNE Y7
+
+X7:
+ MOVW $0, R1
+ MOVW R1, c+28(FP)
+ RET
+
+
+// func shrVU(z, x []Word, s uint) (c Word)
+TEXT ·shrVU(SB),NOSPLIT,$0
+ MOVW z_len+4(FP), R5
+ TEQ $0, R5
+ BEQ X6
+
+ MOVW z+0(FP), R1
+ MOVW x+12(FP), R2
+ ADD R5<<2, R1, R5
+ MOVW s+24(FP), R3
+ TEQ $0, R3 // shift 0 is special
+ BEQ Y6
+ SUB $4, R5 // stop one word early
+ MOVW $32, R4
+ SUB R3, R4
+ MOVW $0, R7
+
+ // first word
+ MOVW.P 4(R2), R6
+ MOVW R6>>R3, R7
+ MOVW R6<<R4, R6
+ MOVW R6, c+28(FP)
+ B E6
+
+ // word loop
+L6:
+ MOVW.P 4(R2), R6
+ ORR R6<<R4, R7
+ MOVW.P R7, 4(R1)
+ MOVW R6>>R3, R7
+E6:
+ TEQ R1, R5
+ BNE L6
+
+ MOVW R7, 0(R1)
+ RET
+
+Y6: // copy loop, because shift 0 == shift 32
+ MOVW.P 4(R2), R6
+ MOVW.P R6, 4(R1)
+ TEQ R1, R5
+ BNE Y6
+
+X6:
+ MOVW $0, R1
+ MOVW R1, c+28(FP)
+ RET
+
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ MOVW $0, R0
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R5
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ MOVW r+28(FP), R4
+ ADD R5<<2, R1, R5
+ B E8
+
+ // word loop
+L8:
+ MOVW.P 4(R2), R6
+ MULLU R6, R3, (R7, R6)
+ ADD.S R4, R6
+ ADC R0, R7
+ MOVW.P R6, 4(R1)
+ MOVW R7, R4
+E8:
+ TEQ R1, R5
+ BNE L8
+
+ MOVW R4, c+32(FP)
+ RET
+
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ MOVW $0, R0
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R5
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ ADD R5<<2, R1, R5
+ MOVW $0, R4
+ B E9
+
+ // word loop
+L9:
+ MOVW.P 4(R2), R6
+ MULLU R6, R3, (R7, R6)
+ ADD.S R4, R6
+ ADC R0, R7
+ MOVW 0(R1), R4
+ ADD.S R4, R6
+ ADC R0, R7
+ MOVW.P R6, 4(R1)
+ MOVW R7, R4
+E9:
+ TEQ R1, R5
+ BNE L9
+
+ MOVW R4, c+28(FP)
+ RET
+
+
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+ MOVW x+0(FP), R1
+ MOVW y+4(FP), R2
+ MULLU R1, R2, (R4, R3)
+ MOVW R4, z1+8(FP)
+ MOVW R3, z0+12(FP)
+ RET
diff --git a/src/math/big/arith_arm64.s b/src/math/big/arith_arm64.s
new file mode 100644
index 0000000..22357d0
--- /dev/null
+++ b/src/math/big/arith_arm64.s
@@ -0,0 +1,583 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// TODO: Consider re-implementing using Advanced SIMD
+// once the assembler supports those instructions.
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+ MOVD x+0(FP), R0
+ MOVD y+8(FP), R1
+ MUL R0, R1, R2
+ UMULH R0, R1, R3
+ MOVD R3, z1+16(FP)
+ MOVD R2, z0+24(FP)
+ RET
+
+
+// func addVV(z, x, y []Word) (c Word)
+TEXT ·addVV(SB),NOSPLIT,$0
+ MOVD z_len+8(FP), R0
+ MOVD x+24(FP), R8
+ MOVD y+48(FP), R9
+ MOVD z+0(FP), R10
+ ADDS $0, R0 // clear carry flag
+ TBZ $0, R0, two
+ MOVD.P 8(R8), R11
+ MOVD.P 8(R9), R15
+ ADCS R15, R11
+ MOVD.P R11, 8(R10)
+ SUB $1, R0
+two:
+ TBZ $1, R0, loop
+ LDP.P 16(R8), (R11, R12)
+ LDP.P 16(R9), (R15, R16)
+ ADCS R15, R11
+ ADCS R16, R12
+ STP.P (R11, R12), 16(R10)
+ SUB $2, R0
+loop:
+ CBZ R0, done // careful not to touch the carry flag
+ LDP.P 32(R8), (R11, R12)
+ LDP -16(R8), (R13, R14)
+ LDP.P 32(R9), (R15, R16)
+ LDP -16(R9), (R17, R19)
+ ADCS R15, R11
+ ADCS R16, R12
+ ADCS R17, R13
+ ADCS R19, R14
+ STP.P (R11, R12), 32(R10)
+ STP (R13, R14), -16(R10)
+ SUB $4, R0
+ B loop
+done:
+ CSET HS, R0 // extract carry flag
+ MOVD R0, c+72(FP)
+ RET
+
+
+// func subVV(z, x, y []Word) (c Word)
+TEXT ·subVV(SB),NOSPLIT,$0
+ MOVD z_len+8(FP), R0
+ MOVD x+24(FP), R8
+ MOVD y+48(FP), R9
+ MOVD z+0(FP), R10
+ CMP R0, R0 // set carry flag
+ TBZ $0, R0, two
+ MOVD.P 8(R8), R11
+ MOVD.P 8(R9), R15
+ SBCS R15, R11
+ MOVD.P R11, 8(R10)
+ SUB $1, R0
+two:
+ TBZ $1, R0, loop
+ LDP.P 16(R8), (R11, R12)
+ LDP.P 16(R9), (R15, R16)
+ SBCS R15, R11
+ SBCS R16, R12
+ STP.P (R11, R12), 16(R10)
+ SUB $2, R0
+loop:
+ CBZ R0, done // careful not to touch the carry flag
+ LDP.P 32(R8), (R11, R12)
+ LDP -16(R8), (R13, R14)
+ LDP.P 32(R9), (R15, R16)
+ LDP -16(R9), (R17, R19)
+ SBCS R15, R11
+ SBCS R16, R12
+ SBCS R17, R13
+ SBCS R19, R14
+ STP.P (R11, R12), 32(R10)
+ STP (R13, R14), -16(R10)
+ SUB $4, R0
+ B loop
+done:
+ CSET LO, R0 // extract carry flag
+ MOVD R0, c+72(FP)
+ RET
+
+#define vwOneOp(instr, op1) \
+ MOVD.P 8(R1), R4; \
+ instr op1, R4; \
+ MOVD.P R4, 8(R3);
+
+// handle the first 1~4 elements before starting iteration in addVW/subVW
+#define vwPreIter(instr1, instr2, counter, target) \
+ vwOneOp(instr1, R2); \
+ SUB $1, counter; \
+ CBZ counter, target; \
+ vwOneOp(instr2, $0); \
+ SUB $1, counter; \
+ CBZ counter, target; \
+ vwOneOp(instr2, $0); \
+ SUB $1, counter; \
+ CBZ counter, target; \
+ vwOneOp(instr2, $0);
+
+// do one iteration of add or sub in addVW/subVW
+#define vwOneIter(instr, counter, exit) \
+ CBZ counter, exit; \ // careful not to touch the carry flag
+ LDP.P 32(R1), (R4, R5); \
+ LDP -16(R1), (R6, R7); \
+ instr $0, R4, R8; \
+ instr $0, R5, R9; \
+ instr $0, R6, R10; \
+ instr $0, R7, R11; \
+ STP.P (R8, R9), 32(R3); \
+ STP (R10, R11), -16(R3); \
+ SUB $4, counter;
+
+// do one iteration of copy in addVW/subVW
+#define vwOneIterCopy(counter, exit) \
+ CBZ counter, exit; \
+ LDP.P 32(R1), (R4, R5); \
+ LDP -16(R1), (R6, R7); \
+ STP.P (R4, R5), 32(R3); \
+ STP (R6, R7), -16(R3); \
+ SUB $4, counter;
+
+// func addVW(z, x []Word, y Word) (c Word)
+// The 'large' branch handles large 'z'. It checks the carry flag on every iteration
+// and switches to copy if we are done with carries. The copying is skipped as well
+// if 'x' and 'z' happen to share the same underlying storage.
+// The overhead of the checking and branching is visible when 'z' are small (~5%),
+// so set a threshold of 32, and remain the small-sized part entirely untouched.
+TEXT ·addVW(SB),NOSPLIT,$0
+ MOVD z+0(FP), R3
+ MOVD z_len+8(FP), R0
+ MOVD x+24(FP), R1
+ MOVD y+48(FP), R2
+ CMP $32, R0
+ BGE large // large-sized 'z' and 'x'
+ CBZ R0, len0 // the length of z is 0
+ MOVD.P 8(R1), R4
+ ADDS R2, R4 // z[0] = x[0] + y, set carry
+ MOVD.P R4, 8(R3)
+ SUB $1, R0
+ CBZ R0, len1 // the length of z is 1
+ TBZ $0, R0, two
+ MOVD.P 8(R1), R4 // do it once
+ ADCS $0, R4
+ MOVD.P R4, 8(R3)
+ SUB $1, R0
+two: // do it twice
+ TBZ $1, R0, loop
+ LDP.P 16(R1), (R4, R5)
+ ADCS $0, R4, R8 // c, z[i] = x[i] + c
+ ADCS $0, R5, R9
+ STP.P (R8, R9), 16(R3)
+ SUB $2, R0
+loop: // do four times per round
+ vwOneIter(ADCS, R0, len1)
+ B loop
+len1:
+ CSET HS, R2 // extract carry flag
+len0:
+ MOVD R2, c+56(FP)
+done:
+ RET
+large:
+ AND $0x3, R0, R10
+ AND $~0x3, R0
+ // unrolling for the first 1~4 elements to avoid saving the carry
+ // flag in each step, adjust $R0 if we unrolled 4 elements
+ vwPreIter(ADDS, ADCS, R10, add4)
+ SUB $4, R0
+add4:
+ BCC copy
+ vwOneIter(ADCS, R0, len1)
+ B add4
+copy:
+ MOVD ZR, c+56(FP)
+ CMP R1, R3
+ BEQ done
+copy_4: // no carry flag, copy the rest
+ vwOneIterCopy(R0, done)
+ B copy_4
+
+// func subVW(z, x []Word, y Word) (c Word)
+// The 'large' branch handles large 'z'. It checks the carry flag on every iteration
+// and switches to copy if we are done with carries. The copying is skipped as well
+// if 'x' and 'z' happen to share the same underlying storage.
+// The overhead of the checking and branching is visible when 'z' are small (~5%),
+// so set a threshold of 32, and remain the small-sized part entirely untouched.
+TEXT ·subVW(SB),NOSPLIT,$0
+ MOVD z+0(FP), R3
+ MOVD z_len+8(FP), R0
+ MOVD x+24(FP), R1
+ MOVD y+48(FP), R2
+ CMP $32, R0
+ BGE large // large-sized 'z' and 'x'
+ CBZ R0, len0 // the length of z is 0
+ MOVD.P 8(R1), R4
+ SUBS R2, R4 // z[0] = x[0] - y, set carry
+ MOVD.P R4, 8(R3)
+ SUB $1, R0
+ CBZ R0, len1 // the length of z is 1
+ TBZ $0, R0, two // do it once
+ MOVD.P 8(R1), R4
+ SBCS $0, R4
+ MOVD.P R4, 8(R3)
+ SUB $1, R0
+two: // do it twice
+ TBZ $1, R0, loop
+ LDP.P 16(R1), (R4, R5)
+ SBCS $0, R4, R8 // c, z[i] = x[i] + c
+ SBCS $0, R5, R9
+ STP.P (R8, R9), 16(R3)
+ SUB $2, R0
+loop: // do four times per round
+ vwOneIter(SBCS, R0, len1)
+ B loop
+len1:
+ CSET LO, R2 // extract carry flag
+len0:
+ MOVD R2, c+56(FP)
+done:
+ RET
+large:
+ AND $0x3, R0, R10
+ AND $~0x3, R0
+ // unrolling for the first 1~4 elements to avoid saving the carry
+ // flag in each step, adjust $R0 if we unrolled 4 elements
+ vwPreIter(SUBS, SBCS, R10, sub4)
+ SUB $4, R0
+sub4:
+ BCS copy
+ vwOneIter(SBCS, R0, len1)
+ B sub4
+copy:
+ MOVD ZR, c+56(FP)
+ CMP R1, R3
+ BEQ done
+copy_4: // no carry flag, copy the rest
+ vwOneIterCopy(R0, done)
+ B copy_4
+
+// func shlVU(z, x []Word, s uint) (c Word)
+// This implementation handles the shift operation from the high word to the low word,
+// which may be an error for the case where the low word of x overlaps with the high
+// word of z. When calling this function directly, you need to pay attention to this
+// situation.
+TEXT ·shlVU(SB),NOSPLIT,$0
+ LDP z+0(FP), (R0, R1) // R0 = z.ptr, R1 = len(z)
+ MOVD x+24(FP), R2
+ MOVD s+48(FP), R3
+ ADD R1<<3, R0 // R0 = &z[n]
+ ADD R1<<3, R2 // R2 = &x[n]
+ CBZ R1, len0
+ CBZ R3, copy // if the number of shift is 0, just copy x to z
+ MOVD $64, R4
+ SUB R3, R4
+ // handling the most significant element x[n-1]
+ MOVD.W -8(R2), R6
+ LSR R4, R6, R5 // return value
+ LSL R3, R6, R8 // x[i] << s
+ SUB $1, R1
+one: TBZ $0, R1, two
+ MOVD.W -8(R2), R6
+ LSR R4, R6, R7
+ ORR R8, R7
+ LSL R3, R6, R8
+ SUB $1, R1
+ MOVD.W R7, -8(R0)
+two:
+ TBZ $1, R1, loop
+ LDP.W -16(R2), (R6, R7)
+ LSR R4, R7, R10
+ ORR R8, R10
+ LSL R3, R7
+ LSR R4, R6, R9
+ ORR R7, R9
+ LSL R3, R6, R8
+ SUB $2, R1
+ STP.W (R9, R10), -16(R0)
+loop:
+ CBZ R1, done
+ LDP.W -32(R2), (R10, R11)
+ LDP 16(R2), (R12, R13)
+ LSR R4, R13, R23
+ ORR R8, R23 // z[i] = (x[i] << s) | (x[i-1] >> (64 - s))
+ LSL R3, R13
+ LSR R4, R12, R22
+ ORR R13, R22
+ LSL R3, R12
+ LSR R4, R11, R21
+ ORR R12, R21
+ LSL R3, R11
+ LSR R4, R10, R20
+ ORR R11, R20
+ LSL R3, R10, R8
+ STP.W (R20, R21), -32(R0)
+ STP (R22, R23), 16(R0)
+ SUB $4, R1
+ B loop
+done:
+ MOVD.W R8, -8(R0) // the first element x[0]
+ MOVD R5, c+56(FP) // the part moved out from x[n-1]
+ RET
+copy:
+ CMP R0, R2
+ BEQ len0
+ TBZ $0, R1, ctwo
+ MOVD.W -8(R2), R4
+ MOVD.W R4, -8(R0)
+ SUB $1, R1
+ctwo:
+ TBZ $1, R1, cloop
+ LDP.W -16(R2), (R4, R5)
+ STP.W (R4, R5), -16(R0)
+ SUB $2, R1
+cloop:
+ CBZ R1, len0
+ LDP.W -32(R2), (R4, R5)
+ LDP 16(R2), (R6, R7)
+ STP.W (R4, R5), -32(R0)
+ STP (R6, R7), 16(R0)
+ SUB $4, R1
+ B cloop
+len0:
+ MOVD $0, c+56(FP)
+ RET
+
+// func shrVU(z, x []Word, s uint) (c Word)
+// This implementation handles the shift operation from the low word to the high word,
+// which may be an error for the case where the high word of x overlaps with the low
+// word of z. When calling this function directly, you need to pay attention to this
+// situation.
+TEXT ·shrVU(SB),NOSPLIT,$0
+ MOVD z+0(FP), R0
+ MOVD z_len+8(FP), R1
+ MOVD x+24(FP), R2
+ MOVD s+48(FP), R3
+ MOVD $0, R8
+ MOVD $64, R4
+ SUB R3, R4
+ CBZ R1, len0
+ CBZ R3, copy // if the number of shift is 0, just copy x to z
+
+ MOVD.P 8(R2), R20
+ LSR R3, R20, R8
+ LSL R4, R20
+ MOVD R20, c+56(FP) // deal with the first element
+ SUB $1, R1
+
+ TBZ $0, R1, two
+ MOVD.P 8(R2), R6
+ LSL R4, R6, R20
+ ORR R8, R20
+ LSR R3, R6, R8
+ MOVD.P R20, 8(R0)
+ SUB $1, R1
+two:
+ TBZ $1, R1, loop
+ LDP.P 16(R2), (R6, R7)
+ LSL R4, R6, R20
+ LSR R3, R6
+ ORR R8, R20
+ LSL R4, R7, R21
+ LSR R3, R7, R8
+ ORR R6, R21
+ STP.P (R20, R21), 16(R0)
+ SUB $2, R1
+loop:
+ CBZ R1, done
+ LDP.P 32(R2), (R10, R11)
+ LDP -16(R2), (R12, R13)
+ LSL R4, R10, R20
+ LSR R3, R10
+ ORR R8, R20 // z[i] = (x[i] >> s) | (x[i+1] << (64 - s))
+ LSL R4, R11, R21
+ LSR R3, R11
+ ORR R10, R21
+ LSL R4, R12, R22
+ LSR R3, R12
+ ORR R11, R22
+ LSL R4, R13, R23
+ LSR R3, R13, R8
+ ORR R12, R23
+ STP.P (R20, R21), 32(R0)
+ STP (R22, R23), -16(R0)
+ SUB $4, R1
+ B loop
+done:
+ MOVD R8, (R0) // deal with the last element
+ RET
+copy:
+ CMP R0, R2
+ BEQ len0
+ TBZ $0, R1, ctwo
+ MOVD.P 8(R2), R3
+ MOVD.P R3, 8(R0)
+ SUB $1, R1
+ctwo:
+ TBZ $1, R1, cloop
+ LDP.P 16(R2), (R4, R5)
+ STP.P (R4, R5), 16(R0)
+ SUB $2, R1
+cloop:
+ CBZ R1, len0
+ LDP.P 32(R2), (R4, R5)
+ LDP -16(R2), (R6, R7)
+ STP.P (R4, R5), 32(R0)
+ STP (R6, R7), -16(R0)
+ SUB $4, R1
+ B cloop
+len0:
+ MOVD $0, c+56(FP)
+ RET
+
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ MOVD z+0(FP), R1
+ MOVD z_len+8(FP), R0
+ MOVD x+24(FP), R2
+ MOVD y+48(FP), R3
+ MOVD r+56(FP), R4
+ // c, z = x * y + r
+ TBZ $0, R0, two
+ MOVD.P 8(R2), R5
+ MUL R3, R5, R7
+ UMULH R3, R5, R8
+ ADDS R4, R7
+ ADC $0, R8, R4 // c, z[i] = x[i] * y + r
+ MOVD.P R7, 8(R1)
+ SUB $1, R0
+two:
+ TBZ $1, R0, loop
+ LDP.P 16(R2), (R5, R6)
+ MUL R3, R5, R10
+ UMULH R3, R5, R11
+ ADDS R4, R10
+ MUL R3, R6, R12
+ UMULH R3, R6, R13
+ ADCS R12, R11
+ ADC $0, R13, R4
+
+ STP.P (R10, R11), 16(R1)
+ SUB $2, R0
+loop:
+ CBZ R0, done
+ LDP.P 32(R2), (R5, R6)
+ LDP -16(R2), (R7, R8)
+
+ MUL R3, R5, R10
+ UMULH R3, R5, R11
+ ADDS R4, R10
+ MUL R3, R6, R12
+ UMULH R3, R6, R13
+ ADCS R11, R12
+
+ MUL R3, R7, R14
+ UMULH R3, R7, R15
+ ADCS R13, R14
+ MUL R3, R8, R16
+ UMULH R3, R8, R17
+ ADCS R15, R16
+ ADC $0, R17, R4
+
+ STP.P (R10, R12), 32(R1)
+ STP (R14, R16), -16(R1)
+ SUB $4, R0
+ B loop
+done:
+ MOVD R4, c+64(FP)
+ RET
+
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ MOVD z+0(FP), R1
+ MOVD z_len+8(FP), R0
+ MOVD x+24(FP), R2
+ MOVD y+48(FP), R3
+ MOVD $0, R4
+
+ TBZ $0, R0, two
+
+ MOVD.P 8(R2), R5
+ MOVD (R1), R6
+
+ MUL R5, R3, R7
+ UMULH R5, R3, R8
+
+ ADDS R7, R6
+ ADC $0, R8, R4
+
+ MOVD.P R6, 8(R1)
+ SUB $1, R0
+
+two:
+ TBZ $1, R0, loop
+
+ LDP.P 16(R2), (R5, R10)
+ LDP (R1), (R6, R11)
+
+ MUL R10, R3, R13
+ UMULH R10, R3, R12
+
+ MUL R5, R3, R7
+ UMULH R5, R3, R8
+
+ ADDS R4, R6
+ ADCS R13, R11
+ ADC $0, R12
+
+ ADDS R7, R6
+ ADCS R8, R11
+ ADC $0, R12, R4
+
+ STP.P (R6, R11), 16(R1)
+ SUB $2, R0
+
+// The main loop of this code operates on a block of 4 words every iteration
+// performing [R4:R12:R11:R10:R9] = R4 + R3 * [R8:R7:R6:R5] + [R12:R11:R10:R9]
+// where R4 is carried from the previous iteration, R8:R7:R6:R5 hold the next
+// 4 words of x, R3 is y and R12:R11:R10:R9 are part of the result z.
+loop:
+ CBZ R0, done
+
+ LDP.P 16(R2), (R5, R6)
+ LDP.P 16(R2), (R7, R8)
+
+ LDP (R1), (R9, R10)
+ ADDS R4, R9
+ MUL R6, R3, R14
+ ADCS R14, R10
+ MUL R7, R3, R15
+ LDP 16(R1), (R11, R12)
+ ADCS R15, R11
+ MUL R8, R3, R16
+ ADCS R16, R12
+ UMULH R8, R3, R20
+ ADC $0, R20
+
+ MUL R5, R3, R13
+ ADDS R13, R9
+ UMULH R5, R3, R17
+ ADCS R17, R10
+ UMULH R6, R3, R21
+ STP.P (R9, R10), 16(R1)
+ ADCS R21, R11
+ UMULH R7, R3, R19
+ ADCS R19, R12
+ STP.P (R11, R12), 16(R1)
+ ADC $0, R20, R4
+
+ SUB $4, R0
+ B loop
+
+done:
+ MOVD R4, c+56(FP)
+ RET
+
+
diff --git a/src/math/big/arith_decl.go b/src/math/big/arith_decl.go
new file mode 100644
index 0000000..d519bdc
--- /dev/null
+++ b/src/math/big/arith_decl.go
@@ -0,0 +1,18 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go
+
+package big
+
+// implemented in arith_$GOARCH.s
+func mulWW(x, y Word) (z1, z0 Word)
+func addVV(z, x, y []Word) (c Word)
+func subVV(z, x, y []Word) (c Word)
+func addVW(z, x []Word, y Word) (c Word)
+func subVW(z, x []Word, y Word) (c Word)
+func shlVU(z, x []Word, s uint) (c Word)
+func shrVU(z, x []Word, s uint) (c Word)
+func mulAddVWW(z, x []Word, y, r Word) (c Word)
+func addMulVVW(z, x []Word, y Word) (c Word)
diff --git a/src/math/big/arith_decl_pure.go b/src/math/big/arith_decl_pure.go
new file mode 100644
index 0000000..5faa3bd
--- /dev/null
+++ b/src/math/big/arith_decl_pure.go
@@ -0,0 +1,53 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build math_big_pure_go
+
+package big
+
+func mulWW(x, y Word) (z1, z0 Word) {
+ return mulWW_g(x, y)
+}
+
+func addVV(z, x, y []Word) (c Word) {
+ return addVV_g(z, x, y)
+}
+
+func subVV(z, x, y []Word) (c Word) {
+ return subVV_g(z, x, y)
+}
+
+func addVW(z, x []Word, y Word) (c Word) {
+ // TODO: remove indirect function call when golang.org/issue/30548 is fixed
+ fn := addVW_g
+ if len(z) > 32 {
+ fn = addVWlarge
+ }
+ return fn(z, x, y)
+}
+
+func subVW(z, x []Word, y Word) (c Word) {
+ // TODO: remove indirect function call when golang.org/issue/30548 is fixed
+ fn := subVW_g
+ if len(z) > 32 {
+ fn = subVWlarge
+ }
+ return fn(z, x, y)
+}
+
+func shlVU(z, x []Word, s uint) (c Word) {
+ return shlVU_g(z, x, s)
+}
+
+func shrVU(z, x []Word, s uint) (c Word) {
+ return shrVU_g(z, x, s)
+}
+
+func mulAddVWW(z, x []Word, y, r Word) (c Word) {
+ return mulAddVWW_g(z, x, y, r)
+}
+
+func addMulVVW(z, x []Word, y Word) (c Word) {
+ return addMulVVW_g(z, x, y)
+}
diff --git a/src/math/big/arith_decl_s390x.go b/src/math/big/arith_decl_s390x.go
new file mode 100644
index 0000000..f1a69e1
--- /dev/null
+++ b/src/math/big/arith_decl_s390x.go
@@ -0,0 +1,18 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go
+
+package big
+
+import "internal/cpu"
+
+func addVV_check(z, x, y []Word) (c Word)
+func addVV_vec(z, x, y []Word) (c Word)
+func addVV_novec(z, x, y []Word) (c Word)
+func subVV_check(z, x, y []Word) (c Word)
+func subVV_vec(z, x, y []Word) (c Word)
+func subVV_novec(z, x, y []Word) (c Word)
+
+var hasVX = cpu.S390X.HasVX
diff --git a/src/math/big/arith_mips64x.s b/src/math/big/arith_mips64x.s
new file mode 100644
index 0000000..804b9fe
--- /dev/null
+++ b/src/math/big/arith_mips64x.s
@@ -0,0 +1,38 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go,mips64 !math_big_pure_go,mips64le
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+TEXT ·mulWW(SB),NOSPLIT,$0
+ JMP ·mulWW_g(SB)
+
+TEXT ·addVV(SB),NOSPLIT,$0
+ JMP ·addVV_g(SB)
+
+TEXT ·subVV(SB),NOSPLIT,$0
+ JMP ·subVV_g(SB)
+
+TEXT ·addVW(SB),NOSPLIT,$0
+ JMP ·addVW_g(SB)
+
+TEXT ·subVW(SB),NOSPLIT,$0
+ JMP ·subVW_g(SB)
+
+TEXT ·shlVU(SB),NOSPLIT,$0
+ JMP ·shlVU_g(SB)
+
+TEXT ·shrVU(SB),NOSPLIT,$0
+ JMP ·shrVU_g(SB)
+
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ JMP ·mulAddVWW_g(SB)
+
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ JMP ·addMulVVW_g(SB)
+
diff --git a/src/math/big/arith_mipsx.s b/src/math/big/arith_mipsx.s
new file mode 100644
index 0000000..efdecb8
--- /dev/null
+++ b/src/math/big/arith_mipsx.s
@@ -0,0 +1,38 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go,mips !math_big_pure_go,mipsle
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+TEXT ·mulWW(SB),NOSPLIT,$0
+ JMP ·mulWW_g(SB)
+
+TEXT ·addVV(SB),NOSPLIT,$0
+ JMP ·addVV_g(SB)
+
+TEXT ·subVV(SB),NOSPLIT,$0
+ JMP ·subVV_g(SB)
+
+TEXT ·addVW(SB),NOSPLIT,$0
+ JMP ·addVW_g(SB)
+
+TEXT ·subVW(SB),NOSPLIT,$0
+ JMP ·subVW_g(SB)
+
+TEXT ·shlVU(SB),NOSPLIT,$0
+ JMP ·shlVU_g(SB)
+
+TEXT ·shrVU(SB),NOSPLIT,$0
+ JMP ·shrVU_g(SB)
+
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ JMP ·mulAddVWW_g(SB)
+
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ JMP ·addMulVVW_g(SB)
+
diff --git a/src/math/big/arith_ppc64x.s b/src/math/big/arith_ppc64x.s
new file mode 100644
index 0000000..b299ccc
--- /dev/null
+++ b/src/math/big/arith_ppc64x.s
@@ -0,0 +1,481 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go,ppc64 !math_big_pure_go,ppc64le
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB), NOSPLIT, $0
+ MOVD x+0(FP), R4
+ MOVD y+8(FP), R5
+ MULHDU R4, R5, R6
+ MULLD R4, R5, R7
+ MOVD R6, z1+16(FP)
+ MOVD R7, z0+24(FP)
+ RET
+
+// func addVV(z, y, y []Word) (c Word)
+// z[i] = x[i] + y[i] for all i, carrying
+TEXT ·addVV(SB), NOSPLIT, $0
+ MOVD z_len+8(FP), R7 // R7 = z_len
+ MOVD x+24(FP), R8 // R8 = x[]
+ MOVD y+48(FP), R9 // R9 = y[]
+ MOVD z+0(FP), R10 // R10 = z[]
+
+ // If z_len = 0, we are done
+ CMP R0, R7
+ MOVD R0, R4
+ BEQ done
+
+ // Process the first iteration out of the loop so we can
+ // use MOVDU and avoid 3 index registers updates.
+ MOVD 0(R8), R11 // R11 = x[i]
+ MOVD 0(R9), R12 // R12 = y[i]
+ ADD $-1, R7 // R7 = z_len - 1
+ ADDC R12, R11, R15 // R15 = x[i] + y[i], set CA
+ CMP R0, R7
+ MOVD R15, 0(R10) // z[i]
+ BEQ final // If z_len was 1, we are done
+
+ SRD $2, R7, R5 // R5 = z_len/4
+ CMP R0, R5
+ MOVD R5, CTR // Set up loop counter
+ BEQ tail // If R5 = 0, we can't use the loop
+
+ // Process 4 elements per iteration. Unrolling this loop
+ // means a performance trade-off: we will lose performance
+ // for small values of z_len (0.90x in the worst case), but
+ // gain significant performance as z_len increases (up to
+ // 1.45x).
+loop:
+ MOVD 8(R8), R11 // R11 = x[i]
+ MOVD 16(R8), R12 // R12 = x[i+1]
+ MOVD 24(R8), R14 // R14 = x[i+2]
+ MOVDU 32(R8), R15 // R15 = x[i+3]
+ MOVD 8(R9), R16 // R16 = y[i]
+ MOVD 16(R9), R17 // R17 = y[i+1]
+ MOVD 24(R9), R18 // R18 = y[i+2]
+ MOVDU 32(R9), R19 // R19 = y[i+3]
+ ADDE R11, R16, R20 // R20 = x[i] + y[i] + CA
+ ADDE R12, R17, R21 // R21 = x[i+1] + y[i+1] + CA
+ ADDE R14, R18, R22 // R22 = x[i+2] + y[i+2] + CA
+ ADDE R15, R19, R23 // R23 = x[i+3] + y[i+3] + CA
+ MOVD R20, 8(R10) // z[i]
+ MOVD R21, 16(R10) // z[i+1]
+ MOVD R22, 24(R10) // z[i+2]
+ MOVDU R23, 32(R10) // z[i+3]
+ ADD $-4, R7 // R7 = z_len - 4
+ BC 16, 0, loop // bdnz
+
+ // We may have more elements to read
+ CMP R0, R7
+ BEQ final
+
+ // Process the remaining elements, one at a time
+tail:
+ MOVDU 8(R8), R11 // R11 = x[i]
+ MOVDU 8(R9), R16 // R16 = y[i]
+ ADD $-1, R7 // R7 = z_len - 1
+ ADDE R11, R16, R20 // R20 = x[i] + y[i] + CA
+ CMP R0, R7
+ MOVDU R20, 8(R10) // z[i]
+ BEQ final // If R7 = 0, we are done
+
+ MOVDU 8(R8), R11
+ MOVDU 8(R9), R16
+ ADD $-1, R7
+ ADDE R11, R16, R20
+ CMP R0, R7
+ MOVDU R20, 8(R10)
+ BEQ final
+
+ MOVD 8(R8), R11
+ MOVD 8(R9), R16
+ ADDE R11, R16, R20
+ MOVD R20, 8(R10)
+
+final:
+ ADDZE R4 // Capture CA
+
+done:
+ MOVD R4, c+72(FP)
+ RET
+
+// func subVV(z, x, y []Word) (c Word)
+// z[i] = x[i] - y[i] for all i, carrying
+TEXT ·subVV(SB), NOSPLIT, $0
+ MOVD z_len+8(FP), R7 // R7 = z_len
+ MOVD x+24(FP), R8 // R8 = x[]
+ MOVD y+48(FP), R9 // R9 = y[]
+ MOVD z+0(FP), R10 // R10 = z[]
+
+ // If z_len = 0, we are done
+ CMP R0, R7
+ MOVD R0, R4
+ BEQ done
+
+ // Process the first iteration out of the loop so we can
+ // use MOVDU and avoid 3 index registers updates.
+ MOVD 0(R8), R11 // R11 = x[i]
+ MOVD 0(R9), R12 // R12 = y[i]
+ ADD $-1, R7 // R7 = z_len - 1
+ SUBC R12, R11, R15 // R15 = x[i] - y[i], set CA
+ CMP R0, R7
+ MOVD R15, 0(R10) // z[i]
+ BEQ final // If z_len was 1, we are done
+
+ SRD $2, R7, R5 // R5 = z_len/4
+ CMP R0, R5
+ MOVD R5, CTR // Set up loop counter
+ BEQ tail // If R5 = 0, we can't use the loop
+
+ // Process 4 elements per iteration. Unrolling this loop
+ // means a performance trade-off: we will lose performance
+ // for small values of z_len (0.92x in the worst case), but
+ // gain significant performance as z_len increases (up to
+ // 1.45x).
+loop:
+ MOVD 8(R8), R11 // R11 = x[i]
+ MOVD 16(R8), R12 // R12 = x[i+1]
+ MOVD 24(R8), R14 // R14 = x[i+2]
+ MOVDU 32(R8), R15 // R15 = x[i+3]
+ MOVD 8(R9), R16 // R16 = y[i]
+ MOVD 16(R9), R17 // R17 = y[i+1]
+ MOVD 24(R9), R18 // R18 = y[i+2]
+ MOVDU 32(R9), R19 // R19 = y[i+3]
+ SUBE R16, R11, R20 // R20 = x[i] - y[i] + CA
+ SUBE R17, R12, R21 // R21 = x[i+1] - y[i+1] + CA
+ SUBE R18, R14, R22 // R22 = x[i+2] - y[i+2] + CA
+ SUBE R19, R15, R23 // R23 = x[i+3] - y[i+3] + CA
+ MOVD R20, 8(R10) // z[i]
+ MOVD R21, 16(R10) // z[i+1]
+ MOVD R22, 24(R10) // z[i+2]
+ MOVDU R23, 32(R10) // z[i+3]
+ ADD $-4, R7 // R7 = z_len - 4
+ BC 16, 0, loop // bdnz
+
+ // We may have more elements to read
+ CMP R0, R7
+ BEQ final
+
+ // Process the remaining elements, one at a time
+tail:
+ MOVDU 8(R8), R11 // R11 = x[i]
+ MOVDU 8(R9), R16 // R16 = y[i]
+ ADD $-1, R7 // R7 = z_len - 1
+ SUBE R16, R11, R20 // R20 = x[i] - y[i] + CA
+ CMP R0, R7
+ MOVDU R20, 8(R10) // z[i]
+ BEQ final // If R7 = 0, we are done
+
+ MOVDU 8(R8), R11
+ MOVDU 8(R9), R16
+ ADD $-1, R7
+ SUBE R16, R11, R20
+ CMP R0, R7
+ MOVDU R20, 8(R10)
+ BEQ final
+
+ MOVD 8(R8), R11
+ MOVD 8(R9), R16
+ SUBE R16, R11, R20
+ MOVD R20, 8(R10)
+
+final:
+ ADDZE R4
+ XOR $1, R4
+
+done:
+ MOVD R4, c+72(FP)
+ RET
+
+// func addVW(z, x []Word, y Word) (c Word)
+TEXT ·addVW(SB), NOSPLIT, $0
+ MOVD z+0(FP), R10 // R10 = z[]
+ MOVD x+24(FP), R8 // R8 = x[]
+ MOVD y+48(FP), R4 // R4 = y = c
+ MOVD z_len+8(FP), R11 // R11 = z_len
+
+ CMP R0, R11 // If z_len is zero, return
+ BEQ done
+
+ // We will process the first iteration out of the loop so we capture
+ // the value of c. In the subsequent iterations, we will rely on the
+ // value of CA set here.
+ MOVD 0(R8), R20 // R20 = x[i]
+ ADD $-1, R11 // R11 = z_len - 1
+ ADDC R20, R4, R6 // R6 = x[i] + c
+ CMP R0, R11 // If z_len was 1, we are done
+ MOVD R6, 0(R10) // z[i]
+ BEQ final
+
+ // We will read 4 elements per iteration
+ SRD $2, R11, R9 // R9 = z_len/4
+ DCBT (R8)
+ CMP R0, R9
+ MOVD R9, CTR // Set up the loop counter
+ BEQ tail // If R9 = 0, we can't use the loop
+
+loop:
+ MOVD 8(R8), R20 // R20 = x[i]
+ MOVD 16(R8), R21 // R21 = x[i+1]
+ MOVD 24(R8), R22 // R22 = x[i+2]
+ MOVDU 32(R8), R23 // R23 = x[i+3]
+ ADDZE R20, R24 // R24 = x[i] + CA
+ ADDZE R21, R25 // R25 = x[i+1] + CA
+ ADDZE R22, R26 // R26 = x[i+2] + CA
+ ADDZE R23, R27 // R27 = x[i+3] + CA
+ MOVD R24, 8(R10) // z[i]
+ MOVD R25, 16(R10) // z[i+1]
+ MOVD R26, 24(R10) // z[i+2]
+ MOVDU R27, 32(R10) // z[i+3]
+ ADD $-4, R11 // R11 = z_len - 4
+ BC 16, 0, loop // bdnz
+
+ // We may have some elements to read
+ CMP R0, R11
+ BEQ final
+
+tail:
+ MOVDU 8(R8), R20
+ ADDZE R20, R24
+ ADD $-1, R11
+ MOVDU R24, 8(R10)
+ CMP R0, R11
+ BEQ final
+
+ MOVDU 8(R8), R20
+ ADDZE R20, R24
+ ADD $-1, R11
+ MOVDU R24, 8(R10)
+ CMP R0, R11
+ BEQ final
+
+ MOVD 8(R8), R20
+ ADDZE R20, R24
+ MOVD R24, 8(R10)
+
+final:
+ ADDZE R0, R4 // c = CA
+done:
+ MOVD R4, c+56(FP)
+ RET
+
+// func subVW(z, x []Word, y Word) (c Word)
+TEXT ·subVW(SB), NOSPLIT, $0
+ MOVD z+0(FP), R10 // R10 = z[]
+ MOVD x+24(FP), R8 // R8 = x[]
+ MOVD y+48(FP), R4 // R4 = y = c
+ MOVD z_len+8(FP), R11 // R11 = z_len
+
+ CMP R0, R11 // If z_len is zero, return
+ BEQ done
+
+ // We will process the first iteration out of the loop so we capture
+ // the value of c. In the subsequent iterations, we will rely on the
+ // value of CA set here.
+ MOVD 0(R8), R20 // R20 = x[i]
+ ADD $-1, R11 // R11 = z_len - 1
+ SUBC R4, R20, R6 // R6 = x[i] - c
+ CMP R0, R11 // If z_len was 1, we are done
+ MOVD R6, 0(R10) // z[i]
+ BEQ final
+
+ // We will read 4 elements per iteration
+ SRD $2, R11, R9 // R9 = z_len/4
+ DCBT (R8)
+ CMP R0, R9
+ MOVD R9, CTR // Set up the loop counter
+ BEQ tail // If R9 = 0, we can't use the loop
+
+ // The loop here is almost the same as the one used in s390x, but
+ // we don't need to capture CA every iteration because we've already
+ // done that above.
+loop:
+ MOVD 8(R8), R20
+ MOVD 16(R8), R21
+ MOVD 24(R8), R22
+ MOVDU 32(R8), R23
+ SUBE R0, R20
+ SUBE R0, R21
+ SUBE R0, R22
+ SUBE R0, R23
+ MOVD R20, 8(R10)
+ MOVD R21, 16(R10)
+ MOVD R22, 24(R10)
+ MOVDU R23, 32(R10)
+ ADD $-4, R11
+ BC 16, 0, loop // bdnz
+
+ // We may have some elements to read
+ CMP R0, R11
+ BEQ final
+
+tail:
+ MOVDU 8(R8), R20
+ SUBE R0, R20
+ ADD $-1, R11
+ MOVDU R20, 8(R10)
+ CMP R0, R11
+ BEQ final
+
+ MOVDU 8(R8), R20
+ SUBE R0, R20
+ ADD $-1, R11
+ MOVDU R20, 8(R10)
+ CMP R0, R11
+ BEQ final
+
+ MOVD 8(R8), R20
+ SUBE R0, R20
+ MOVD R20, 8(R10)
+
+final:
+ // Capture CA
+ SUBE R4, R4
+ NEG R4, R4
+
+done:
+ MOVD R4, c+56(FP)
+ RET
+
+TEXT ·shlVU(SB), NOSPLIT, $0
+ BR ·shlVU_g(SB)
+
+TEXT ·shrVU(SB), NOSPLIT, $0
+ BR ·shrVU_g(SB)
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB), NOSPLIT, $0
+ MOVD z+0(FP), R10 // R10 = z[]
+ MOVD x+24(FP), R8 // R8 = x[]
+ MOVD y+48(FP), R9 // R9 = y
+ MOVD r+56(FP), R4 // R4 = r = c
+ MOVD z_len+8(FP), R11 // R11 = z_len
+
+ CMP R0, R11
+ BEQ done
+
+ MOVD 0(R8), R20
+ ADD $-1, R11
+ MULLD R9, R20, R6 // R6 = z0 = Low-order(x[i]*y)
+ MULHDU R9, R20, R7 // R7 = z1 = High-order(x[i]*y)
+ ADDC R4, R6 // R6 = z0 + r
+ ADDZE R7 // R7 = z1 + CA
+ CMP R0, R11
+ MOVD R7, R4 // R4 = c
+ MOVD R6, 0(R10) // z[i]
+ BEQ done
+
+ // We will read 4 elements per iteration
+ SRD $2, R11, R14 // R14 = z_len/4
+ DCBT (R8)
+ CMP R0, R14
+ MOVD R14, CTR // Set up the loop counter
+ BEQ tail // If R9 = 0, we can't use the loop
+
+loop:
+ MOVD 8(R8), R20 // R20 = x[i]
+ MOVD 16(R8), R21 // R21 = x[i+1]
+ MOVD 24(R8), R22 // R22 = x[i+2]
+ MOVDU 32(R8), R23 // R23 = x[i+3]
+ MULLD R9, R20, R24 // R24 = z0[i]
+ MULHDU R9, R20, R20 // R20 = z1[i]
+ ADDC R4, R24 // R24 = z0[i] + c
+ ADDZE R20 // R7 = z1[i] + CA
+ MULLD R9, R21, R25
+ MULHDU R9, R21, R21
+ ADDC R20, R25
+ ADDZE R21
+ MULLD R9, R22, R26
+ MULHDU R9, R22, R22
+ MULLD R9, R23, R27
+ MULHDU R9, R23, R23
+ ADDC R21, R26
+ ADDZE R22
+ MOVD R24, 8(R10) // z[i]
+ MOVD R25, 16(R10) // z[i+1]
+ ADDC R22, R27
+ ADDZE R23,R4 // update carry
+ MOVD R26, 24(R10) // z[i+2]
+ MOVDU R27, 32(R10) // z[i+3]
+ ADD $-4, R11 // R11 = z_len - 4
+ BC 16, 0, loop // bdnz
+
+ // We may have some elements to read
+ CMP R0, R11
+ BEQ done
+
+ // Process the remaining elements, one at a time
+tail:
+ MOVDU 8(R8), R20 // R20 = x[i]
+ MULLD R9, R20, R24 // R24 = z0[i]
+ MULHDU R9, R20, R25 // R25 = z1[i]
+ ADD $-1, R11 // R11 = z_len - 1
+ ADDC R4, R24
+ ADDZE R25
+ MOVDU R24, 8(R10) // z[i]
+ CMP R0, R11
+ MOVD R25, R4 // R4 = c
+ BEQ done // If R11 = 0, we are done
+
+ MOVDU 8(R8), R20
+ MULLD R9, R20, R24
+ MULHDU R9, R20, R25
+ ADD $-1, R11
+ ADDC R4, R24
+ ADDZE R25
+ MOVDU R24, 8(R10)
+ CMP R0, R11
+ MOVD R25, R4
+ BEQ done
+
+ MOVD 8(R8), R20
+ MULLD R9, R20, R24
+ MULHDU R9, R20, R25
+ ADD $-1, R11
+ ADDC R4, R24
+ ADDZE R25
+ MOVD R24, 8(R10)
+ MOVD R25, R4
+
+done:
+ MOVD R4, c+64(FP)
+ RET
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB), NOSPLIT, $0
+ MOVD z+0(FP), R10 // R10 = z[]
+ MOVD x+24(FP), R8 // R8 = x[]
+ MOVD y+48(FP), R9 // R9 = y
+ MOVD z_len+8(FP), R22 // R22 = z_len
+
+ MOVD R0, R3 // R3 will be the index register
+ CMP R0, R22
+ MOVD R0, R4 // R4 = c = 0
+ MOVD R22, CTR // Initialize loop counter
+ BEQ done
+
+loop:
+ MOVD (R8)(R3), R20 // Load x[i]
+ MOVD (R10)(R3), R21 // Load z[i]
+ MULLD R9, R20, R6 // R6 = Low-order(x[i]*y)
+ MULHDU R9, R20, R7 // R7 = High-order(x[i]*y)
+ ADDC R21, R6 // R6 = z0
+ ADDZE R7 // R7 = z1
+ ADDC R4, R6 // R6 = z0 + c + 0
+ ADDZE R7, R4 // c += z1
+ MOVD R6, (R10)(R3) // Store z[i]
+ ADD $8, R3
+ BC 16, 0, loop // bdnz
+
+done:
+ MOVD R4, c+56(FP)
+ RET
+
+
diff --git a/src/math/big/arith_riscv64.s b/src/math/big/arith_riscv64.s
new file mode 100644
index 0000000..a2f7666
--- /dev/null
+++ b/src/math/big/arith_riscv64.s
@@ -0,0 +1,46 @@
+// Copyright 2020 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go,riscv64
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+ MOV x+0(FP), X5
+ MOV y+8(FP), X6
+ MULHU X5, X6, X7
+ MUL X5, X6, X8
+ MOV X7, z1+16(FP)
+ MOV X8, z0+24(FP)
+ RET
+
+
+TEXT ·addVV(SB),NOSPLIT,$0
+ JMP ·addVV_g(SB)
+
+TEXT ·subVV(SB),NOSPLIT,$0
+ JMP ·subVV_g(SB)
+
+TEXT ·addVW(SB),NOSPLIT,$0
+ JMP ·addVW_g(SB)
+
+TEXT ·subVW(SB),NOSPLIT,$0
+ JMP ·subVW_g(SB)
+
+TEXT ·shlVU(SB),NOSPLIT,$0
+ JMP ·shlVU_g(SB)
+
+TEXT ·shrVU(SB),NOSPLIT,$0
+ JMP ·shrVU_g(SB)
+
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ JMP ·mulAddVWW_g(SB)
+
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ JMP ·addMulVVW_g(SB)
+
diff --git a/src/math/big/arith_s390x.s b/src/math/big/arith_s390x.s
new file mode 100644
index 0000000..caa4db0
--- /dev/null
+++ b/src/math/big/arith_s390x.s
@@ -0,0 +1,795 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go,s390x
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+TEXT ·mulWW(SB), NOSPLIT, $0
+ MOVD x+0(FP), R3
+ MOVD y+8(FP), R4
+ MULHDU R3, R4
+ MOVD R10, z1+16(FP)
+ MOVD R11, z0+24(FP)
+ RET
+
+
+// DI = R3, CX = R4, SI = r10, r8 = r8, r9=r9, r10 = r2 , r11 = r5, r12 = r6, r13 = r7, r14 = r1 (R0 set to 0) + use R11
+// func addVV(z, x, y []Word) (c Word)
+
+TEXT ·addVV(SB), NOSPLIT, $0
+ MOVD addvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·addVV_check(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $addvectorfacility+0x00(SB), R1
+ MOVD $·addVV_novec(SB), R2
+ MOVD R2, 0(R1)
+
+ // MOVD $·addVV_novec(SB), 0(R1)
+ BR ·addVV_novec(SB)
+
+vectorimpl:
+ MOVD $addvectorfacility+0x00(SB), R1
+ MOVD $·addVV_vec(SB), R2
+ MOVD R2, 0(R1)
+
+ // MOVD $·addVV_vec(SB), 0(R1)
+ BR ·addVV_vec(SB)
+
+GLOBL addvectorfacility+0x00(SB), NOPTR, $8
+DATA addvectorfacility+0x00(SB)/8, $·addVV_check(SB)
+
+TEXT ·addVV_vec(SB), NOSPLIT, $0
+ MOVD z_len+8(FP), R3
+ MOVD x+24(FP), R8
+ MOVD y+48(FP), R9
+ MOVD z+0(FP), R2
+
+ MOVD $0, R4 // c = 0
+ MOVD $0, R0 // make sure it's zero
+ MOVD $0, R10 // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUB $4, R3
+ BLT v1
+ SUB $12, R3 // n -= 16
+ BLT A1 // if n < 0 goto A1
+
+ MOVD R8, R5
+ MOVD R9, R6
+ MOVD R2, R7
+
+ // n >= 0
+ // regular loop body unrolled 16x
+ VZERO V0 // c = 0
+
+UU1:
+ VLM 0(R5), V1, V4 // 64-bytes into V1..V8
+ ADD $64, R5
+ VPDI $0x4, V1, V1, V1 // flip the doublewords to big-endian order
+ VPDI $0x4, V2, V2, V2 // flip the doublewords to big-endian order
+
+ VLM 0(R6), V9, V12 // 64-bytes into V9..V16
+ ADD $64, R6
+ VPDI $0x4, V9, V9, V9 // flip the doublewords to big-endian order
+ VPDI $0x4, V10, V10, V10 // flip the doublewords to big-endian order
+
+ VACCCQ V1, V9, V0, V25
+ VACQ V1, V9, V0, V17
+ VACCCQ V2, V10, V25, V26
+ VACQ V2, V10, V25, V18
+
+ VLM 0(R5), V5, V6 // 32-bytes into V1..V8
+ VLM 0(R6), V13, V14 // 32-bytes into V9..V16
+ ADD $32, R5
+ ADD $32, R6
+
+ VPDI $0x4, V3, V3, V3 // flip the doublewords to big-endian order
+ VPDI $0x4, V4, V4, V4 // flip the doublewords to big-endian order
+ VPDI $0x4, V11, V11, V11 // flip the doublewords to big-endian order
+ VPDI $0x4, V12, V12, V12 // flip the doublewords to big-endian order
+
+ VACCCQ V3, V11, V26, V27
+ VACQ V3, V11, V26, V19
+ VACCCQ V4, V12, V27, V28
+ VACQ V4, V12, V27, V20
+
+ VLM 0(R5), V7, V8 // 32-bytes into V1..V8
+ VLM 0(R6), V15, V16 // 32-bytes into V9..V16
+ ADD $32, R5
+ ADD $32, R6
+
+ VPDI $0x4, V5, V5, V5 // flip the doublewords to big-endian order
+ VPDI $0x4, V6, V6, V6 // flip the doublewords to big-endian order
+ VPDI $0x4, V13, V13, V13 // flip the doublewords to big-endian order
+ VPDI $0x4, V14, V14, V14 // flip the doublewords to big-endian order
+
+ VACCCQ V5, V13, V28, V29
+ VACQ V5, V13, V28, V21
+ VACCCQ V6, V14, V29, V30
+ VACQ V6, V14, V29, V22
+
+ VPDI $0x4, V7, V7, V7 // flip the doublewords to big-endian order
+ VPDI $0x4, V8, V8, V8 // flip the doublewords to big-endian order
+ VPDI $0x4, V15, V15, V15 // flip the doublewords to big-endian order
+ VPDI $0x4, V16, V16, V16 // flip the doublewords to big-endian order
+
+ VACCCQ V7, V15, V30, V31
+ VACQ V7, V15, V30, V23
+ VACCCQ V8, V16, V31, V0 // V0 has carry-over
+ VACQ V8, V16, V31, V24
+
+ VPDI $0x4, V17, V17, V17 // flip the doublewords to big-endian order
+ VPDI $0x4, V18, V18, V18 // flip the doublewords to big-endian order
+ VPDI $0x4, V19, V19, V19 // flip the doublewords to big-endian order
+ VPDI $0x4, V20, V20, V20 // flip the doublewords to big-endian order
+ VPDI $0x4, V21, V21, V21 // flip the doublewords to big-endian order
+ VPDI $0x4, V22, V22, V22 // flip the doublewords to big-endian order
+ VPDI $0x4, V23, V23, V23 // flip the doublewords to big-endian order
+ VPDI $0x4, V24, V24, V24 // flip the doublewords to big-endian order
+ VSTM V17, V24, 0(R7) // 128-bytes into z
+ ADD $128, R7
+ ADD $128, R10 // i += 16
+ SUB $16, R3 // n -= 16
+ BGE UU1 // if n >= 0 goto U1
+ VLGVG $1, V0, R4 // put cf into R4
+ NEG R4, R4 // save cf
+
+A1:
+ ADD $12, R3 // n += 16
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ BLT v1 // if n < 0 goto v1
+
+U1: // n >= 0
+ // regular loop body unrolled 4x
+ MOVD 0(R8)(R10*1), R5
+ MOVD 8(R8)(R10*1), R6
+ MOVD 16(R8)(R10*1), R7
+ MOVD 24(R8)(R10*1), R1
+ ADDC R4, R4 // restore CF
+ MOVD 0(R9)(R10*1), R11
+ ADDE R11, R5
+ MOVD 8(R9)(R10*1), R11
+ ADDE R11, R6
+ MOVD 16(R9)(R10*1), R11
+ ADDE R11, R7
+ MOVD 24(R9)(R10*1), R11
+ ADDE R11, R1
+ MOVD R0, R4
+ ADDE R4, R4 // save CF
+ NEG R4, R4
+ MOVD R5, 0(R2)(R10*1)
+ MOVD R6, 8(R2)(R10*1)
+ MOVD R7, 16(R2)(R10*1)
+ MOVD R1, 24(R2)(R10*1)
+
+ ADD $32, R10 // i += 4
+ SUB $4, R3 // n -= 4
+ BGE U1 // if n >= 0 goto U1
+
+v1:
+ ADD $4, R3 // n += 4
+ BLE E1 // if n <= 0 goto E1
+
+L1: // n > 0
+ ADDC R4, R4 // restore CF
+ MOVD 0(R8)(R10*1), R5
+ MOVD 0(R9)(R10*1), R11
+ ADDE R11, R5
+ MOVD R5, 0(R2)(R10*1)
+ MOVD R0, R4
+ ADDE R4, R4 // save CF
+ NEG R4, R4
+
+ ADD $8, R10 // i++
+ SUB $1, R3 // n--
+ BGT L1 // if n > 0 goto L1
+
+E1:
+ NEG R4, R4
+ MOVD R4, c+72(FP) // return c
+ RET
+
+TEXT ·addVV_novec(SB), NOSPLIT, $0
+novec:
+ MOVD z_len+8(FP), R3
+ MOVD x+24(FP), R8
+ MOVD y+48(FP), R9
+ MOVD z+0(FP), R2
+
+ MOVD $0, R4 // c = 0
+ MOVD $0, R0 // make sure it's zero
+ MOVD $0, R10 // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUB $4, R3 // n -= 4
+ BLT v1n // if n < 0 goto v1n
+
+U1n: // n >= 0
+ // regular loop body unrolled 4x
+ MOVD 0(R8)(R10*1), R5
+ MOVD 8(R8)(R10*1), R6
+ MOVD 16(R8)(R10*1), R7
+ MOVD 24(R8)(R10*1), R1
+ ADDC R4, R4 // restore CF
+ MOVD 0(R9)(R10*1), R11
+ ADDE R11, R5
+ MOVD 8(R9)(R10*1), R11
+ ADDE R11, R6
+ MOVD 16(R9)(R10*1), R11
+ ADDE R11, R7
+ MOVD 24(R9)(R10*1), R11
+ ADDE R11, R1
+ MOVD R0, R4
+ ADDE R4, R4 // save CF
+ NEG R4, R4
+ MOVD R5, 0(R2)(R10*1)
+ MOVD R6, 8(R2)(R10*1)
+ MOVD R7, 16(R2)(R10*1)
+ MOVD R1, 24(R2)(R10*1)
+
+ ADD $32, R10 // i += 4
+ SUB $4, R3 // n -= 4
+ BGE U1n // if n >= 0 goto U1n
+
+v1n:
+ ADD $4, R3 // n += 4
+ BLE E1n // if n <= 0 goto E1n
+
+L1n: // n > 0
+ ADDC R4, R4 // restore CF
+ MOVD 0(R8)(R10*1), R5
+ MOVD 0(R9)(R10*1), R11
+ ADDE R11, R5
+ MOVD R5, 0(R2)(R10*1)
+ MOVD R0, R4
+ ADDE R4, R4 // save CF
+ NEG R4, R4
+
+ ADD $8, R10 // i++
+ SUB $1, R3 // n--
+ BGT L1n // if n > 0 goto L1n
+
+E1n:
+ NEG R4, R4
+ MOVD R4, c+72(FP) // return c
+ RET
+
+TEXT ·subVV(SB), NOSPLIT, $0
+ MOVD subvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·subVV_check(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $subvectorfacility+0x00(SB), R1
+ MOVD $·subVV_novec(SB), R2
+ MOVD R2, 0(R1)
+
+ // MOVD $·subVV_novec(SB), 0(R1)
+ BR ·subVV_novec(SB)
+
+vectorimpl:
+ MOVD $subvectorfacility+0x00(SB), R1
+ MOVD $·subVV_vec(SB), R2
+ MOVD R2, 0(R1)
+
+ // MOVD $·subVV_vec(SB), 0(R1)
+ BR ·subVV_vec(SB)
+
+GLOBL subvectorfacility+0x00(SB), NOPTR, $8
+DATA subvectorfacility+0x00(SB)/8, $·subVV_check(SB)
+
+// DI = R3, CX = R4, SI = r10, r8 = r8, r9=r9, r10 = r2 , r11 = r5, r12 = r6, r13 = r7, r14 = r1 (R0 set to 0) + use R11
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SUBC/SUBE instead of ADDC/ADDE and label names)
+TEXT ·subVV_vec(SB), NOSPLIT, $0
+ MOVD z_len+8(FP), R3
+ MOVD x+24(FP), R8
+ MOVD y+48(FP), R9
+ MOVD z+0(FP), R2
+ MOVD $0, R4 // c = 0
+ MOVD $0, R0 // make sure it's zero
+ MOVD $0, R10 // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUB $4, R3 // n -= 4
+ BLT v1 // if n < 0 goto v1
+ SUB $12, R3 // n -= 16
+ BLT A1 // if n < 0 goto A1
+
+ MOVD R8, R5
+ MOVD R9, R6
+ MOVD R2, R7
+
+ // n >= 0
+ // regular loop body unrolled 16x
+ VZERO V0 // cf = 0
+ MOVD $1, R4 // for 390 subtraction cf starts as 1 (no borrow)
+ VLVGG $1, R4, V0 // put carry into V0
+
+UU1:
+ VLM 0(R5), V1, V4 // 64-bytes into V1..V8
+ ADD $64, R5
+ VPDI $0x4, V1, V1, V1 // flip the doublewords to big-endian order
+ VPDI $0x4, V2, V2, V2 // flip the doublewords to big-endian order
+
+ VLM 0(R6), V9, V12 // 64-bytes into V9..V16
+ ADD $64, R6
+ VPDI $0x4, V9, V9, V9 // flip the doublewords to big-endian order
+ VPDI $0x4, V10, V10, V10 // flip the doublewords to big-endian order
+
+ VSBCBIQ V1, V9, V0, V25
+ VSBIQ V1, V9, V0, V17
+ VSBCBIQ V2, V10, V25, V26
+ VSBIQ V2, V10, V25, V18
+
+ VLM 0(R5), V5, V6 // 32-bytes into V1..V8
+ VLM 0(R6), V13, V14 // 32-bytes into V9..V16
+ ADD $32, R5
+ ADD $32, R6
+
+ VPDI $0x4, V3, V3, V3 // flip the doublewords to big-endian order
+ VPDI $0x4, V4, V4, V4 // flip the doublewords to big-endian order
+ VPDI $0x4, V11, V11, V11 // flip the doublewords to big-endian order
+ VPDI $0x4, V12, V12, V12 // flip the doublewords to big-endian order
+
+ VSBCBIQ V3, V11, V26, V27
+ VSBIQ V3, V11, V26, V19
+ VSBCBIQ V4, V12, V27, V28
+ VSBIQ V4, V12, V27, V20
+
+ VLM 0(R5), V7, V8 // 32-bytes into V1..V8
+ VLM 0(R6), V15, V16 // 32-bytes into V9..V16
+ ADD $32, R5
+ ADD $32, R6
+
+ VPDI $0x4, V5, V5, V5 // flip the doublewords to big-endian order
+ VPDI $0x4, V6, V6, V6 // flip the doublewords to big-endian order
+ VPDI $0x4, V13, V13, V13 // flip the doublewords to big-endian order
+ VPDI $0x4, V14, V14, V14 // flip the doublewords to big-endian order
+
+ VSBCBIQ V5, V13, V28, V29
+ VSBIQ V5, V13, V28, V21
+ VSBCBIQ V6, V14, V29, V30
+ VSBIQ V6, V14, V29, V22
+
+ VPDI $0x4, V7, V7, V7 // flip the doublewords to big-endian order
+ VPDI $0x4, V8, V8, V8 // flip the doublewords to big-endian order
+ VPDI $0x4, V15, V15, V15 // flip the doublewords to big-endian order
+ VPDI $0x4, V16, V16, V16 // flip the doublewords to big-endian order
+
+ VSBCBIQ V7, V15, V30, V31
+ VSBIQ V7, V15, V30, V23
+ VSBCBIQ V8, V16, V31, V0 // V0 has carry-over
+ VSBIQ V8, V16, V31, V24
+
+ VPDI $0x4, V17, V17, V17 // flip the doublewords to big-endian order
+ VPDI $0x4, V18, V18, V18 // flip the doublewords to big-endian order
+ VPDI $0x4, V19, V19, V19 // flip the doublewords to big-endian order
+ VPDI $0x4, V20, V20, V20 // flip the doublewords to big-endian order
+ VPDI $0x4, V21, V21, V21 // flip the doublewords to big-endian order
+ VPDI $0x4, V22, V22, V22 // flip the doublewords to big-endian order
+ VPDI $0x4, V23, V23, V23 // flip the doublewords to big-endian order
+ VPDI $0x4, V24, V24, V24 // flip the doublewords to big-endian order
+ VSTM V17, V24, 0(R7) // 128-bytes into z
+ ADD $128, R7
+ ADD $128, R10 // i += 16
+ SUB $16, R3 // n -= 16
+ BGE UU1 // if n >= 0 goto U1
+ VLGVG $1, V0, R4 // put cf into R4
+ SUB $1, R4 // save cf
+
+A1:
+ ADD $12, R3 // n += 16
+ BLT v1 // if n < 0 goto v1
+
+U1: // n >= 0
+ // regular loop body unrolled 4x
+ MOVD 0(R8)(R10*1), R5
+ MOVD 8(R8)(R10*1), R6
+ MOVD 16(R8)(R10*1), R7
+ MOVD 24(R8)(R10*1), R1
+ MOVD R0, R11
+ SUBC R4, R11 // restore CF
+ MOVD 0(R9)(R10*1), R11
+ SUBE R11, R5
+ MOVD 8(R9)(R10*1), R11
+ SUBE R11, R6
+ MOVD 16(R9)(R10*1), R11
+ SUBE R11, R7
+ MOVD 24(R9)(R10*1), R11
+ SUBE R11, R1
+ MOVD R0, R4
+ SUBE R4, R4 // save CF
+ MOVD R5, 0(R2)(R10*1)
+ MOVD R6, 8(R2)(R10*1)
+ MOVD R7, 16(R2)(R10*1)
+ MOVD R1, 24(R2)(R10*1)
+
+ ADD $32, R10 // i += 4
+ SUB $4, R3 // n -= 4
+ BGE U1 // if n >= 0 goto U1n
+
+v1:
+ ADD $4, R3 // n += 4
+ BLE E1 // if n <= 0 goto E1
+
+L1: // n > 0
+ MOVD R0, R11
+ SUBC R4, R11 // restore CF
+ MOVD 0(R8)(R10*1), R5
+ MOVD 0(R9)(R10*1), R11
+ SUBE R11, R5
+ MOVD R5, 0(R2)(R10*1)
+ MOVD R0, R4
+ SUBE R4, R4 // save CF
+
+ ADD $8, R10 // i++
+ SUB $1, R3 // n--
+ BGT L1 // if n > 0 goto L1n
+
+E1:
+ NEG R4, R4
+ MOVD R4, c+72(FP) // return c
+ RET
+
+// DI = R3, CX = R4, SI = r10, r8 = r8, r9=r9, r10 = r2 , r11 = r5, r12 = r6, r13 = r7, r14 = r1 (R0 set to 0) + use R11
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SUBC/SUBE instead of ADDC/ADDE and label names)
+TEXT ·subVV_novec(SB), NOSPLIT, $0
+ MOVD z_len+8(FP), R3
+ MOVD x+24(FP), R8
+ MOVD y+48(FP), R9
+ MOVD z+0(FP), R2
+
+ MOVD $0, R4 // c = 0
+ MOVD $0, R0 // make sure it's zero
+ MOVD $0, R10 // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUB $4, R3 // n -= 4
+ BLT v1 // if n < 0 goto v1
+
+U1: // n >= 0
+ // regular loop body unrolled 4x
+ MOVD 0(R8)(R10*1), R5
+ MOVD 8(R8)(R10*1), R6
+ MOVD 16(R8)(R10*1), R7
+ MOVD 24(R8)(R10*1), R1
+ MOVD R0, R11
+ SUBC R4, R11 // restore CF
+ MOVD 0(R9)(R10*1), R11
+ SUBE R11, R5
+ MOVD 8(R9)(R10*1), R11
+ SUBE R11, R6
+ MOVD 16(R9)(R10*1), R11
+ SUBE R11, R7
+ MOVD 24(R9)(R10*1), R11
+ SUBE R11, R1
+ MOVD R0, R4
+ SUBE R4, R4 // save CF
+ MOVD R5, 0(R2)(R10*1)
+ MOVD R6, 8(R2)(R10*1)
+ MOVD R7, 16(R2)(R10*1)
+ MOVD R1, 24(R2)(R10*1)
+
+ ADD $32, R10 // i += 4
+ SUB $4, R3 // n -= 4
+ BGE U1 // if n >= 0 goto U1
+
+v1:
+ ADD $4, R3 // n += 4
+ BLE E1 // if n <= 0 goto E1
+
+L1: // n > 0
+ MOVD R0, R11
+ SUBC R4, R11 // restore CF
+ MOVD 0(R8)(R10*1), R5
+ MOVD 0(R9)(R10*1), R11
+ SUBE R11, R5
+ MOVD R5, 0(R2)(R10*1)
+ MOVD R0, R4
+ SUBE R4, R4 // save CF
+
+ ADD $8, R10 // i++
+ SUB $1, R3 // n--
+ BGT L1 // if n > 0 goto L1
+
+E1:
+ NEG R4, R4
+ MOVD R4, c+72(FP) // return c
+ RET
+
+TEXT ·addVW(SB), NOSPLIT, $0
+ MOVD z_len+8(FP), R5 // length of z
+ MOVD x+24(FP), R6
+ MOVD y+48(FP), R7 // c = y
+ MOVD z+0(FP), R8
+
+ CMPBEQ R5, $0, returnC // if len(z) == 0, we can have an early return
+
+ // Add the first two words, and determine which path (copy path or loop path) to take based on the carry flag.
+ ADDC 0(R6), R7
+ MOVD R7, 0(R8)
+ CMPBEQ R5, $1, returnResult // len(z) == 1
+ MOVD $0, R9
+ ADDE 8(R6), R9
+ MOVD R9, 8(R8)
+ CMPBEQ R5, $2, returnResult // len(z) == 2
+
+ // Update the counters
+ MOVD $16, R12 // i = 2
+ MOVD $-2(R5), R5 // n = n - 2
+
+loopOverEachWord:
+ BRC $12, copySetup // carry = 0, copy the rest
+ MOVD $1, R9
+
+ // Originally we used the carry flag generated in the previous iteration
+ // (i.e: ADDE could be used here to do the addition). However, since we
+ // already know carry is 1 (otherwise we will go to copy section), we can use
+ // ADDC here so the current iteration does not depend on the carry flag
+ // generated in the previous iteration. This could be useful when branch prediction happens.
+ ADDC 0(R6)(R12*1), R9
+ MOVD R9, 0(R8)(R12*1) // z[i] = x[i] + c
+
+ MOVD $8(R12), R12 // i++
+ BRCTG R5, loopOverEachWord // n--
+
+// Return the current carry value
+returnResult:
+ MOVD $0, R0
+ ADDE R0, R0
+ MOVD R0, c+56(FP)
+ RET
+
+// Update position of x(R6) and z(R8) based on the current counter value and perform copying.
+// With the assumption that x and z will not overlap with each other or x and z will
+// point to same memory region, we can use a faster version of copy using only MVC here.
+// In the following implementation, we have three copy loops, each copying a word, 4 words, and
+// 32 words at a time. Via benchmarking, this implementation is faster than calling runtime·memmove.
+copySetup:
+ ADD R12, R6
+ ADD R12, R8
+
+ CMPBGE R5, $4, mediumLoop
+
+smallLoop: // does a loop unrolling to copy word when n < 4
+ CMPBEQ R5, $0, returnZero
+ MVC $8, 0(R6), 0(R8)
+ CMPBEQ R5, $1, returnZero
+ MVC $8, 8(R6), 8(R8)
+ CMPBEQ R5, $2, returnZero
+ MVC $8, 16(R6), 16(R8)
+
+returnZero:
+ MOVD $0, c+56(FP) // return 0 as carry
+ RET
+
+mediumLoop:
+ CMPBLT R5, $4, smallLoop
+ CMPBLT R5, $32, mediumLoopBody
+
+largeLoop: // Copying 256 bytes at a time.
+ MVC $256, 0(R6), 0(R8)
+ MOVD $256(R6), R6
+ MOVD $256(R8), R8
+ MOVD $-32(R5), R5
+ CMPBGE R5, $32, largeLoop
+ BR mediumLoop
+
+mediumLoopBody: // Copying 32 bytes at a time
+ MVC $32, 0(R6), 0(R8)
+ MOVD $32(R6), R6
+ MOVD $32(R8), R8
+ MOVD $-4(R5), R5
+ CMPBGE R5, $4, mediumLoopBody
+ BR smallLoop
+
+returnC:
+ MOVD R7, c+56(FP)
+ RET
+
+TEXT ·subVW(SB), NOSPLIT, $0
+ MOVD z_len+8(FP), R5
+ MOVD x+24(FP), R6
+ MOVD y+48(FP), R7 // The borrow bit passed in
+ MOVD z+0(FP), R8
+ MOVD $0, R0 // R0 is a temporary variable used during computation. Ensure it has zero in it.
+
+ CMPBEQ R5, $0, returnC // len(z) == 0, have an early return
+
+ // Subtract the first two words, and determine which path (copy path or loop path) to take based on the borrow flag
+ MOVD 0(R6), R9
+ SUBC R7, R9
+ MOVD R9, 0(R8)
+ CMPBEQ R5, $1, returnResult
+ MOVD 8(R6), R9
+ SUBE R0, R9
+ MOVD R9, 8(R8)
+ CMPBEQ R5, $2, returnResult
+
+ // Update the counters
+ MOVD $16, R12 // i = 2
+ MOVD $-2(R5), R5 // n = n - 2
+
+loopOverEachWord:
+ BRC $3, copySetup // no borrow, copy the rest
+ MOVD 0(R6)(R12*1), R9
+
+ // Originally we used the borrow flag generated in the previous iteration
+ // (i.e: SUBE could be used here to do the subtraction). However, since we
+ // already know borrow is 1 (otherwise we will go to copy section), we can
+ // use SUBC here so the current iteration does not depend on the borrow flag
+ // generated in the previous iteration. This could be useful when branch prediction happens.
+ SUBC $1, R9
+ MOVD R9, 0(R8)(R12*1) // z[i] = x[i] - 1
+
+ MOVD $8(R12), R12 // i++
+ BRCTG R5, loopOverEachWord // n--
+
+// return the current borrow value
+returnResult:
+ SUBE R0, R0
+ NEG R0, R0
+ MOVD R0, c+56(FP)
+ RET
+
+// Update position of x(R6) and z(R8) based on the current counter value and perform copying.
+// With the assumption that x and z will not overlap with each other or x and z will
+// point to same memory region, we can use a faster version of copy using only MVC here.
+// In the following implementation, we have three copy loops, each copying a word, 4 words, and
+// 32 words at a time. Via benchmarking, this implementation is faster than calling runtime·memmove.
+copySetup:
+ ADD R12, R6
+ ADD R12, R8
+
+ CMPBGE R5, $4, mediumLoop
+
+smallLoop: // does a loop unrolling to copy word when n < 4
+ CMPBEQ R5, $0, returnZero
+ MVC $8, 0(R6), 0(R8)
+ CMPBEQ R5, $1, returnZero
+ MVC $8, 8(R6), 8(R8)
+ CMPBEQ R5, $2, returnZero
+ MVC $8, 16(R6), 16(R8)
+
+returnZero:
+ MOVD $0, c+56(FP) // return 0 as borrow
+ RET
+
+mediumLoop:
+ CMPBLT R5, $4, smallLoop
+ CMPBLT R5, $32, mediumLoopBody
+
+largeLoop: // Copying 256 bytes at a time
+ MVC $256, 0(R6), 0(R8)
+ MOVD $256(R6), R6
+ MOVD $256(R8), R8
+ MOVD $-32(R5), R5
+ CMPBGE R5, $32, largeLoop
+ BR mediumLoop
+
+mediumLoopBody: // Copying 32 bytes at a time
+ MVC $32, 0(R6), 0(R8)
+ MOVD $32(R6), R6
+ MOVD $32(R8), R8
+ MOVD $-4(R5), R5
+ CMPBGE R5, $4, mediumLoopBody
+ BR smallLoop
+
+returnC:
+ MOVD R7, c+56(FP)
+ RET
+
+// func shlVU(z, x []Word, s uint) (c Word)
+TEXT ·shlVU(SB), NOSPLIT, $0
+ BR ·shlVU_g(SB)
+
+// func shrVU(z, x []Word, s uint) (c Word)
+TEXT ·shrVU(SB), NOSPLIT, $0
+ BR ·shrVU_g(SB)
+
+// CX = R4, r8 = r8, r9=r9, r10 = r2 , r11 = r5, DX = r3, AX = r6 , BX = R1 , (R0 set to 0) + use R11 + use R7 for i
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB), NOSPLIT, $0
+ MOVD z+0(FP), R2
+ MOVD x+24(FP), R8
+ MOVD y+48(FP), R9
+ MOVD r+56(FP), R4 // c = r
+ MOVD z_len+8(FP), R5
+ MOVD $0, R1 // i = 0
+ MOVD $0, R7 // i*8 = 0
+ MOVD $0, R0 // make sure it's zero
+ BR E5
+
+L5:
+ MOVD (R8)(R1*1), R6
+ MULHDU R9, R6
+ ADDC R4, R11 // add to low order bits
+ ADDE R0, R6
+ MOVD R11, (R2)(R1*1)
+ MOVD R6, R4
+ ADD $8, R1 // i*8 + 8
+ ADD $1, R7 // i++
+
+E5:
+ CMPBLT R7, R5, L5 // i < n
+
+ MOVD R4, c+64(FP)
+ RET
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+// CX = R4, r8 = r8, r9=r9, r10 = r2 , r11 = r5, AX = r11, DX = R6, r12=r12, BX = R1 , (R0 set to 0) + use R11 + use R7 for i
+TEXT ·addMulVVW(SB), NOSPLIT, $0
+ MOVD z+0(FP), R2
+ MOVD x+24(FP), R8
+ MOVD y+48(FP), R9
+ MOVD z_len+8(FP), R5
+
+ MOVD $0, R1 // i*8 = 0
+ MOVD $0, R7 // i = 0
+ MOVD $0, R0 // make sure it's zero
+ MOVD $0, R4 // c = 0
+
+ MOVD R5, R12
+ AND $-2, R12
+ CMPBGE R5, $2, A6
+ BR E6
+
+A6:
+ MOVD (R8)(R1*1), R6
+ MULHDU R9, R6
+ MOVD (R2)(R1*1), R10
+ ADDC R10, R11 // add to low order bits
+ ADDE R0, R6
+ ADDC R4, R11
+ ADDE R0, R6
+ MOVD R6, R4
+ MOVD R11, (R2)(R1*1)
+
+ MOVD (8)(R8)(R1*1), R6
+ MULHDU R9, R6
+ MOVD (8)(R2)(R1*1), R10
+ ADDC R10, R11 // add to low order bits
+ ADDE R0, R6
+ ADDC R4, R11
+ ADDE R0, R6
+ MOVD R6, R4
+ MOVD R11, (8)(R2)(R1*1)
+
+ ADD $16, R1 // i*8 + 8
+ ADD $2, R7 // i++
+
+ CMPBLT R7, R12, A6
+ BR E6
+
+L6:
+ MOVD (R8)(R1*1), R6
+ MULHDU R9, R6
+ MOVD (R2)(R1*1), R10
+ ADDC R10, R11 // add to low order bits
+ ADDE R0, R6
+ ADDC R4, R11
+ ADDE R0, R6
+ MOVD R6, R4
+ MOVD R11, (R2)(R1*1)
+
+ ADD $8, R1 // i*8 + 8
+ ADD $1, R7 // i++
+
+E6:
+ CMPBLT R7, R5, L6 // i < n
+
+ MOVD R4, c+56(FP)
+ RET
+
diff --git a/src/math/big/arith_s390x_test.go b/src/math/big/arith_s390x_test.go
new file mode 100644
index 0000000..ce6bca8
--- /dev/null
+++ b/src/math/big/arith_s390x_test.go
@@ -0,0 +1,32 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build s390x,!math_big_pure_go
+
+package big
+
+import (
+ "testing"
+)
+
+// Tests whether the non vector routines are working, even when the tests are run on a
+// vector-capable machine
+
+func TestFunVVnovec(t *testing.T) {
+ if hasVX == true {
+ for _, a := range sumVV {
+ arg := a
+ testFunVV(t, "addVV_novec", addVV_novec, arg)
+
+ arg = argVV{a.z, a.y, a.x, a.c}
+ testFunVV(t, "addVV_novec symmetric", addVV_novec, arg)
+
+ arg = argVV{a.x, a.z, a.y, a.c}
+ testFunVV(t, "subVV_novec", subVV_novec, arg)
+
+ arg = argVV{a.y, a.z, a.x, a.c}
+ testFunVV(t, "subVV_novec symmetric", subVV_novec, arg)
+ }
+ }
+}
diff --git a/src/math/big/arith_test.go b/src/math/big/arith_test.go
new file mode 100644
index 0000000..2aca0ef
--- /dev/null
+++ b/src/math/big/arith_test.go
@@ -0,0 +1,673 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "fmt"
+ "internal/testenv"
+ "math/bits"
+ "math/rand"
+ "strings"
+ "testing"
+)
+
+var isRaceBuilder = strings.HasSuffix(testenv.Builder(), "-race")
+
+type funVV func(z, x, y []Word) (c Word)
+type argVV struct {
+ z, x, y nat
+ c Word
+}
+
+var sumVV = []argVV{
+ {},
+ {nat{0}, nat{0}, nat{0}, 0},
+ {nat{1}, nat{1}, nat{0}, 0},
+ {nat{0}, nat{_M}, nat{1}, 1},
+ {nat{80235}, nat{12345}, nat{67890}, 0},
+ {nat{_M - 1}, nat{_M}, nat{_M}, 1},
+ {nat{0, 0, 0, 0}, nat{_M, _M, _M, _M}, nat{1, 0, 0, 0}, 1},
+ {nat{0, 0, 0, _M}, nat{_M, _M, _M, _M - 1}, nat{1, 0, 0, 0}, 0},
+ {nat{0, 0, 0, 0}, nat{_M, 0, _M, 0}, nat{1, _M, 0, _M}, 1},
+}
+
+func testFunVV(t *testing.T, msg string, f funVV, a argVV) {
+ z := make(nat, len(a.z))
+ c := f(z, a.x, a.y)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if c != a.c {
+ t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+ }
+}
+
+func TestFunVV(t *testing.T) {
+ for _, a := range sumVV {
+ arg := a
+ testFunVV(t, "addVV_g", addVV_g, arg)
+ testFunVV(t, "addVV", addVV, arg)
+
+ arg = argVV{a.z, a.y, a.x, a.c}
+ testFunVV(t, "addVV_g symmetric", addVV_g, arg)
+ testFunVV(t, "addVV symmetric", addVV, arg)
+
+ arg = argVV{a.x, a.z, a.y, a.c}
+ testFunVV(t, "subVV_g", subVV_g, arg)
+ testFunVV(t, "subVV", subVV, arg)
+
+ arg = argVV{a.y, a.z, a.x, a.c}
+ testFunVV(t, "subVV_g symmetric", subVV_g, arg)
+ testFunVV(t, "subVV symmetric", subVV, arg)
+ }
+}
+
+// Always the same seed for reproducible results.
+var rnd = rand.New(rand.NewSource(0))
+
+func rndW() Word {
+ return Word(rnd.Int63()<<1 | rnd.Int63n(2))
+}
+
+func rndV(n int) []Word {
+ v := make([]Word, n)
+ for i := range v {
+ v[i] = rndW()
+ }
+ return v
+}
+
+var benchSizes = []int{1, 2, 3, 4, 5, 1e1, 1e2, 1e3, 1e4, 1e5}
+
+func BenchmarkAddVV(b *testing.B) {
+ for _, n := range benchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ x := rndV(n)
+ y := rndV(n)
+ z := make([]Word, n)
+ b.Run(fmt.Sprint(n), func(b *testing.B) {
+ b.SetBytes(int64(n * _W))
+ for i := 0; i < b.N; i++ {
+ addVV(z, x, y)
+ }
+ })
+ }
+}
+
+func BenchmarkSubVV(b *testing.B) {
+ for _, n := range benchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ x := rndV(n)
+ y := rndV(n)
+ z := make([]Word, n)
+ b.Run(fmt.Sprint(n), func(b *testing.B) {
+ b.SetBytes(int64(n * _W))
+ for i := 0; i < b.N; i++ {
+ subVV(z, x, y)
+ }
+ })
+ }
+}
+
+type funVW func(z, x []Word, y Word) (c Word)
+type argVW struct {
+ z, x nat
+ y Word
+ c Word
+}
+
+var sumVW = []argVW{
+ {},
+ {nil, nil, 2, 2},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{1}, nat{0}, 1, 0},
+ {nat{1}, nat{1}, 0, 0},
+ {nat{0}, nat{_M}, 1, 1},
+ {nat{0, 0, 0, 0}, nat{_M, _M, _M, _M}, 1, 1},
+ {nat{585}, nat{314}, 271, 0},
+}
+
+var lshVW = []argVW{
+ {},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{0}, nat{0}, 1, 0},
+ {nat{0}, nat{0}, 20, 0},
+
+ {nat{_M}, nat{_M}, 0, 0},
+ {nat{_M << 1 & _M}, nat{_M}, 1, 1},
+ {nat{_M << 20 & _M}, nat{_M}, 20, _M >> (_W - 20)},
+
+ {nat{_M, _M, _M}, nat{_M, _M, _M}, 0, 0},
+ {nat{_M << 1 & _M, _M, _M}, nat{_M, _M, _M}, 1, 1},
+ {nat{_M << 20 & _M, _M, _M}, nat{_M, _M, _M}, 20, _M >> (_W - 20)},
+}
+
+var rshVW = []argVW{
+ {},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{0}, nat{0}, 1, 0},
+ {nat{0}, nat{0}, 20, 0},
+
+ {nat{_M}, nat{_M}, 0, 0},
+ {nat{_M >> 1}, nat{_M}, 1, _M << (_W - 1) & _M},
+ {nat{_M >> 20}, nat{_M}, 20, _M << (_W - 20) & _M},
+
+ {nat{_M, _M, _M}, nat{_M, _M, _M}, 0, 0},
+ {nat{_M, _M, _M >> 1}, nat{_M, _M, _M}, 1, _M << (_W - 1) & _M},
+ {nat{_M, _M, _M >> 20}, nat{_M, _M, _M}, 20, _M << (_W - 20) & _M},
+}
+
+func testFunVW(t *testing.T, msg string, f funVW, a argVW) {
+ z := make(nat, len(a.z))
+ c := f(z, a.x, a.y)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if c != a.c {
+ t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+ }
+}
+
+func testFunVWext(t *testing.T, msg string, f funVW, f_g funVW, a argVW) {
+ // using the result of addVW_g/subVW_g as golden
+ z_g := make(nat, len(a.z))
+ c_g := f_g(z_g, a.x, a.y)
+ c := f(a.z, a.x, a.y)
+
+ for i, zi := range a.z {
+ if zi != z_g[i] {
+ t.Errorf("%s\n\tgot z[%d] = %#x; want %#x", msg, i, zi, z_g[i])
+ break
+ }
+ }
+ if c != c_g {
+ t.Errorf("%s\n\tgot c = %#x; want %#x", msg, c, c_g)
+ }
+}
+
+func makeFunVW(f func(z, x []Word, s uint) (c Word)) funVW {
+ return func(z, x []Word, s Word) (c Word) {
+ return f(z, x, uint(s))
+ }
+}
+
+func TestFunVW(t *testing.T) {
+ for _, a := range sumVW {
+ arg := a
+ testFunVW(t, "addVW_g", addVW_g, arg)
+ testFunVW(t, "addVW", addVW, arg)
+
+ arg = argVW{a.x, a.z, a.y, a.c}
+ testFunVW(t, "subVW_g", subVW_g, arg)
+ testFunVW(t, "subVW", subVW, arg)
+ }
+
+ shlVW_g := makeFunVW(shlVU_g)
+ shlVW := makeFunVW(shlVU)
+ for _, a := range lshVW {
+ arg := a
+ testFunVW(t, "shlVU_g", shlVW_g, arg)
+ testFunVW(t, "shlVU", shlVW, arg)
+ }
+
+ shrVW_g := makeFunVW(shrVU_g)
+ shrVW := makeFunVW(shrVU)
+ for _, a := range rshVW {
+ arg := a
+ testFunVW(t, "shrVU_g", shrVW_g, arg)
+ testFunVW(t, "shrVU", shrVW, arg)
+ }
+}
+
+// Construct a vector comprising the same word, usually '0' or 'maximum uint'
+func makeWordVec(e Word, n int) []Word {
+ v := make([]Word, n)
+ for i := range v {
+ v[i] = e
+ }
+ return v
+}
+
+// Extended testing to addVW and subVW using various kinds of input data.
+// We utilize the results of addVW_g and subVW_g as golden reference to check
+// correctness.
+func TestFunVWExt(t *testing.T) {
+ // 32 is the current threshold that triggers an optimized version of
+ // calculation for large-sized vector, ensure we have sizes around it tested.
+ var vwSizes = []int{0, 1, 3, 4, 5, 8, 9, 23, 31, 32, 33, 34, 35, 36, 50, 120}
+ for _, n := range vwSizes {
+ // vector of random numbers, using the result of addVW_g/subVW_g as golden
+ x := rndV(n)
+ y := rndW()
+ z := make(nat, n)
+ arg := argVW{z, x, y, 0}
+ testFunVWext(t, "addVW, random inputs", addVW, addVW_g, arg)
+ testFunVWext(t, "subVW, random inputs", subVW, subVW_g, arg)
+
+ // vector of random numbers, but make 'x' and 'z' share storage
+ arg = argVW{x, x, y, 0}
+ testFunVWext(t, "addVW, random inputs, sharing storage", addVW, addVW_g, arg)
+ testFunVWext(t, "subVW, random inputs, sharing storage", subVW, subVW_g, arg)
+
+ // vector of maximum uint, to force carry flag set in each 'add'
+ y = ^Word(0)
+ x = makeWordVec(y, n)
+ arg = argVW{z, x, y, 0}
+ testFunVWext(t, "addVW, vector of max uint", addVW, addVW_g, arg)
+
+ // vector of '0', to force carry flag set in each 'sub'
+ x = makeWordVec(0, n)
+ arg = argVW{z, x, 1, 0}
+ testFunVWext(t, "subVW, vector of zero", subVW, subVW_g, arg)
+ }
+}
+
+type argVU struct {
+ d []Word // d is a Word slice, the input parameters x and z come from this array.
+ l uint // l is the length of the input parameters x and z.
+ xp uint // xp is the starting position of the input parameter x, x := d[xp:xp+l].
+ zp uint // zp is the starting position of the input parameter z, z := d[zp:zp+l].
+ s uint // s is the shift number.
+ r []Word // r is the expected output result z.
+ c Word // c is the expected return value.
+ m string // message.
+}
+
+var argshlVUIn = []Word{1, 2, 4, 8, 16, 32, 64, 0, 0, 0}
+var argshlVUr0 = []Word{1, 2, 4, 8, 16, 32, 64}
+var argshlVUr1 = []Word{2, 4, 8, 16, 32, 64, 128}
+var argshlVUrWm1 = []Word{1 << (_W - 1), 0, 1, 2, 4, 8, 16}
+
+var argshlVU = []argVU{
+ // test cases for shlVU
+ {[]Word{1, _M, _M, _M, _M, _M, 3 << (_W - 2), 0}, 7, 0, 0, 1, []Word{2, _M - 1, _M, _M, _M, _M, 1<<(_W-1) + 1}, 1, "complete overlap of shlVU"},
+ {[]Word{1, _M, _M, _M, _M, _M, 3 << (_W - 2), 0, 0, 0, 0}, 7, 0, 3, 1, []Word{2, _M - 1, _M, _M, _M, _M, 1<<(_W-1) + 1}, 1, "partial overlap by half of shlVU"},
+ {[]Word{1, _M, _M, _M, _M, _M, 3 << (_W - 2), 0, 0, 0, 0, 0, 0, 0}, 7, 0, 6, 1, []Word{2, _M - 1, _M, _M, _M, _M, 1<<(_W-1) + 1}, 1, "partial overlap by 1 Word of shlVU"},
+ {[]Word{1, _M, _M, _M, _M, _M, 3 << (_W - 2), 0, 0, 0, 0, 0, 0, 0, 0}, 7, 0, 7, 1, []Word{2, _M - 1, _M, _M, _M, _M, 1<<(_W-1) + 1}, 1, "no overlap of shlVU"},
+ // additional test cases with shift values of 0, 1 and (_W-1)
+ {argshlVUIn, 7, 0, 0, 0, argshlVUr0, 0, "complete overlap of shlVU and shift of 0"},
+ {argshlVUIn, 7, 0, 0, 1, argshlVUr1, 0, "complete overlap of shlVU and shift of 1"},
+ {argshlVUIn, 7, 0, 0, _W - 1, argshlVUrWm1, 32, "complete overlap of shlVU and shift of _W - 1"},
+ {argshlVUIn, 7, 0, 1, 0, argshlVUr0, 0, "partial overlap by 6 Words of shlVU and shift of 0"},
+ {argshlVUIn, 7, 0, 1, 1, argshlVUr1, 0, "partial overlap by 6 Words of shlVU and shift of 1"},
+ {argshlVUIn, 7, 0, 1, _W - 1, argshlVUrWm1, 32, "partial overlap by 6 Words of shlVU and shift of _W - 1"},
+ {argshlVUIn, 7, 0, 2, 0, argshlVUr0, 0, "partial overlap by 5 Words of shlVU and shift of 0"},
+ {argshlVUIn, 7, 0, 2, 1, argshlVUr1, 0, "partial overlap by 5 Words of shlVU and shift of 1"},
+ {argshlVUIn, 7, 0, 2, _W - 1, argshlVUrWm1, 32, "partial overlap by 5 Words of shlVU abd shift of _W - 1"},
+ {argshlVUIn, 7, 0, 3, 0, argshlVUr0, 0, "partial overlap by 4 Words of shlVU and shift of 0"},
+ {argshlVUIn, 7, 0, 3, 1, argshlVUr1, 0, "partial overlap by 4 Words of shlVU and shift of 1"},
+ {argshlVUIn, 7, 0, 3, _W - 1, argshlVUrWm1, 32, "partial overlap by 4 Words of shlVU and shift of _W - 1"},
+}
+
+var argshrVUIn = []Word{0, 0, 0, 1, 2, 4, 8, 16, 32, 64}
+var argshrVUr0 = []Word{1, 2, 4, 8, 16, 32, 64}
+var argshrVUr1 = []Word{0, 1, 2, 4, 8, 16, 32}
+var argshrVUrWm1 = []Word{4, 8, 16, 32, 64, 128, 0}
+
+var argshrVU = []argVU{
+ // test cases for shrVU
+ {[]Word{0, 3, _M, _M, _M, _M, _M, 1 << (_W - 1)}, 7, 1, 1, 1, []Word{1<<(_W-1) + 1, _M, _M, _M, _M, _M >> 1, 1 << (_W - 2)}, 1 << (_W - 1), "complete overlap of shrVU"},
+ {[]Word{0, 0, 0, 0, 3, _M, _M, _M, _M, _M, 1 << (_W - 1)}, 7, 4, 1, 1, []Word{1<<(_W-1) + 1, _M, _M, _M, _M, _M >> 1, 1 << (_W - 2)}, 1 << (_W - 1), "partial overlap by half of shrVU"},
+ {[]Word{0, 0, 0, 0, 0, 0, 0, 3, _M, _M, _M, _M, _M, 1 << (_W - 1)}, 7, 7, 1, 1, []Word{1<<(_W-1) + 1, _M, _M, _M, _M, _M >> 1, 1 << (_W - 2)}, 1 << (_W - 1), "partial overlap by 1 Word of shrVU"},
+ {[]Word{0, 0, 0, 0, 0, 0, 0, 0, 3, _M, _M, _M, _M, _M, 1 << (_W - 1)}, 7, 8, 1, 1, []Word{1<<(_W-1) + 1, _M, _M, _M, _M, _M >> 1, 1 << (_W - 2)}, 1 << (_W - 1), "no overlap of shrVU"},
+ // additional test cases with shift values of 0, 1 and (_W-1)
+ {argshrVUIn, 7, 3, 3, 0, argshrVUr0, 0, "complete overlap of shrVU and shift of 0"},
+ {argshrVUIn, 7, 3, 3, 1, argshrVUr1, 1 << (_W - 1), "complete overlap of shrVU and shift of 1"},
+ {argshrVUIn, 7, 3, 3, _W - 1, argshrVUrWm1, 2, "complete overlap of shrVU and shift of _W - 1"},
+ {argshrVUIn, 7, 3, 2, 0, argshrVUr0, 0, "partial overlap by 6 Words of shrVU and shift of 0"},
+ {argshrVUIn, 7, 3, 2, 1, argshrVUr1, 1 << (_W - 1), "partial overlap by 6 Words of shrVU and shift of 1"},
+ {argshrVUIn, 7, 3, 2, _W - 1, argshrVUrWm1, 2, "partial overlap by 6 Words of shrVU and shift of _W - 1"},
+ {argshrVUIn, 7, 3, 1, 0, argshrVUr0, 0, "partial overlap by 5 Words of shrVU and shift of 0"},
+ {argshrVUIn, 7, 3, 1, 1, argshrVUr1, 1 << (_W - 1), "partial overlap by 5 Words of shrVU and shift of 1"},
+ {argshrVUIn, 7, 3, 1, _W - 1, argshrVUrWm1, 2, "partial overlap by 5 Words of shrVU and shift of _W - 1"},
+ {argshrVUIn, 7, 3, 0, 0, argshrVUr0, 0, "partial overlap by 4 Words of shrVU and shift of 0"},
+ {argshrVUIn, 7, 3, 0, 1, argshrVUr1, 1 << (_W - 1), "partial overlap by 4 Words of shrVU and shift of 1"},
+ {argshrVUIn, 7, 3, 0, _W - 1, argshrVUrWm1, 2, "partial overlap by 4 Words of shrVU and shift of _W - 1"},
+}
+
+func testShiftFunc(t *testing.T, f func(z, x []Word, s uint) Word, a argVU) {
+ // work on copy of a.d to preserve the original data.
+ b := make([]Word, len(a.d))
+ copy(b, a.d)
+ z := b[a.zp : a.zp+a.l]
+ x := b[a.xp : a.xp+a.l]
+ c := f(z, x, a.s)
+ for i, zi := range z {
+ if zi != a.r[i] {
+ t.Errorf("d := %v, %s(d[%d:%d], d[%d:%d], %d)\n\tgot z[%d] = %#x; want %#x", a.d, a.m, a.zp, a.zp+a.l, a.xp, a.xp+a.l, a.s, i, zi, a.r[i])
+ break
+ }
+ }
+ if c != a.c {
+ t.Errorf("d := %v, %s(d[%d:%d], d[%d:%d], %d)\n\tgot c = %#x; want %#x", a.d, a.m, a.zp, a.zp+a.l, a.xp, a.xp+a.l, a.s, c, a.c)
+ }
+}
+
+func TestShiftOverlap(t *testing.T) {
+ for _, a := range argshlVU {
+ arg := a
+ testShiftFunc(t, shlVU, arg)
+ }
+
+ for _, a := range argshrVU {
+ arg := a
+ testShiftFunc(t, shrVU, arg)
+ }
+}
+
+func TestIssue31084(t *testing.T) {
+ // compute 10^n via 5^n << n.
+ const n = 165
+ p := nat(nil).expNN(nat{5}, nat{n}, nil)
+ p = p.shl(p, n)
+ got := string(p.utoa(10))
+ want := "1" + strings.Repeat("0", n)
+ if got != want {
+ t.Errorf("shl(%v, %v)\n\tgot %s\n\twant %s", p, n, got, want)
+ }
+}
+
+const issue42838Value = "159309191113245227702888039776771180559110455519261878607388585338616290151305816094308987472018268594098344692611135542392730712890625"
+
+func TestIssue42838(t *testing.T) {
+ const s = 192
+ z, _, _, _ := nat(nil).scan(strings.NewReader(issue42838Value), 0, false)
+ z = z.shl(z, s)
+ got := string(z.utoa(10))
+ want := "1" + strings.Repeat("0", s)
+ if got != want {
+ t.Errorf("shl(%v, %v)\n\tgot %s\n\twant %s", z, s, got, want)
+ }
+}
+
+func BenchmarkAddVW(b *testing.B) {
+ for _, n := range benchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ x := rndV(n)
+ y := rndW()
+ z := make([]Word, n)
+ b.Run(fmt.Sprint(n), func(b *testing.B) {
+ b.SetBytes(int64(n * _S))
+ for i := 0; i < b.N; i++ {
+ addVW(z, x, y)
+ }
+ })
+ }
+}
+
+// Benchmarking addVW using vector of maximum uint to force carry flag set
+func BenchmarkAddVWext(b *testing.B) {
+ for _, n := range benchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ y := ^Word(0)
+ x := makeWordVec(y, n)
+ z := make([]Word, n)
+ b.Run(fmt.Sprint(n), func(b *testing.B) {
+ b.SetBytes(int64(n * _S))
+ for i := 0; i < b.N; i++ {
+ addVW(z, x, y)
+ }
+ })
+ }
+}
+
+func BenchmarkSubVW(b *testing.B) {
+ for _, n := range benchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ x := rndV(n)
+ y := rndW()
+ z := make([]Word, n)
+ b.Run(fmt.Sprint(n), func(b *testing.B) {
+ b.SetBytes(int64(n * _S))
+ for i := 0; i < b.N; i++ {
+ subVW(z, x, y)
+ }
+ })
+ }
+}
+
+// Benchmarking subVW using vector of zero to force carry flag set
+func BenchmarkSubVWext(b *testing.B) {
+ for _, n := range benchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ x := makeWordVec(0, n)
+ y := Word(1)
+ z := make([]Word, n)
+ b.Run(fmt.Sprint(n), func(b *testing.B) {
+ b.SetBytes(int64(n * _S))
+ for i := 0; i < b.N; i++ {
+ subVW(z, x, y)
+ }
+ })
+ }
+}
+
+type funVWW func(z, x []Word, y, r Word) (c Word)
+type argVWW struct {
+ z, x nat
+ y, r Word
+ c Word
+}
+
+var prodVWW = []argVWW{
+ {},
+ {nat{0}, nat{0}, 0, 0, 0},
+ {nat{991}, nat{0}, 0, 991, 0},
+ {nat{0}, nat{_M}, 0, 0, 0},
+ {nat{991}, nat{_M}, 0, 991, 0},
+ {nat{0}, nat{0}, _M, 0, 0},
+ {nat{991}, nat{0}, _M, 991, 0},
+ {nat{1}, nat{1}, 1, 0, 0},
+ {nat{992}, nat{1}, 1, 991, 0},
+ {nat{22793}, nat{991}, 23, 0, 0},
+ {nat{22800}, nat{991}, 23, 7, 0},
+ {nat{0, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 0, 0},
+ {nat{7, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 7, 0},
+ {nat{0, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 0, 0},
+ {nat{991, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 991, 0},
+ {nat{0, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 0, 0},
+ {nat{991, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 991, 0},
+ {nat{_M << 1 & _M}, nat{_M}, 1 << 1, 0, _M >> (_W - 1)},
+ {nat{_M<<1&_M + 1}, nat{_M}, 1 << 1, 1, _M >> (_W - 1)},
+ {nat{_M << 7 & _M}, nat{_M}, 1 << 7, 0, _M >> (_W - 7)},
+ {nat{_M<<7&_M + 1<<6}, nat{_M}, 1 << 7, 1 << 6, _M >> (_W - 7)},
+ {nat{_M << 7 & _M, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, 0, _M >> (_W - 7)},
+ {nat{_M<<7&_M + 1<<6, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, 1 << 6, _M >> (_W - 7)},
+}
+
+func testFunVWW(t *testing.T, msg string, f funVWW, a argVWW) {
+ z := make(nat, len(a.z))
+ c := f(z, a.x, a.y, a.r)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if c != a.c {
+ t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+ }
+}
+
+// TODO(gri) mulAddVWW and divWVW are symmetric operations but
+// their signature is not symmetric. Try to unify.
+
+type funWVW func(z []Word, xn Word, x []Word, y Word) (r Word)
+type argWVW struct {
+ z nat
+ xn Word
+ x nat
+ y Word
+ r Word
+}
+
+func testFunWVW(t *testing.T, msg string, f funWVW, a argWVW) {
+ z := make(nat, len(a.z))
+ r := f(z, a.xn, a.x, a.y)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if r != a.r {
+ t.Errorf("%s%+v\n\tgot r = %#x; want %#x", msg, a, r, a.r)
+ }
+}
+
+func TestFunVWW(t *testing.T) {
+ for _, a := range prodVWW {
+ arg := a
+ testFunVWW(t, "mulAddVWW_g", mulAddVWW_g, arg)
+ testFunVWW(t, "mulAddVWW", mulAddVWW, arg)
+
+ if a.y != 0 && a.r < a.y {
+ arg := argWVW{a.x, a.c, a.z, a.y, a.r}
+ testFunWVW(t, "divWVW", divWVW, arg)
+ }
+ }
+}
+
+var mulWWTests = []struct {
+ x, y Word
+ q, r Word
+}{
+ {_M, _M, _M - 1, 1},
+ // 32 bit only: {0xc47dfa8c, 50911, 0x98a4, 0x998587f4},
+}
+
+func TestMulWW(t *testing.T) {
+ for i, test := range mulWWTests {
+ q, r := mulWW_g(test.x, test.y)
+ if q != test.q || r != test.r {
+ t.Errorf("#%d got (%x, %x) want (%x, %x)", i, q, r, test.q, test.r)
+ }
+ }
+}
+
+var mulAddWWWTests = []struct {
+ x, y, c Word
+ q, r Word
+}{
+ // TODO(agl): These will only work on 64-bit platforms.
+ // {15064310297182388543, 0xe7df04d2d35d5d80, 13537600649892366549, 13644450054494335067, 10832252001440893781},
+ // {15064310297182388543, 0xdab2f18048baa68d, 13644450054494335067, 12869334219691522700, 14233854684711418382},
+ {_M, _M, 0, _M - 1, 1},
+ {_M, _M, _M, _M, 0},
+}
+
+func TestMulAddWWW(t *testing.T) {
+ for i, test := range mulAddWWWTests {
+ q, r := mulAddWWW_g(test.x, test.y, test.c)
+ if q != test.q || r != test.r {
+ t.Errorf("#%d got (%x, %x) want (%x, %x)", i, q, r, test.q, test.r)
+ }
+ }
+}
+
+var divWWTests = []struct {
+ x1, x0, y Word
+ q, r Word
+}{
+ {_M >> 1, 0, _M, _M >> 1, _M >> 1},
+ {_M - (1 << (_W - 2)), _M, 3 << (_W - 2), _M, _M - (1 << (_W - 2))},
+}
+
+const testsNumber = 1 << 16
+
+func TestDivWW(t *testing.T) {
+ i := 0
+ for i, test := range divWWTests {
+ rec := reciprocalWord(test.y)
+ q, r := divWW(test.x1, test.x0, test.y, rec)
+ if q != test.q || r != test.r {
+ t.Errorf("#%d got (%x, %x) want (%x, %x)", i, q, r, test.q, test.r)
+ }
+ }
+ //random tests
+ for ; i < testsNumber; i++ {
+ x1 := rndW()
+ x0 := rndW()
+ y := rndW()
+ if x1 >= y {
+ continue
+ }
+ rec := reciprocalWord(y)
+ qGot, rGot := divWW(x1, x0, y, rec)
+ qWant, rWant := bits.Div(uint(x1), uint(x0), uint(y))
+ if uint(qGot) != qWant || uint(rGot) != rWant {
+ t.Errorf("#%d got (%x, %x) want (%x, %x)", i, qGot, rGot, qWant, rWant)
+ }
+ }
+}
+
+func BenchmarkMulAddVWW(b *testing.B) {
+ for _, n := range benchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ z := make([]Word, n+1)
+ x := rndV(n)
+ y := rndW()
+ r := rndW()
+ b.Run(fmt.Sprint(n), func(b *testing.B) {
+ b.SetBytes(int64(n * _W))
+ for i := 0; i < b.N; i++ {
+ mulAddVWW(z, x, y, r)
+ }
+ })
+ }
+}
+
+func BenchmarkAddMulVVW(b *testing.B) {
+ for _, n := range benchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ x := rndV(n)
+ y := rndW()
+ z := make([]Word, n)
+ b.Run(fmt.Sprint(n), func(b *testing.B) {
+ b.SetBytes(int64(n * _W))
+ for i := 0; i < b.N; i++ {
+ addMulVVW(z, x, y)
+ }
+ })
+ }
+}
+func BenchmarkDivWVW(b *testing.B) {
+ for _, n := range benchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ x := rndV(n)
+ y := rndW()
+ z := make([]Word, n)
+ b.Run(fmt.Sprint(n), func(b *testing.B) {
+ b.SetBytes(int64(n * _W))
+ for i := 0; i < b.N; i++ {
+ divWVW(z, 0, x, y)
+ }
+ })
+ }
+}
diff --git a/src/math/big/arith_wasm.s b/src/math/big/arith_wasm.s
new file mode 100644
index 0000000..add1064
--- /dev/null
+++ b/src/math/big/arith_wasm.s
@@ -0,0 +1,35 @@
+// Copyright 2018 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !math_big_pure_go
+
+#include "textflag.h"
+
+TEXT ·mulWW(SB),NOSPLIT,$0
+ JMP ·mulWW_g(SB)
+
+TEXT ·addVV(SB),NOSPLIT,$0
+ JMP ·addVV_g(SB)
+
+TEXT ·subVV(SB),NOSPLIT,$0
+ JMP ·subVV_g(SB)
+
+TEXT ·addVW(SB),NOSPLIT,$0
+ JMP ·addVW_g(SB)
+
+TEXT ·subVW(SB),NOSPLIT,$0
+ JMP ·subVW_g(SB)
+
+TEXT ·shlVU(SB),NOSPLIT,$0
+ JMP ·shlVU_g(SB)
+
+TEXT ·shrVU(SB),NOSPLIT,$0
+ JMP ·shrVU_g(SB)
+
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ JMP ·mulAddVWW_g(SB)
+
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ JMP ·addMulVVW_g(SB)
+
diff --git a/src/math/big/bits_test.go b/src/math/big/bits_test.go
new file mode 100644
index 0000000..985b60b
--- /dev/null
+++ b/src/math/big/bits_test.go
@@ -0,0 +1,224 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements the Bits type used for testing Float operations
+// via an independent (albeit slower) representations for floating-point
+// numbers.
+
+package big
+
+import (
+ "fmt"
+ "sort"
+ "testing"
+)
+
+// A Bits value b represents a finite floating-point number x of the form
+//
+// x = 2**b[0] + 2**b[1] + ... 2**b[len(b)-1]
+//
+// The order of slice elements is not significant. Negative elements may be
+// used to form fractions. A Bits value is normalized if each b[i] occurs at
+// most once. For instance Bits{0, 0, 1} is not normalized but represents the
+// same floating-point number as Bits{2}, which is normalized. The zero (nil)
+// value of Bits is a ready to use Bits value and represents the value 0.
+type Bits []int
+
+func (x Bits) add(y Bits) Bits {
+ return append(x, y...)
+}
+
+func (x Bits) mul(y Bits) Bits {
+ var p Bits
+ for _, x := range x {
+ for _, y := range y {
+ p = append(p, x+y)
+ }
+ }
+ return p
+}
+
+func TestMulBits(t *testing.T) {
+ for _, test := range []struct {
+ x, y, want Bits
+ }{
+ {nil, nil, nil},
+ {Bits{}, Bits{}, nil},
+ {Bits{0}, Bits{0}, Bits{0}},
+ {Bits{0}, Bits{1}, Bits{1}},
+ {Bits{1}, Bits{1, 2, 3}, Bits{2, 3, 4}},
+ {Bits{-1}, Bits{1}, Bits{0}},
+ {Bits{-10, -1, 0, 1, 10}, Bits{1, 2, 3}, Bits{-9, -8, -7, 0, 1, 2, 1, 2, 3, 2, 3, 4, 11, 12, 13}},
+ } {
+ got := fmt.Sprintf("%v", test.x.mul(test.y))
+ want := fmt.Sprintf("%v", test.want)
+ if got != want {
+ t.Errorf("%v * %v = %s; want %s", test.x, test.y, got, want)
+ }
+
+ }
+}
+
+// norm returns the normalized bits for x: It removes multiple equal entries
+// by treating them as an addition (e.g., Bits{5, 5} => Bits{6}), and it sorts
+// the result list for reproducible results.
+func (x Bits) norm() Bits {
+ m := make(map[int]bool)
+ for _, b := range x {
+ for m[b] {
+ m[b] = false
+ b++
+ }
+ m[b] = true
+ }
+ var z Bits
+ for b, set := range m {
+ if set {
+ z = append(z, b)
+ }
+ }
+ sort.Ints([]int(z))
+ return z
+}
+
+func TestNormBits(t *testing.T) {
+ for _, test := range []struct {
+ x, want Bits
+ }{
+ {nil, nil},
+ {Bits{}, Bits{}},
+ {Bits{0}, Bits{0}},
+ {Bits{0, 0}, Bits{1}},
+ {Bits{3, 1, 1}, Bits{2, 3}},
+ {Bits{10, 9, 8, 7, 6, 6}, Bits{11}},
+ } {
+ got := fmt.Sprintf("%v", test.x.norm())
+ want := fmt.Sprintf("%v", test.want)
+ if got != want {
+ t.Errorf("normBits(%v) = %s; want %s", test.x, got, want)
+ }
+
+ }
+}
+
+// round returns the Float value corresponding to x after rounding x
+// to prec bits according to mode.
+func (x Bits) round(prec uint, mode RoundingMode) *Float {
+ x = x.norm()
+
+ // determine range
+ var min, max int
+ for i, b := range x {
+ if i == 0 || b < min {
+ min = b
+ }
+ if i == 0 || b > max {
+ max = b
+ }
+ }
+ prec0 := uint(max + 1 - min)
+ if prec >= prec0 {
+ return x.Float()
+ }
+ // prec < prec0
+
+ // determine bit 0, rounding, and sticky bit, and result bits z
+ var bit0, rbit, sbit uint
+ var z Bits
+ r := max - int(prec)
+ for _, b := range x {
+ switch {
+ case b == r:
+ rbit = 1
+ case b < r:
+ sbit = 1
+ default:
+ // b > r
+ if b == r+1 {
+ bit0 = 1
+ }
+ z = append(z, b)
+ }
+ }
+
+ // round
+ f := z.Float() // rounded to zero
+ if mode == ToNearestAway {
+ panic("not yet implemented")
+ }
+ if mode == ToNearestEven && rbit == 1 && (sbit == 1 || sbit == 0 && bit0 != 0) || mode == AwayFromZero {
+ // round away from zero
+ f.SetMode(ToZero).SetPrec(prec)
+ f.Add(f, Bits{int(r) + 1}.Float())
+ }
+ return f
+}
+
+// Float returns the *Float z of the smallest possible precision such that
+// z = sum(2**bits[i]), with i = range bits. If multiple bits[i] are equal,
+// they are added: Bits{0, 1, 0}.Float() == 2**0 + 2**1 + 2**0 = 4.
+func (bits Bits) Float() *Float {
+ // handle 0
+ if len(bits) == 0 {
+ return new(Float)
+ }
+ // len(bits) > 0
+
+ // determine lsb exponent
+ var min int
+ for i, b := range bits {
+ if i == 0 || b < min {
+ min = b
+ }
+ }
+
+ // create bit pattern
+ x := NewInt(0)
+ for _, b := range bits {
+ badj := b - min
+ // propagate carry if necessary
+ for x.Bit(badj) != 0 {
+ x.SetBit(x, badj, 0)
+ badj++
+ }
+ x.SetBit(x, badj, 1)
+ }
+
+ // create corresponding float
+ z := new(Float).SetInt(x) // normalized
+ if e := int64(z.exp) + int64(min); MinExp <= e && e <= MaxExp {
+ z.exp = int32(e)
+ } else {
+ // this should never happen for our test cases
+ panic("exponent out of range")
+ }
+ return z
+}
+
+func TestFromBits(t *testing.T) {
+ for _, test := range []struct {
+ bits Bits
+ want string
+ }{
+ // all different bit numbers
+ {nil, "0"},
+ {Bits{0}, "0x.8p+1"},
+ {Bits{1}, "0x.8p+2"},
+ {Bits{-1}, "0x.8p+0"},
+ {Bits{63}, "0x.8p+64"},
+ {Bits{33, -30}, "0x.8000000000000001p+34"},
+ {Bits{255, 0}, "0x.8000000000000000000000000000000000000000000000000000000000000001p+256"},
+
+ // multiple equal bit numbers
+ {Bits{0, 0}, "0x.8p+2"},
+ {Bits{0, 0, 0, 0}, "0x.8p+3"},
+ {Bits{0, 1, 0}, "0x.8p+3"},
+ {append(Bits{2, 1, 0} /* 7 */, Bits{3, 1} /* 10 */ ...), "0x.88p+5" /* 17 */},
+ } {
+ f := test.bits.Float()
+ if got := f.Text('p', 0); got != test.want {
+ t.Errorf("setBits(%v) = %s; want %s", test.bits, got, test.want)
+ }
+ }
+}
diff --git a/src/math/big/calibrate_test.go b/src/math/big/calibrate_test.go
new file mode 100644
index 0000000..4fa663f
--- /dev/null
+++ b/src/math/big/calibrate_test.go
@@ -0,0 +1,173 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Calibration used to determine thresholds for using
+// different algorithms. Ideally, this would be converted
+// to go generate to create thresholds.go
+
+// This file prints execution times for the Mul benchmark
+// given different Karatsuba thresholds. The result may be
+// used to manually fine-tune the threshold constant. The
+// results are somewhat fragile; use repeated runs to get
+// a clear picture.
+
+// Calculates lower and upper thresholds for when basicSqr
+// is faster than standard multiplication.
+
+// Usage: go test -run=TestCalibrate -v -calibrate
+
+package big
+
+import (
+ "flag"
+ "fmt"
+ "testing"
+ "time"
+)
+
+var calibrate = flag.Bool("calibrate", false, "run calibration test")
+
+const (
+ sqrModeMul = "mul(x, x)"
+ sqrModeBasic = "basicSqr(x)"
+ sqrModeKaratsuba = "karatsubaSqr(x)"
+)
+
+func TestCalibrate(t *testing.T) {
+ if !*calibrate {
+ return
+ }
+
+ computeKaratsubaThresholds()
+
+ // compute basicSqrThreshold where overhead becomes negligible
+ minSqr := computeSqrThreshold(10, 30, 1, 3, sqrModeMul, sqrModeBasic)
+ // compute karatsubaSqrThreshold where karatsuba is faster
+ maxSqr := computeSqrThreshold(200, 500, 10, 3, sqrModeBasic, sqrModeKaratsuba)
+ if minSqr != 0 {
+ fmt.Printf("found basicSqrThreshold = %d\n", minSqr)
+ } else {
+ fmt.Println("no basicSqrThreshold found")
+ }
+ if maxSqr != 0 {
+ fmt.Printf("found karatsubaSqrThreshold = %d\n", maxSqr)
+ } else {
+ fmt.Println("no karatsubaSqrThreshold found")
+ }
+}
+
+func karatsubaLoad(b *testing.B) {
+ BenchmarkMul(b)
+}
+
+// measureKaratsuba returns the time to run a Karatsuba-relevant benchmark
+// given Karatsuba threshold th.
+func measureKaratsuba(th int) time.Duration {
+ th, karatsubaThreshold = karatsubaThreshold, th
+ res := testing.Benchmark(karatsubaLoad)
+ karatsubaThreshold = th
+ return time.Duration(res.NsPerOp())
+}
+
+func computeKaratsubaThresholds() {
+ fmt.Printf("Multiplication times for varying Karatsuba thresholds\n")
+ fmt.Printf("(run repeatedly for good results)\n")
+
+ // determine Tk, the work load execution time using basic multiplication
+ Tb := measureKaratsuba(1e9) // th == 1e9 => Karatsuba multiplication disabled
+ fmt.Printf("Tb = %10s\n", Tb)
+
+ // thresholds
+ th := 4
+ th1 := -1
+ th2 := -1
+
+ var deltaOld time.Duration
+ for count := -1; count != 0 && th < 128; count-- {
+ // determine Tk, the work load execution time using Karatsuba multiplication
+ Tk := measureKaratsuba(th)
+
+ // improvement over Tb
+ delta := (Tb - Tk) * 100 / Tb
+
+ fmt.Printf("th = %3d Tk = %10s %4d%%", th, Tk, delta)
+
+ // determine break-even point
+ if Tk < Tb && th1 < 0 {
+ th1 = th
+ fmt.Print(" break-even point")
+ }
+
+ // determine diminishing return
+ if 0 < delta && delta < deltaOld && th2 < 0 {
+ th2 = th
+ fmt.Print(" diminishing return")
+ }
+ deltaOld = delta
+
+ fmt.Println()
+
+ // trigger counter
+ if th1 >= 0 && th2 >= 0 && count < 0 {
+ count = 10 // this many extra measurements after we got both thresholds
+ }
+
+ th++
+ }
+}
+
+func measureSqr(words, nruns int, mode string) time.Duration {
+ // more runs for better statistics
+ initBasicSqr, initKaratsubaSqr := basicSqrThreshold, karatsubaSqrThreshold
+
+ switch mode {
+ case sqrModeMul:
+ basicSqrThreshold = words + 1
+ case sqrModeBasic:
+ basicSqrThreshold, karatsubaSqrThreshold = words-1, words+1
+ case sqrModeKaratsuba:
+ karatsubaSqrThreshold = words - 1
+ }
+
+ var testval int64
+ for i := 0; i < nruns; i++ {
+ res := testing.Benchmark(func(b *testing.B) { benchmarkNatSqr(b, words) })
+ testval += res.NsPerOp()
+ }
+ testval /= int64(nruns)
+
+ basicSqrThreshold, karatsubaSqrThreshold = initBasicSqr, initKaratsubaSqr
+
+ return time.Duration(testval)
+}
+
+func computeSqrThreshold(from, to, step, nruns int, lower, upper string) int {
+ fmt.Printf("Calibrating threshold between %s and %s\n", lower, upper)
+ fmt.Printf("Looking for a timing difference for x between %d - %d words by %d step\n", from, to, step)
+ var initPos bool
+ var threshold int
+ for i := from; i <= to; i += step {
+ baseline := measureSqr(i, nruns, lower)
+ testval := measureSqr(i, nruns, upper)
+ pos := baseline > testval
+ delta := baseline - testval
+ percent := delta * 100 / baseline
+ fmt.Printf("words = %3d deltaT = %10s (%4d%%) is %s better: %v", i, delta, percent, upper, pos)
+ if i == from {
+ initPos = pos
+ }
+ if threshold == 0 && pos != initPos {
+ threshold = i
+ fmt.Printf(" threshold found")
+ }
+ fmt.Println()
+
+ }
+ if threshold != 0 {
+ fmt.Printf("Found threshold = %d between %d - %d\n", threshold, from, to)
+ } else {
+ fmt.Printf("Found NO threshold between %d - %d\n", from, to)
+ }
+ return threshold
+}
diff --git a/src/math/big/decimal.go b/src/math/big/decimal.go
new file mode 100644
index 0000000..716f03b
--- /dev/null
+++ b/src/math/big/decimal.go
@@ -0,0 +1,270 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements multi-precision decimal numbers.
+// The implementation is for float to decimal conversion only;
+// not general purpose use.
+// The only operations are precise conversion from binary to
+// decimal and rounding.
+//
+// The key observation and some code (shr) is borrowed from
+// strconv/decimal.go: conversion of binary fractional values can be done
+// precisely in multi-precision decimal because 2 divides 10 (required for
+// >> of mantissa); but conversion of decimal floating-point values cannot
+// be done precisely in binary representation.
+//
+// In contrast to strconv/decimal.go, only right shift is implemented in
+// decimal format - left shift can be done precisely in binary format.
+
+package big
+
+// A decimal represents an unsigned floating-point number in decimal representation.
+// The value of a non-zero decimal d is d.mant * 10**d.exp with 0.1 <= d.mant < 1,
+// with the most-significant mantissa digit at index 0. For the zero decimal, the
+// mantissa length and exponent are 0.
+// The zero value for decimal represents a ready-to-use 0.0.
+type decimal struct {
+ mant []byte // mantissa ASCII digits, big-endian
+ exp int // exponent
+}
+
+// at returns the i'th mantissa digit, starting with the most significant digit at 0.
+func (d *decimal) at(i int) byte {
+ if 0 <= i && i < len(d.mant) {
+ return d.mant[i]
+ }
+ return '0'
+}
+
+// Maximum shift amount that can be done in one pass without overflow.
+// A Word has _W bits and (1<<maxShift - 1)*10 + 9 must fit into Word.
+const maxShift = _W - 4
+
+// TODO(gri) Since we know the desired decimal precision when converting
+// a floating-point number, we may be able to limit the number of decimal
+// digits that need to be computed by init by providing an additional
+// precision argument and keeping track of when a number was truncated early
+// (equivalent of "sticky bit" in binary rounding).
+
+// TODO(gri) Along the same lines, enforce some limit to shift magnitudes
+// to avoid "infinitely" long running conversions (until we run out of space).
+
+// Init initializes x to the decimal representation of m << shift (for
+// shift >= 0), or m >> -shift (for shift < 0).
+func (x *decimal) init(m nat, shift int) {
+ // special case 0
+ if len(m) == 0 {
+ x.mant = x.mant[:0]
+ x.exp = 0
+ return
+ }
+
+ // Optimization: If we need to shift right, first remove any trailing
+ // zero bits from m to reduce shift amount that needs to be done in
+ // decimal format (since that is likely slower).
+ if shift < 0 {
+ ntz := m.trailingZeroBits()
+ s := uint(-shift)
+ if s >= ntz {
+ s = ntz // shift at most ntz bits
+ }
+ m = nat(nil).shr(m, s)
+ shift += int(s)
+ }
+
+ // Do any shift left in binary representation.
+ if shift > 0 {
+ m = nat(nil).shl(m, uint(shift))
+ shift = 0
+ }
+
+ // Convert mantissa into decimal representation.
+ s := m.utoa(10)
+ n := len(s)
+ x.exp = n
+ // Trim trailing zeros; instead the exponent is tracking
+ // the decimal point independent of the number of digits.
+ for n > 0 && s[n-1] == '0' {
+ n--
+ }
+ x.mant = append(x.mant[:0], s[:n]...)
+
+ // Do any (remaining) shift right in decimal representation.
+ if shift < 0 {
+ for shift < -maxShift {
+ shr(x, maxShift)
+ shift += maxShift
+ }
+ shr(x, uint(-shift))
+ }
+}
+
+// shr implements x >> s, for s <= maxShift.
+func shr(x *decimal, s uint) {
+ // Division by 1<<s using shift-and-subtract algorithm.
+
+ // pick up enough leading digits to cover first shift
+ r := 0 // read index
+ var n Word
+ for n>>s == 0 && r < len(x.mant) {
+ ch := Word(x.mant[r])
+ r++
+ n = n*10 + ch - '0'
+ }
+ if n == 0 {
+ // x == 0; shouldn't get here, but handle anyway
+ x.mant = x.mant[:0]
+ return
+ }
+ for n>>s == 0 {
+ r++
+ n *= 10
+ }
+ x.exp += 1 - r
+
+ // read a digit, write a digit
+ w := 0 // write index
+ mask := Word(1)<<s - 1
+ for r < len(x.mant) {
+ ch := Word(x.mant[r])
+ r++
+ d := n >> s
+ n &= mask // n -= d << s
+ x.mant[w] = byte(d + '0')
+ w++
+ n = n*10 + ch - '0'
+ }
+
+ // write extra digits that still fit
+ for n > 0 && w < len(x.mant) {
+ d := n >> s
+ n &= mask
+ x.mant[w] = byte(d + '0')
+ w++
+ n = n * 10
+ }
+ x.mant = x.mant[:w] // the number may be shorter (e.g. 1024 >> 10)
+
+ // append additional digits that didn't fit
+ for n > 0 {
+ d := n >> s
+ n &= mask
+ x.mant = append(x.mant, byte(d+'0'))
+ n = n * 10
+ }
+
+ trim(x)
+}
+
+func (x *decimal) String() string {
+ if len(x.mant) == 0 {
+ return "0"
+ }
+
+ var buf []byte
+ switch {
+ case x.exp <= 0:
+ // 0.00ddd
+ buf = make([]byte, 0, 2+(-x.exp)+len(x.mant))
+ buf = append(buf, "0."...)
+ buf = appendZeros(buf, -x.exp)
+ buf = append(buf, x.mant...)
+
+ case /* 0 < */ x.exp < len(x.mant):
+ // dd.ddd
+ buf = make([]byte, 0, 1+len(x.mant))
+ buf = append(buf, x.mant[:x.exp]...)
+ buf = append(buf, '.')
+ buf = append(buf, x.mant[x.exp:]...)
+
+ default: // len(x.mant) <= x.exp
+ // ddd00
+ buf = make([]byte, 0, x.exp)
+ buf = append(buf, x.mant...)
+ buf = appendZeros(buf, x.exp-len(x.mant))
+ }
+
+ return string(buf)
+}
+
+// appendZeros appends n 0 digits to buf and returns buf.
+func appendZeros(buf []byte, n int) []byte {
+ for ; n > 0; n-- {
+ buf = append(buf, '0')
+ }
+ return buf
+}
+
+// shouldRoundUp reports if x should be rounded up
+// if shortened to n digits. n must be a valid index
+// for x.mant.
+func shouldRoundUp(x *decimal, n int) bool {
+ if x.mant[n] == '5' && n+1 == len(x.mant) {
+ // exactly halfway - round to even
+ return n > 0 && (x.mant[n-1]-'0')&1 != 0
+ }
+ // not halfway - digit tells all (x.mant has no trailing zeros)
+ return x.mant[n] >= '5'
+}
+
+// round sets x to (at most) n mantissa digits by rounding it
+// to the nearest even value with n (or fever) mantissa digits.
+// If n < 0, x remains unchanged.
+func (x *decimal) round(n int) {
+ if n < 0 || n >= len(x.mant) {
+ return // nothing to do
+ }
+
+ if shouldRoundUp(x, n) {
+ x.roundUp(n)
+ } else {
+ x.roundDown(n)
+ }
+}
+
+func (x *decimal) roundUp(n int) {
+ if n < 0 || n >= len(x.mant) {
+ return // nothing to do
+ }
+ // 0 <= n < len(x.mant)
+
+ // find first digit < '9'
+ for n > 0 && x.mant[n-1] >= '9' {
+ n--
+ }
+
+ if n == 0 {
+ // all digits are '9's => round up to '1' and update exponent
+ x.mant[0] = '1' // ok since len(x.mant) > n
+ x.mant = x.mant[:1]
+ x.exp++
+ return
+ }
+
+ // n > 0 && x.mant[n-1] < '9'
+ x.mant[n-1]++
+ x.mant = x.mant[:n]
+ // x already trimmed
+}
+
+func (x *decimal) roundDown(n int) {
+ if n < 0 || n >= len(x.mant) {
+ return // nothing to do
+ }
+ x.mant = x.mant[:n]
+ trim(x)
+}
+
+// trim cuts off any trailing zeros from x's mantissa;
+// they are meaningless for the value of x.
+func trim(x *decimal) {
+ i := len(x.mant)
+ for i > 0 && x.mant[i-1] == '0' {
+ i--
+ }
+ x.mant = x.mant[:i]
+ if i == 0 {
+ x.exp = 0
+ }
+}
diff --git a/src/math/big/decimal_test.go b/src/math/big/decimal_test.go
new file mode 100644
index 0000000..424811e
--- /dev/null
+++ b/src/math/big/decimal_test.go
@@ -0,0 +1,134 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "fmt"
+ "testing"
+)
+
+func TestDecimalString(t *testing.T) {
+ for _, test := range []struct {
+ x decimal
+ want string
+ }{
+ {want: "0"},
+ {decimal{nil, 1000}, "0"}, // exponent of 0 is ignored
+ {decimal{[]byte("12345"), 0}, "0.12345"},
+ {decimal{[]byte("12345"), -3}, "0.00012345"},
+ {decimal{[]byte("12345"), +3}, "123.45"},
+ {decimal{[]byte("12345"), +10}, "1234500000"},
+ } {
+ if got := test.x.String(); got != test.want {
+ t.Errorf("%v == %s; want %s", test.x, got, test.want)
+ }
+ }
+}
+
+func TestDecimalInit(t *testing.T) {
+ for _, test := range []struct {
+ x Word
+ shift int
+ want string
+ }{
+ {0, 0, "0"},
+ {0, -100, "0"},
+ {0, 100, "0"},
+ {1, 0, "1"},
+ {1, 10, "1024"},
+ {1, 100, "1267650600228229401496703205376"},
+ {1, -100, "0.0000000000000000000000000000007888609052210118054117285652827862296732064351090230047702789306640625"},
+ {12345678, 8, "3160493568"},
+ {12345678, -8, "48225.3046875"},
+ {195312, 9, "99999744"},
+ {1953125, 9, "1000000000"},
+ } {
+ var d decimal
+ d.init(nat{test.x}.norm(), test.shift)
+ if got := d.String(); got != test.want {
+ t.Errorf("%d << %d == %s; want %s", test.x, test.shift, got, test.want)
+ }
+ }
+}
+
+func TestDecimalRounding(t *testing.T) {
+ for _, test := range []struct {
+ x uint64
+ n int
+ down, even, up string
+ }{
+ {0, 0, "0", "0", "0"},
+ {0, 1, "0", "0", "0"},
+
+ {1, 0, "0", "0", "10"},
+ {5, 0, "0", "0", "10"},
+ {9, 0, "0", "10", "10"},
+
+ {15, 1, "10", "20", "20"},
+ {45, 1, "40", "40", "50"},
+ {95, 1, "90", "100", "100"},
+
+ {12344999, 4, "12340000", "12340000", "12350000"},
+ {12345000, 4, "12340000", "12340000", "12350000"},
+ {12345001, 4, "12340000", "12350000", "12350000"},
+ {23454999, 4, "23450000", "23450000", "23460000"},
+ {23455000, 4, "23450000", "23460000", "23460000"},
+ {23455001, 4, "23450000", "23460000", "23460000"},
+
+ {99994999, 4, "99990000", "99990000", "100000000"},
+ {99995000, 4, "99990000", "100000000", "100000000"},
+ {99999999, 4, "99990000", "100000000", "100000000"},
+
+ {12994999, 4, "12990000", "12990000", "13000000"},
+ {12995000, 4, "12990000", "13000000", "13000000"},
+ {12999999, 4, "12990000", "13000000", "13000000"},
+ } {
+ x := nat(nil).setUint64(test.x)
+
+ var d decimal
+ d.init(x, 0)
+ d.roundDown(test.n)
+ if got := d.String(); got != test.down {
+ t.Errorf("roundDown(%d, %d) = %s; want %s", test.x, test.n, got, test.down)
+ }
+
+ d.init(x, 0)
+ d.round(test.n)
+ if got := d.String(); got != test.even {
+ t.Errorf("round(%d, %d) = %s; want %s", test.x, test.n, got, test.even)
+ }
+
+ d.init(x, 0)
+ d.roundUp(test.n)
+ if got := d.String(); got != test.up {
+ t.Errorf("roundUp(%d, %d) = %s; want %s", test.x, test.n, got, test.up)
+ }
+ }
+}
+
+var sink string
+
+func BenchmarkDecimalConversion(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ for shift := -100; shift <= +100; shift++ {
+ var d decimal
+ d.init(natOne, shift)
+ sink = d.String()
+ }
+ }
+}
+
+func BenchmarkFloatString(b *testing.B) {
+ x := new(Float)
+ for _, prec := range []uint{1e2, 1e3, 1e4, 1e5} {
+ x.SetPrec(prec).SetRat(NewRat(1, 3))
+ b.Run(fmt.Sprintf("%v", prec), func(b *testing.B) {
+ b.ReportAllocs()
+ for i := 0; i < b.N; i++ {
+ sink = x.String()
+ }
+ })
+ }
+}
diff --git a/src/math/big/doc.go b/src/math/big/doc.go
new file mode 100644
index 0000000..65ed019
--- /dev/null
+++ b/src/math/big/doc.go
@@ -0,0 +1,99 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+/*
+Package big implements arbitrary-precision arithmetic (big numbers).
+The following numeric types are supported:
+
+ Int signed integers
+ Rat rational numbers
+ Float floating-point numbers
+
+The zero value for an Int, Rat, or Float correspond to 0. Thus, new
+values can be declared in the usual ways and denote 0 without further
+initialization:
+
+ var x Int // &x is an *Int of value 0
+ var r = &Rat{} // r is a *Rat of value 0
+ y := new(Float) // y is a *Float of value 0
+
+Alternatively, new values can be allocated and initialized with factory
+functions of the form:
+
+ func NewT(v V) *T
+
+For instance, NewInt(x) returns an *Int set to the value of the int64
+argument x, NewRat(a, b) returns a *Rat set to the fraction a/b where
+a and b are int64 values, and NewFloat(f) returns a *Float initialized
+to the float64 argument f. More flexibility is provided with explicit
+setters, for instance:
+
+ var z1 Int
+ z1.SetUint64(123) // z1 := 123
+ z2 := new(Rat).SetFloat64(1.25) // z2 := 5/4
+ z3 := new(Float).SetInt(z1) // z3 := 123.0
+
+Setters, numeric operations and predicates are represented as methods of
+the form:
+
+ func (z *T) SetV(v V) *T // z = v
+ func (z *T) Unary(x *T) *T // z = unary x
+ func (z *T) Binary(x, y *T) *T // z = x binary y
+ func (x *T) Pred() P // p = pred(x)
+
+with T one of Int, Rat, or Float. For unary and binary operations, the
+result is the receiver (usually named z in that case; see below); if it
+is one of the operands x or y it may be safely overwritten (and its memory
+reused).
+
+Arithmetic expressions are typically written as a sequence of individual
+method calls, with each call corresponding to an operation. The receiver
+denotes the result and the method arguments are the operation's operands.
+For instance, given three *Int values a, b and c, the invocation
+
+ c.Add(a, b)
+
+computes the sum a + b and stores the result in c, overwriting whatever
+value was held in c before. Unless specified otherwise, operations permit
+aliasing of parameters, so it is perfectly ok to write
+
+ sum.Add(sum, x)
+
+to accumulate values x in a sum.
+
+(By always passing in a result value via the receiver, memory use can be
+much better controlled. Instead of having to allocate new memory for each
+result, an operation can reuse the space allocated for the result value,
+and overwrite that value with the new result in the process.)
+
+Notational convention: Incoming method parameters (including the receiver)
+are named consistently in the API to clarify their use. Incoming operands
+are usually named x, y, a, b, and so on, but never z. A parameter specifying
+the result is named z (typically the receiver).
+
+For instance, the arguments for (*Int).Add are named x and y, and because
+the receiver specifies the result destination, it is called z:
+
+ func (z *Int) Add(x, y *Int) *Int
+
+Methods of this form typically return the incoming receiver as well, to
+enable simple call chaining.
+
+Methods which don't require a result value to be passed in (for instance,
+Int.Sign), simply return the result. In this case, the receiver is typically
+the first operand, named x:
+
+ func (x *Int) Sign() int
+
+Various methods support conversions between strings and corresponding
+numeric values, and vice versa: *Int, *Rat, and *Float values implement
+the Stringer interface for a (default) string representation of the value,
+but also provide SetString methods to initialize a value from a string in
+a variety of supported formats (see the respective SetString documentation).
+
+Finally, *Int, *Rat, and *Float satisfy the fmt package's Scanner interface
+for scanning and (except for *Rat) the Formatter interface for formatted
+printing.
+*/
+package big
diff --git a/src/math/big/example_rat_test.go b/src/math/big/example_rat_test.go
new file mode 100644
index 0000000..a971170
--- /dev/null
+++ b/src/math/big/example_rat_test.go
@@ -0,0 +1,65 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big_test
+
+import (
+ "fmt"
+ "math/big"
+)
+
+// Use the classic continued fraction for e
+// e = [1; 0, 1, 1, 2, 1, 1, ... 2n, 1, 1, ...]
+// i.e., for the nth term, use
+// 1 if n mod 3 != 1
+// (n-1)/3 * 2 if n mod 3 == 1
+func recur(n, lim int64) *big.Rat {
+ term := new(big.Rat)
+ if n%3 != 1 {
+ term.SetInt64(1)
+ } else {
+ term.SetInt64((n - 1) / 3 * 2)
+ }
+
+ if n > lim {
+ return term
+ }
+
+ // Directly initialize frac as the fractional
+ // inverse of the result of recur.
+ frac := new(big.Rat).Inv(recur(n+1, lim))
+
+ return term.Add(term, frac)
+}
+
+// This example demonstrates how to use big.Rat to compute the
+// first 15 terms in the sequence of rational convergents for
+// the constant e (base of natural logarithm).
+func Example_eConvergents() {
+ for i := 1; i <= 15; i++ {
+ r := recur(0, int64(i))
+
+ // Print r both as a fraction and as a floating-point number.
+ // Since big.Rat implements fmt.Formatter, we can use %-13s to
+ // get a left-aligned string representation of the fraction.
+ fmt.Printf("%-13s = %s\n", r, r.FloatString(8))
+ }
+
+ // Output:
+ // 2/1 = 2.00000000
+ // 3/1 = 3.00000000
+ // 8/3 = 2.66666667
+ // 11/4 = 2.75000000
+ // 19/7 = 2.71428571
+ // 87/32 = 2.71875000
+ // 106/39 = 2.71794872
+ // 193/71 = 2.71830986
+ // 1264/465 = 2.71827957
+ // 1457/536 = 2.71828358
+ // 2721/1001 = 2.71828172
+ // 23225/8544 = 2.71828184
+ // 25946/9545 = 2.71828182
+ // 49171/18089 = 2.71828183
+ // 517656/190435 = 2.71828183
+}
diff --git a/src/math/big/example_test.go b/src/math/big/example_test.go
new file mode 100644
index 0000000..31ca784
--- /dev/null
+++ b/src/math/big/example_test.go
@@ -0,0 +1,148 @@
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big_test
+
+import (
+ "fmt"
+ "log"
+ "math"
+ "math/big"
+)
+
+func ExampleRat_SetString() {
+ r := new(big.Rat)
+ r.SetString("355/113")
+ fmt.Println(r.FloatString(3))
+ // Output: 3.142
+}
+
+func ExampleInt_SetString() {
+ i := new(big.Int)
+ i.SetString("644", 8) // octal
+ fmt.Println(i)
+ // Output: 420
+}
+
+func ExampleFloat_SetString() {
+ f := new(big.Float)
+ f.SetString("3.14159")
+ fmt.Println(f)
+ // Output: 3.14159
+}
+
+func ExampleRat_Scan() {
+ // The Scan function is rarely used directly;
+ // the fmt package recognizes it as an implementation of fmt.Scanner.
+ r := new(big.Rat)
+ _, err := fmt.Sscan("1.5000", r)
+ if err != nil {
+ log.Println("error scanning value:", err)
+ } else {
+ fmt.Println(r)
+ }
+ // Output: 3/2
+}
+
+func ExampleInt_Scan() {
+ // The Scan function is rarely used directly;
+ // the fmt package recognizes it as an implementation of fmt.Scanner.
+ i := new(big.Int)
+ _, err := fmt.Sscan("18446744073709551617", i)
+ if err != nil {
+ log.Println("error scanning value:", err)
+ } else {
+ fmt.Println(i)
+ }
+ // Output: 18446744073709551617
+}
+
+func ExampleFloat_Scan() {
+ // The Scan function is rarely used directly;
+ // the fmt package recognizes it as an implementation of fmt.Scanner.
+ f := new(big.Float)
+ _, err := fmt.Sscan("1.19282e99", f)
+ if err != nil {
+ log.Println("error scanning value:", err)
+ } else {
+ fmt.Println(f)
+ }
+ // Output: 1.19282e+99
+}
+
+// This example demonstrates how to use big.Int to compute the smallest
+// Fibonacci number with 100 decimal digits and to test whether it is prime.
+func Example_fibonacci() {
+ // Initialize two big ints with the first two numbers in the sequence.
+ a := big.NewInt(0)
+ b := big.NewInt(1)
+
+ // Initialize limit as 10^99, the smallest integer with 100 digits.
+ var limit big.Int
+ limit.Exp(big.NewInt(10), big.NewInt(99), nil)
+
+ // Loop while a is smaller than 1e100.
+ for a.Cmp(&limit) < 0 {
+ // Compute the next Fibonacci number, storing it in a.
+ a.Add(a, b)
+ // Swap a and b so that b is the next number in the sequence.
+ a, b = b, a
+ }
+ fmt.Println(a) // 100-digit Fibonacci number
+
+ // Test a for primality.
+ // (ProbablyPrimes' argument sets the number of Miller-Rabin
+ // rounds to be performed. 20 is a good value.)
+ fmt.Println(a.ProbablyPrime(20))
+
+ // Output:
+ // 1344719667586153181419716641724567886890850696275767987106294472017884974410332069524504824747437757
+ // false
+}
+
+// This example shows how to use big.Float to compute the square root of 2 with
+// a precision of 200 bits, and how to print the result as a decimal number.
+func Example_sqrt2() {
+ // We'll do computations with 200 bits of precision in the mantissa.
+ const prec = 200
+
+ // Compute the square root of 2 using Newton's Method. We start with
+ // an initial estimate for sqrt(2), and then iterate:
+ // x_{n+1} = 1/2 * ( x_n + (2.0 / x_n) )
+
+ // Since Newton's Method doubles the number of correct digits at each
+ // iteration, we need at least log_2(prec) steps.
+ steps := int(math.Log2(prec))
+
+ // Initialize values we need for the computation.
+ two := new(big.Float).SetPrec(prec).SetInt64(2)
+ half := new(big.Float).SetPrec(prec).SetFloat64(0.5)
+
+ // Use 1 as the initial estimate.
+ x := new(big.Float).SetPrec(prec).SetInt64(1)
+
+ // We use t as a temporary variable. There's no need to set its precision
+ // since big.Float values with unset (== 0) precision automatically assume
+ // the largest precision of the arguments when used as the result (receiver)
+ // of a big.Float operation.
+ t := new(big.Float)
+
+ // Iterate.
+ for i := 0; i <= steps; i++ {
+ t.Quo(two, x) // t = 2.0 / x_n
+ t.Add(x, t) // t = x_n + (2.0 / x_n)
+ x.Mul(half, t) // x_{n+1} = 0.5 * t
+ }
+
+ // We can use the usual fmt.Printf verbs since big.Float implements fmt.Formatter
+ fmt.Printf("sqrt(2) = %.50f\n", x)
+
+ // Print the error between 2 and x*x.
+ t.Mul(x, x) // t = x*x
+ fmt.Printf("error = %e\n", t.Sub(two, t))
+
+ // Output:
+ // sqrt(2) = 1.41421356237309504880168872420969807856967187537695
+ // error = 0.000000e+00
+}
diff --git a/src/math/big/float.go b/src/math/big/float.go
new file mode 100644
index 0000000..42050e2
--- /dev/null
+++ b/src/math/big/float.go
@@ -0,0 +1,1730 @@
+// Copyright 2014 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements multi-precision floating-point numbers.
+// Like in the GNU MPFR library (https://www.mpfr.org/), operands
+// can be of mixed precision. Unlike MPFR, the rounding mode is
+// not specified with each operation, but with each operand. The
+// rounding mode of the result operand determines the rounding
+// mode of an operation. This is a from-scratch implementation.
+
+package big
+
+import (
+ "fmt"
+ "math"
+ "math/bits"
+)
+
+const debugFloat = false // enable for debugging
+
+// A nonzero finite Float represents a multi-precision floating point number
+//
+// sign × mantissa × 2**exponent
+//
+// with 0.5 <= mantissa < 1.0, and MinExp <= exponent <= MaxExp.
+// A Float may also be zero (+0, -0) or infinite (+Inf, -Inf).
+// All Floats are ordered, and the ordering of two Floats x and y
+// is defined by x.Cmp(y).
+//
+// Each Float value also has a precision, rounding mode, and accuracy.
+// The precision is the maximum number of mantissa bits available to
+// represent the value. The rounding mode specifies how a result should
+// be rounded to fit into the mantissa bits, and accuracy describes the
+// rounding error with respect to the exact result.
+//
+// Unless specified otherwise, all operations (including setters) that
+// specify a *Float variable for the result (usually via the receiver
+// with the exception of MantExp), round the numeric result according
+// to the precision and rounding mode of the result variable.
+//
+// If the provided result precision is 0 (see below), it is set to the
+// precision of the argument with the largest precision value before any
+// rounding takes place, and the rounding mode remains unchanged. Thus,
+// uninitialized Floats provided as result arguments will have their
+// precision set to a reasonable value determined by the operands, and
+// their mode is the zero value for RoundingMode (ToNearestEven).
+//
+// By setting the desired precision to 24 or 53 and using matching rounding
+// mode (typically ToNearestEven), Float operations produce the same results
+// as the corresponding float32 or float64 IEEE-754 arithmetic for operands
+// that correspond to normal (i.e., not denormal) float32 or float64 numbers.
+// Exponent underflow and overflow lead to a 0 or an Infinity for different
+// values than IEEE-754 because Float exponents have a much larger range.
+//
+// The zero (uninitialized) value for a Float is ready to use and represents
+// the number +0.0 exactly, with precision 0 and rounding mode ToNearestEven.
+//
+// Operations always take pointer arguments (*Float) rather
+// than Float values, and each unique Float value requires
+// its own unique *Float pointer. To "copy" a Float value,
+// an existing (or newly allocated) Float must be set to
+// a new value using the Float.Set method; shallow copies
+// of Floats are not supported and may lead to errors.
+type Float struct {
+ prec uint32
+ mode RoundingMode
+ acc Accuracy
+ form form
+ neg bool
+ mant nat
+ exp int32
+}
+
+// An ErrNaN panic is raised by a Float operation that would lead to
+// a NaN under IEEE-754 rules. An ErrNaN implements the error interface.
+type ErrNaN struct {
+ msg string
+}
+
+func (err ErrNaN) Error() string {
+ return err.msg
+}
+
+// NewFloat allocates and returns a new Float set to x,
+// with precision 53 and rounding mode ToNearestEven.
+// NewFloat panics with ErrNaN if x is a NaN.
+func NewFloat(x float64) *Float {
+ if math.IsNaN(x) {
+ panic(ErrNaN{"NewFloat(NaN)"})
+ }
+ return new(Float).SetFloat64(x)
+}
+
+// Exponent and precision limits.
+const (
+ MaxExp = math.MaxInt32 // largest supported exponent
+ MinExp = math.MinInt32 // smallest supported exponent
+ MaxPrec = math.MaxUint32 // largest (theoretically) supported precision; likely memory-limited
+)
+
+// Internal representation: The mantissa bits x.mant of a nonzero finite
+// Float x are stored in a nat slice long enough to hold up to x.prec bits;
+// the slice may (but doesn't have to) be shorter if the mantissa contains
+// trailing 0 bits. x.mant is normalized if the msb of x.mant == 1 (i.e.,
+// the msb is shifted all the way "to the left"). Thus, if the mantissa has
+// trailing 0 bits or x.prec is not a multiple of the Word size _W,
+// x.mant[0] has trailing zero bits. The msb of the mantissa corresponds
+// to the value 0.5; the exponent x.exp shifts the binary point as needed.
+//
+// A zero or non-finite Float x ignores x.mant and x.exp.
+//
+// x form neg mant exp
+// ----------------------------------------------------------
+// ±0 zero sign - -
+// 0 < |x| < +Inf finite sign mantissa exponent
+// ±Inf inf sign - -
+
+// A form value describes the internal representation.
+type form byte
+
+// The form value order is relevant - do not change!
+const (
+ zero form = iota
+ finite
+ inf
+)
+
+// RoundingMode determines how a Float value is rounded to the
+// desired precision. Rounding may change the Float value; the
+// rounding error is described by the Float's Accuracy.
+type RoundingMode byte
+
+// These constants define supported rounding modes.
+const (
+ ToNearestEven RoundingMode = iota // == IEEE 754-2008 roundTiesToEven
+ ToNearestAway // == IEEE 754-2008 roundTiesToAway
+ ToZero // == IEEE 754-2008 roundTowardZero
+ AwayFromZero // no IEEE 754-2008 equivalent
+ ToNegativeInf // == IEEE 754-2008 roundTowardNegative
+ ToPositiveInf // == IEEE 754-2008 roundTowardPositive
+)
+
+//go:generate stringer -type=RoundingMode
+
+// Accuracy describes the rounding error produced by the most recent
+// operation that generated a Float value, relative to the exact value.
+type Accuracy int8
+
+// Constants describing the Accuracy of a Float.
+const (
+ Below Accuracy = -1
+ Exact Accuracy = 0
+ Above Accuracy = +1
+)
+
+//go:generate stringer -type=Accuracy
+
+// SetPrec sets z's precision to prec and returns the (possibly) rounded
+// value of z. Rounding occurs according to z's rounding mode if the mantissa
+// cannot be represented in prec bits without loss of precision.
+// SetPrec(0) maps all finite values to ±0; infinite values remain unchanged.
+// If prec > MaxPrec, it is set to MaxPrec.
+func (z *Float) SetPrec(prec uint) *Float {
+ z.acc = Exact // optimistically assume no rounding is needed
+
+ // special case
+ if prec == 0 {
+ z.prec = 0
+ if z.form == finite {
+ // truncate z to 0
+ z.acc = makeAcc(z.neg)
+ z.form = zero
+ }
+ return z
+ }
+
+ // general case
+ if prec > MaxPrec {
+ prec = MaxPrec
+ }
+ old := z.prec
+ z.prec = uint32(prec)
+ if z.prec < old {
+ z.round(0)
+ }
+ return z
+}
+
+func makeAcc(above bool) Accuracy {
+ if above {
+ return Above
+ }
+ return Below
+}
+
+// SetMode sets z's rounding mode to mode and returns an exact z.
+// z remains unchanged otherwise.
+// z.SetMode(z.Mode()) is a cheap way to set z's accuracy to Exact.
+func (z *Float) SetMode(mode RoundingMode) *Float {
+ z.mode = mode
+ z.acc = Exact
+ return z
+}
+
+// Prec returns the mantissa precision of x in bits.
+// The result may be 0 for |x| == 0 and |x| == Inf.
+func (x *Float) Prec() uint {
+ return uint(x.prec)
+}
+
+// MinPrec returns the minimum precision required to represent x exactly
+// (i.e., the smallest prec before x.SetPrec(prec) would start rounding x).
+// The result is 0 for |x| == 0 and |x| == Inf.
+func (x *Float) MinPrec() uint {
+ if x.form != finite {
+ return 0
+ }
+ return uint(len(x.mant))*_W - x.mant.trailingZeroBits()
+}
+
+// Mode returns the rounding mode of x.
+func (x *Float) Mode() RoundingMode {
+ return x.mode
+}
+
+// Acc returns the accuracy of x produced by the most recent
+// operation, unless explicitly documented otherwise by that
+// operation.
+func (x *Float) Acc() Accuracy {
+ return x.acc
+}
+
+// Sign returns:
+//
+// -1 if x < 0
+// 0 if x is ±0
+// +1 if x > 0
+//
+func (x *Float) Sign() int {
+ if debugFloat {
+ x.validate()
+ }
+ if x.form == zero {
+ return 0
+ }
+ if x.neg {
+ return -1
+ }
+ return 1
+}
+
+// MantExp breaks x into its mantissa and exponent components
+// and returns the exponent. If a non-nil mant argument is
+// provided its value is set to the mantissa of x, with the
+// same precision and rounding mode as x. The components
+// satisfy x == mant × 2**exp, with 0.5 <= |mant| < 1.0.
+// Calling MantExp with a nil argument is an efficient way to
+// get the exponent of the receiver.
+//
+// Special cases are:
+//
+// ( ±0).MantExp(mant) = 0, with mant set to ±0
+// (±Inf).MantExp(mant) = 0, with mant set to ±Inf
+//
+// x and mant may be the same in which case x is set to its
+// mantissa value.
+func (x *Float) MantExp(mant *Float) (exp int) {
+ if debugFloat {
+ x.validate()
+ }
+ if x.form == finite {
+ exp = int(x.exp)
+ }
+ if mant != nil {
+ mant.Copy(x)
+ if mant.form == finite {
+ mant.exp = 0
+ }
+ }
+ return
+}
+
+func (z *Float) setExpAndRound(exp int64, sbit uint) {
+ if exp < MinExp {
+ // underflow
+ z.acc = makeAcc(z.neg)
+ z.form = zero
+ return
+ }
+
+ if exp > MaxExp {
+ // overflow
+ z.acc = makeAcc(!z.neg)
+ z.form = inf
+ return
+ }
+
+ z.form = finite
+ z.exp = int32(exp)
+ z.round(sbit)
+}
+
+// SetMantExp sets z to mant × 2**exp and returns z.
+// The result z has the same precision and rounding mode
+// as mant. SetMantExp is an inverse of MantExp but does
+// not require 0.5 <= |mant| < 1.0. Specifically:
+//
+// mant := new(Float)
+// new(Float).SetMantExp(mant, x.MantExp(mant)).Cmp(x) == 0
+//
+// Special cases are:
+//
+// z.SetMantExp( ±0, exp) = ±0
+// z.SetMantExp(±Inf, exp) = ±Inf
+//
+// z and mant may be the same in which case z's exponent
+// is set to exp.
+func (z *Float) SetMantExp(mant *Float, exp int) *Float {
+ if debugFloat {
+ z.validate()
+ mant.validate()
+ }
+ z.Copy(mant)
+
+ if z.form == finite {
+ // 0 < |mant| < +Inf
+ z.setExpAndRound(int64(z.exp)+int64(exp), 0)
+ }
+ return z
+}
+
+// Signbit reports whether x is negative or negative zero.
+func (x *Float) Signbit() bool {
+ return x.neg
+}
+
+// IsInf reports whether x is +Inf or -Inf.
+func (x *Float) IsInf() bool {
+ return x.form == inf
+}
+
+// IsInt reports whether x is an integer.
+// ±Inf values are not integers.
+func (x *Float) IsInt() bool {
+ if debugFloat {
+ x.validate()
+ }
+ // special cases
+ if x.form != finite {
+ return x.form == zero
+ }
+ // x.form == finite
+ if x.exp <= 0 {
+ return false
+ }
+ // x.exp > 0
+ return x.prec <= uint32(x.exp) || x.MinPrec() <= uint(x.exp) // not enough bits for fractional mantissa
+}
+
+// debugging support
+func (x *Float) validate() {
+ if !debugFloat {
+ // avoid performance bugs
+ panic("validate called but debugFloat is not set")
+ }
+ if x.form != finite {
+ return
+ }
+ m := len(x.mant)
+ if m == 0 {
+ panic("nonzero finite number with empty mantissa")
+ }
+ const msb = 1 << (_W - 1)
+ if x.mant[m-1]&msb == 0 {
+ panic(fmt.Sprintf("msb not set in last word %#x of %s", x.mant[m-1], x.Text('p', 0)))
+ }
+ if x.prec == 0 {
+ panic("zero precision finite number")
+ }
+}
+
+// round rounds z according to z.mode to z.prec bits and sets z.acc accordingly.
+// sbit must be 0 or 1 and summarizes any "sticky bit" information one might
+// have before calling round. z's mantissa must be normalized (with the msb set)
+// or empty.
+//
+// CAUTION: The rounding modes ToNegativeInf, ToPositiveInf are affected by the
+// sign of z. For correct rounding, the sign of z must be set correctly before
+// calling round.
+func (z *Float) round(sbit uint) {
+ if debugFloat {
+ z.validate()
+ }
+
+ z.acc = Exact
+ if z.form != finite {
+ // ±0 or ±Inf => nothing left to do
+ return
+ }
+ // z.form == finite && len(z.mant) > 0
+ // m > 0 implies z.prec > 0 (checked by validate)
+
+ m := uint32(len(z.mant)) // present mantissa length in words
+ bits := m * _W // present mantissa bits; bits > 0
+ if bits <= z.prec {
+ // mantissa fits => nothing to do
+ return
+ }
+ // bits > z.prec
+
+ // Rounding is based on two bits: the rounding bit (rbit) and the
+ // sticky bit (sbit). The rbit is the bit immediately before the
+ // z.prec leading mantissa bits (the "0.5"). The sbit is set if any
+ // of the bits before the rbit are set (the "0.25", "0.125", etc.):
+ //
+ // rbit sbit => "fractional part"
+ //
+ // 0 0 == 0
+ // 0 1 > 0 , < 0.5
+ // 1 0 == 0.5
+ // 1 1 > 0.5, < 1.0
+
+ // bits > z.prec: mantissa too large => round
+ r := uint(bits - z.prec - 1) // rounding bit position; r >= 0
+ rbit := z.mant.bit(r) & 1 // rounding bit; be safe and ensure it's a single bit
+ // The sticky bit is only needed for rounding ToNearestEven
+ // or when the rounding bit is zero. Avoid computation otherwise.
+ if sbit == 0 && (rbit == 0 || z.mode == ToNearestEven) {
+ sbit = z.mant.sticky(r)
+ }
+ sbit &= 1 // be safe and ensure it's a single bit
+
+ // cut off extra words
+ n := (z.prec + (_W - 1)) / _W // mantissa length in words for desired precision
+ if m > n {
+ copy(z.mant, z.mant[m-n:]) // move n last words to front
+ z.mant = z.mant[:n]
+ }
+
+ // determine number of trailing zero bits (ntz) and compute lsb mask of mantissa's least-significant word
+ ntz := n*_W - z.prec // 0 <= ntz < _W
+ lsb := Word(1) << ntz
+
+ // round if result is inexact
+ if rbit|sbit != 0 {
+ // Make rounding decision: The result mantissa is truncated ("rounded down")
+ // by default. Decide if we need to increment, or "round up", the (unsigned)
+ // mantissa.
+ inc := false
+ switch z.mode {
+ case ToNegativeInf:
+ inc = z.neg
+ case ToZero:
+ // nothing to do
+ case ToNearestEven:
+ inc = rbit != 0 && (sbit != 0 || z.mant[0]&lsb != 0)
+ case ToNearestAway:
+ inc = rbit != 0
+ case AwayFromZero:
+ inc = true
+ case ToPositiveInf:
+ inc = !z.neg
+ default:
+ panic("unreachable")
+ }
+
+ // A positive result (!z.neg) is Above the exact result if we increment,
+ // and it's Below if we truncate (Exact results require no rounding).
+ // For a negative result (z.neg) it is exactly the opposite.
+ z.acc = makeAcc(inc != z.neg)
+
+ if inc {
+ // add 1 to mantissa
+ if addVW(z.mant, z.mant, lsb) != 0 {
+ // mantissa overflow => adjust exponent
+ if z.exp >= MaxExp {
+ // exponent overflow
+ z.form = inf
+ return
+ }
+ z.exp++
+ // adjust mantissa: divide by 2 to compensate for exponent adjustment
+ shrVU(z.mant, z.mant, 1)
+ // set msb == carry == 1 from the mantissa overflow above
+ const msb = 1 << (_W - 1)
+ z.mant[n-1] |= msb
+ }
+ }
+ }
+
+ // zero out trailing bits in least-significant word
+ z.mant[0] &^= lsb - 1
+
+ if debugFloat {
+ z.validate()
+ }
+}
+
+func (z *Float) setBits64(neg bool, x uint64) *Float {
+ if z.prec == 0 {
+ z.prec = 64
+ }
+ z.acc = Exact
+ z.neg = neg
+ if x == 0 {
+ z.form = zero
+ return z
+ }
+ // x != 0
+ z.form = finite
+ s := bits.LeadingZeros64(x)
+ z.mant = z.mant.setUint64(x << uint(s))
+ z.exp = int32(64 - s) // always fits
+ if z.prec < 64 {
+ z.round(0)
+ }
+ return z
+}
+
+// SetUint64 sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to 64 (and rounding will have
+// no effect).
+func (z *Float) SetUint64(x uint64) *Float {
+ return z.setBits64(false, x)
+}
+
+// SetInt64 sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to 64 (and rounding will have
+// no effect).
+func (z *Float) SetInt64(x int64) *Float {
+ u := x
+ if u < 0 {
+ u = -u
+ }
+ // We cannot simply call z.SetUint64(uint64(u)) and change
+ // the sign afterwards because the sign affects rounding.
+ return z.setBits64(x < 0, uint64(u))
+}
+
+// SetFloat64 sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to 53 (and rounding will have
+// no effect). SetFloat64 panics with ErrNaN if x is a NaN.
+func (z *Float) SetFloat64(x float64) *Float {
+ if z.prec == 0 {
+ z.prec = 53
+ }
+ if math.IsNaN(x) {
+ panic(ErrNaN{"Float.SetFloat64(NaN)"})
+ }
+ z.acc = Exact
+ z.neg = math.Signbit(x) // handle -0, -Inf correctly
+ if x == 0 {
+ z.form = zero
+ return z
+ }
+ if math.IsInf(x, 0) {
+ z.form = inf
+ return z
+ }
+ // normalized x != 0
+ z.form = finite
+ fmant, exp := math.Frexp(x) // get normalized mantissa
+ z.mant = z.mant.setUint64(1<<63 | math.Float64bits(fmant)<<11)
+ z.exp = int32(exp) // always fits
+ if z.prec < 53 {
+ z.round(0)
+ }
+ return z
+}
+
+// fnorm normalizes mantissa m by shifting it to the left
+// such that the msb of the most-significant word (msw) is 1.
+// It returns the shift amount. It assumes that len(m) != 0.
+func fnorm(m nat) int64 {
+ if debugFloat && (len(m) == 0 || m[len(m)-1] == 0) {
+ panic("msw of mantissa is 0")
+ }
+ s := nlz(m[len(m)-1])
+ if s > 0 {
+ c := shlVU(m, m, s)
+ if debugFloat && c != 0 {
+ panic("nlz or shlVU incorrect")
+ }
+ }
+ return int64(s)
+}
+
+// SetInt sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to the larger of x.BitLen()
+// or 64 (and rounding will have no effect).
+func (z *Float) SetInt(x *Int) *Float {
+ // TODO(gri) can be more efficient if z.prec > 0
+ // but small compared to the size of x, or if there
+ // are many trailing 0's.
+ bits := uint32(x.BitLen())
+ if z.prec == 0 {
+ z.prec = umax32(bits, 64)
+ }
+ z.acc = Exact
+ z.neg = x.neg
+ if len(x.abs) == 0 {
+ z.form = zero
+ return z
+ }
+ // x != 0
+ z.mant = z.mant.set(x.abs)
+ fnorm(z.mant)
+ z.setExpAndRound(int64(bits), 0)
+ return z
+}
+
+// SetRat sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to the largest of a.BitLen(),
+// b.BitLen(), or 64; with x = a/b.
+func (z *Float) SetRat(x *Rat) *Float {
+ if x.IsInt() {
+ return z.SetInt(x.Num())
+ }
+ var a, b Float
+ a.SetInt(x.Num())
+ b.SetInt(x.Denom())
+ if z.prec == 0 {
+ z.prec = umax32(a.prec, b.prec)
+ }
+ return z.Quo(&a, &b)
+}
+
+// SetInf sets z to the infinite Float -Inf if signbit is
+// set, or +Inf if signbit is not set, and returns z. The
+// precision of z is unchanged and the result is always
+// Exact.
+func (z *Float) SetInf(signbit bool) *Float {
+ z.acc = Exact
+ z.form = inf
+ z.neg = signbit
+ return z
+}
+
+// Set sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to the precision of x
+// before setting z (and rounding will have no effect).
+// Rounding is performed according to z's precision and rounding
+// mode; and z's accuracy reports the result error relative to the
+// exact (not rounded) result.
+func (z *Float) Set(x *Float) *Float {
+ if debugFloat {
+ x.validate()
+ }
+ z.acc = Exact
+ if z != x {
+ z.form = x.form
+ z.neg = x.neg
+ if x.form == finite {
+ z.exp = x.exp
+ z.mant = z.mant.set(x.mant)
+ }
+ if z.prec == 0 {
+ z.prec = x.prec
+ } else if z.prec < x.prec {
+ z.round(0)
+ }
+ }
+ return z
+}
+
+// Copy sets z to x, with the same precision, rounding mode, and
+// accuracy as x, and returns z. x is not changed even if z and
+// x are the same.
+func (z *Float) Copy(x *Float) *Float {
+ if debugFloat {
+ x.validate()
+ }
+ if z != x {
+ z.prec = x.prec
+ z.mode = x.mode
+ z.acc = x.acc
+ z.form = x.form
+ z.neg = x.neg
+ if z.form == finite {
+ z.mant = z.mant.set(x.mant)
+ z.exp = x.exp
+ }
+ }
+ return z
+}
+
+// msb32 returns the 32 most significant bits of x.
+func msb32(x nat) uint32 {
+ i := len(x) - 1
+ if i < 0 {
+ return 0
+ }
+ if debugFloat && x[i]&(1<<(_W-1)) == 0 {
+ panic("x not normalized")
+ }
+ switch _W {
+ case 32:
+ return uint32(x[i])
+ case 64:
+ return uint32(x[i] >> 32)
+ }
+ panic("unreachable")
+}
+
+// msb64 returns the 64 most significant bits of x.
+func msb64(x nat) uint64 {
+ i := len(x) - 1
+ if i < 0 {
+ return 0
+ }
+ if debugFloat && x[i]&(1<<(_W-1)) == 0 {
+ panic("x not normalized")
+ }
+ switch _W {
+ case 32:
+ v := uint64(x[i]) << 32
+ if i > 0 {
+ v |= uint64(x[i-1])
+ }
+ return v
+ case 64:
+ return uint64(x[i])
+ }
+ panic("unreachable")
+}
+
+// Uint64 returns the unsigned integer resulting from truncating x
+// towards zero. If 0 <= x <= math.MaxUint64, the result is Exact
+// if x is an integer and Below otherwise.
+// The result is (0, Above) for x < 0, and (math.MaxUint64, Below)
+// for x > math.MaxUint64.
+func (x *Float) Uint64() (uint64, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ switch x.form {
+ case finite:
+ if x.neg {
+ return 0, Above
+ }
+ // 0 < x < +Inf
+ if x.exp <= 0 {
+ // 0 < x < 1
+ return 0, Below
+ }
+ // 1 <= x < Inf
+ if x.exp <= 64 {
+ // u = trunc(x) fits into a uint64
+ u := msb64(x.mant) >> (64 - uint32(x.exp))
+ if x.MinPrec() <= 64 {
+ return u, Exact
+ }
+ return u, Below // x truncated
+ }
+ // x too large
+ return math.MaxUint64, Below
+
+ case zero:
+ return 0, Exact
+
+ case inf:
+ if x.neg {
+ return 0, Above
+ }
+ return math.MaxUint64, Below
+ }
+
+ panic("unreachable")
+}
+
+// Int64 returns the integer resulting from truncating x towards zero.
+// If math.MinInt64 <= x <= math.MaxInt64, the result is Exact if x is
+// an integer, and Above (x < 0) or Below (x > 0) otherwise.
+// The result is (math.MinInt64, Above) for x < math.MinInt64,
+// and (math.MaxInt64, Below) for x > math.MaxInt64.
+func (x *Float) Int64() (int64, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ switch x.form {
+ case finite:
+ // 0 < |x| < +Inf
+ acc := makeAcc(x.neg)
+ if x.exp <= 0 {
+ // 0 < |x| < 1
+ return 0, acc
+ }
+ // x.exp > 0
+
+ // 1 <= |x| < +Inf
+ if x.exp <= 63 {
+ // i = trunc(x) fits into an int64 (excluding math.MinInt64)
+ i := int64(msb64(x.mant) >> (64 - uint32(x.exp)))
+ if x.neg {
+ i = -i
+ }
+ if x.MinPrec() <= uint(x.exp) {
+ return i, Exact
+ }
+ return i, acc // x truncated
+ }
+ if x.neg {
+ // check for special case x == math.MinInt64 (i.e., x == -(0.5 << 64))
+ if x.exp == 64 && x.MinPrec() == 1 {
+ acc = Exact
+ }
+ return math.MinInt64, acc
+ }
+ // x too large
+ return math.MaxInt64, Below
+
+ case zero:
+ return 0, Exact
+
+ case inf:
+ if x.neg {
+ return math.MinInt64, Above
+ }
+ return math.MaxInt64, Below
+ }
+
+ panic("unreachable")
+}
+
+// Float32 returns the float32 value nearest to x. If x is too small to be
+// represented by a float32 (|x| < math.SmallestNonzeroFloat32), the result
+// is (0, Below) or (-0, Above), respectively, depending on the sign of x.
+// If x is too large to be represented by a float32 (|x| > math.MaxFloat32),
+// the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.
+func (x *Float) Float32() (float32, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ switch x.form {
+ case finite:
+ // 0 < |x| < +Inf
+
+ const (
+ fbits = 32 // float size
+ mbits = 23 // mantissa size (excluding implicit msb)
+ ebits = fbits - mbits - 1 // 8 exponent size
+ bias = 1<<(ebits-1) - 1 // 127 exponent bias
+ dmin = 1 - bias - mbits // -149 smallest unbiased exponent (denormal)
+ emin = 1 - bias // -126 smallest unbiased exponent (normal)
+ emax = bias // 127 largest unbiased exponent (normal)
+ )
+
+ // Float mantissa m is 0.5 <= m < 1.0; compute exponent e for float32 mantissa.
+ e := x.exp - 1 // exponent for normal mantissa m with 1.0 <= m < 2.0
+
+ // Compute precision p for float32 mantissa.
+ // If the exponent is too small, we have a denormal number before
+ // rounding and fewer than p mantissa bits of precision available
+ // (the exponent remains fixed but the mantissa gets shifted right).
+ p := mbits + 1 // precision of normal float
+ if e < emin {
+ // recompute precision
+ p = mbits + 1 - emin + int(e)
+ // If p == 0, the mantissa of x is shifted so much to the right
+ // that its msb falls immediately to the right of the float32
+ // mantissa space. In other words, if the smallest denormal is
+ // considered "1.0", for p == 0, the mantissa value m is >= 0.5.
+ // If m > 0.5, it is rounded up to 1.0; i.e., the smallest denormal.
+ // If m == 0.5, it is rounded down to even, i.e., 0.0.
+ // If p < 0, the mantissa value m is <= "0.25" which is never rounded up.
+ if p < 0 /* m <= 0.25 */ || p == 0 && x.mant.sticky(uint(len(x.mant))*_W-1) == 0 /* m == 0.5 */ {
+ // underflow to ±0
+ if x.neg {
+ var z float32
+ return -z, Above
+ }
+ return 0.0, Below
+ }
+ // otherwise, round up
+ // We handle p == 0 explicitly because it's easy and because
+ // Float.round doesn't support rounding to 0 bits of precision.
+ if p == 0 {
+ if x.neg {
+ return -math.SmallestNonzeroFloat32, Below
+ }
+ return math.SmallestNonzeroFloat32, Above
+ }
+ }
+ // p > 0
+
+ // round
+ var r Float
+ r.prec = uint32(p)
+ r.Set(x)
+ e = r.exp - 1
+
+ // Rounding may have caused r to overflow to ±Inf
+ // (rounding never causes underflows to 0).
+ // If the exponent is too large, also overflow to ±Inf.
+ if r.form == inf || e > emax {
+ // overflow
+ if x.neg {
+ return float32(math.Inf(-1)), Below
+ }
+ return float32(math.Inf(+1)), Above
+ }
+ // e <= emax
+
+ // Determine sign, biased exponent, and mantissa.
+ var sign, bexp, mant uint32
+ if x.neg {
+ sign = 1 << (fbits - 1)
+ }
+
+ // Rounding may have caused a denormal number to
+ // become normal. Check again.
+ if e < emin {
+ // denormal number: recompute precision
+ // Since rounding may have at best increased precision
+ // and we have eliminated p <= 0 early, we know p > 0.
+ // bexp == 0 for denormals
+ p = mbits + 1 - emin + int(e)
+ mant = msb32(r.mant) >> uint(fbits-p)
+ } else {
+ // normal number: emin <= e <= emax
+ bexp = uint32(e+bias) << mbits
+ mant = msb32(r.mant) >> ebits & (1<<mbits - 1) // cut off msb (implicit 1 bit)
+ }
+
+ return math.Float32frombits(sign | bexp | mant), r.acc
+
+ case zero:
+ if x.neg {
+ var z float32
+ return -z, Exact
+ }
+ return 0.0, Exact
+
+ case inf:
+ if x.neg {
+ return float32(math.Inf(-1)), Exact
+ }
+ return float32(math.Inf(+1)), Exact
+ }
+
+ panic("unreachable")
+}
+
+// Float64 returns the float64 value nearest to x. If x is too small to be
+// represented by a float64 (|x| < math.SmallestNonzeroFloat64), the result
+// is (0, Below) or (-0, Above), respectively, depending on the sign of x.
+// If x is too large to be represented by a float64 (|x| > math.MaxFloat64),
+// the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.
+func (x *Float) Float64() (float64, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ switch x.form {
+ case finite:
+ // 0 < |x| < +Inf
+
+ const (
+ fbits = 64 // float size
+ mbits = 52 // mantissa size (excluding implicit msb)
+ ebits = fbits - mbits - 1 // 11 exponent size
+ bias = 1<<(ebits-1) - 1 // 1023 exponent bias
+ dmin = 1 - bias - mbits // -1074 smallest unbiased exponent (denormal)
+ emin = 1 - bias // -1022 smallest unbiased exponent (normal)
+ emax = bias // 1023 largest unbiased exponent (normal)
+ )
+
+ // Float mantissa m is 0.5 <= m < 1.0; compute exponent e for float64 mantissa.
+ e := x.exp - 1 // exponent for normal mantissa m with 1.0 <= m < 2.0
+
+ // Compute precision p for float64 mantissa.
+ // If the exponent is too small, we have a denormal number before
+ // rounding and fewer than p mantissa bits of precision available
+ // (the exponent remains fixed but the mantissa gets shifted right).
+ p := mbits + 1 // precision of normal float
+ if e < emin {
+ // recompute precision
+ p = mbits + 1 - emin + int(e)
+ // If p == 0, the mantissa of x is shifted so much to the right
+ // that its msb falls immediately to the right of the float64
+ // mantissa space. In other words, if the smallest denormal is
+ // considered "1.0", for p == 0, the mantissa value m is >= 0.5.
+ // If m > 0.5, it is rounded up to 1.0; i.e., the smallest denormal.
+ // If m == 0.5, it is rounded down to even, i.e., 0.0.
+ // If p < 0, the mantissa value m is <= "0.25" which is never rounded up.
+ if p < 0 /* m <= 0.25 */ || p == 0 && x.mant.sticky(uint(len(x.mant))*_W-1) == 0 /* m == 0.5 */ {
+ // underflow to ±0
+ if x.neg {
+ var z float64
+ return -z, Above
+ }
+ return 0.0, Below
+ }
+ // otherwise, round up
+ // We handle p == 0 explicitly because it's easy and because
+ // Float.round doesn't support rounding to 0 bits of precision.
+ if p == 0 {
+ if x.neg {
+ return -math.SmallestNonzeroFloat64, Below
+ }
+ return math.SmallestNonzeroFloat64, Above
+ }
+ }
+ // p > 0
+
+ // round
+ var r Float
+ r.prec = uint32(p)
+ r.Set(x)
+ e = r.exp - 1
+
+ // Rounding may have caused r to overflow to ±Inf
+ // (rounding never causes underflows to 0).
+ // If the exponent is too large, also overflow to ±Inf.
+ if r.form == inf || e > emax {
+ // overflow
+ if x.neg {
+ return math.Inf(-1), Below
+ }
+ return math.Inf(+1), Above
+ }
+ // e <= emax
+
+ // Determine sign, biased exponent, and mantissa.
+ var sign, bexp, mant uint64
+ if x.neg {
+ sign = 1 << (fbits - 1)
+ }
+
+ // Rounding may have caused a denormal number to
+ // become normal. Check again.
+ if e < emin {
+ // denormal number: recompute precision
+ // Since rounding may have at best increased precision
+ // and we have eliminated p <= 0 early, we know p > 0.
+ // bexp == 0 for denormals
+ p = mbits + 1 - emin + int(e)
+ mant = msb64(r.mant) >> uint(fbits-p)
+ } else {
+ // normal number: emin <= e <= emax
+ bexp = uint64(e+bias) << mbits
+ mant = msb64(r.mant) >> ebits & (1<<mbits - 1) // cut off msb (implicit 1 bit)
+ }
+
+ return math.Float64frombits(sign | bexp | mant), r.acc
+
+ case zero:
+ if x.neg {
+ var z float64
+ return -z, Exact
+ }
+ return 0.0, Exact
+
+ case inf:
+ if x.neg {
+ return math.Inf(-1), Exact
+ }
+ return math.Inf(+1), Exact
+ }
+
+ panic("unreachable")
+}
+
+// Int returns the result of truncating x towards zero;
+// or nil if x is an infinity.
+// The result is Exact if x.IsInt(); otherwise it is Below
+// for x > 0, and Above for x < 0.
+// If a non-nil *Int argument z is provided, Int stores
+// the result in z instead of allocating a new Int.
+func (x *Float) Int(z *Int) (*Int, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ if z == nil && x.form <= finite {
+ z = new(Int)
+ }
+
+ switch x.form {
+ case finite:
+ // 0 < |x| < +Inf
+ acc := makeAcc(x.neg)
+ if x.exp <= 0 {
+ // 0 < |x| < 1
+ return z.SetInt64(0), acc
+ }
+ // x.exp > 0
+
+ // 1 <= |x| < +Inf
+ // determine minimum required precision for x
+ allBits := uint(len(x.mant)) * _W
+ exp := uint(x.exp)
+ if x.MinPrec() <= exp {
+ acc = Exact
+ }
+ // shift mantissa as needed
+ if z == nil {
+ z = new(Int)
+ }
+ z.neg = x.neg
+ switch {
+ case exp > allBits:
+ z.abs = z.abs.shl(x.mant, exp-allBits)
+ default:
+ z.abs = z.abs.set(x.mant)
+ case exp < allBits:
+ z.abs = z.abs.shr(x.mant, allBits-exp)
+ }
+ return z, acc
+
+ case zero:
+ return z.SetInt64(0), Exact
+
+ case inf:
+ return nil, makeAcc(x.neg)
+ }
+
+ panic("unreachable")
+}
+
+// Rat returns the rational number corresponding to x;
+// or nil if x is an infinity.
+// The result is Exact if x is not an Inf.
+// If a non-nil *Rat argument z is provided, Rat stores
+// the result in z instead of allocating a new Rat.
+func (x *Float) Rat(z *Rat) (*Rat, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ if z == nil && x.form <= finite {
+ z = new(Rat)
+ }
+
+ switch x.form {
+ case finite:
+ // 0 < |x| < +Inf
+ allBits := int32(len(x.mant)) * _W
+ // build up numerator and denominator
+ z.a.neg = x.neg
+ switch {
+ case x.exp > allBits:
+ z.a.abs = z.a.abs.shl(x.mant, uint(x.exp-allBits))
+ z.b.abs = z.b.abs[:0] // == 1 (see Rat)
+ // z already in normal form
+ default:
+ z.a.abs = z.a.abs.set(x.mant)
+ z.b.abs = z.b.abs[:0] // == 1 (see Rat)
+ // z already in normal form
+ case x.exp < allBits:
+ z.a.abs = z.a.abs.set(x.mant)
+ t := z.b.abs.setUint64(1)
+ z.b.abs = t.shl(t, uint(allBits-x.exp))
+ z.norm()
+ }
+ return z, Exact
+
+ case zero:
+ return z.SetInt64(0), Exact
+
+ case inf:
+ return nil, makeAcc(x.neg)
+ }
+
+ panic("unreachable")
+}
+
+// Abs sets z to the (possibly rounded) value |x| (the absolute value of x)
+// and returns z.
+func (z *Float) Abs(x *Float) *Float {
+ z.Set(x)
+ z.neg = false
+ return z
+}
+
+// Neg sets z to the (possibly rounded) value of x with its sign negated,
+// and returns z.
+func (z *Float) Neg(x *Float) *Float {
+ z.Set(x)
+ z.neg = !z.neg
+ return z
+}
+
+func validateBinaryOperands(x, y *Float) {
+ if !debugFloat {
+ // avoid performance bugs
+ panic("validateBinaryOperands called but debugFloat is not set")
+ }
+ if len(x.mant) == 0 {
+ panic("empty mantissa for x")
+ }
+ if len(y.mant) == 0 {
+ panic("empty mantissa for y")
+ }
+}
+
+// z = x + y, ignoring signs of x and y for the addition
+// but using the sign of z for rounding the result.
+// x and y must have a non-empty mantissa and valid exponent.
+func (z *Float) uadd(x, y *Float) {
+ // Note: This implementation requires 2 shifts most of the
+ // time. It is also inefficient if exponents or precisions
+ // differ by wide margins. The following article describes
+ // an efficient (but much more complicated) implementation
+ // compatible with the internal representation used here:
+ //
+ // Vincent Lefèvre: "The Generic Multiple-Precision Floating-
+ // Point Addition With Exact Rounding (as in the MPFR Library)"
+ // http://www.vinc17.net/research/papers/rnc6.pdf
+
+ if debugFloat {
+ validateBinaryOperands(x, y)
+ }
+
+ // compute exponents ex, ey for mantissa with "binary point"
+ // on the right (mantissa.0) - use int64 to avoid overflow
+ ex := int64(x.exp) - int64(len(x.mant))*_W
+ ey := int64(y.exp) - int64(len(y.mant))*_W
+
+ al := alias(z.mant, x.mant) || alias(z.mant, y.mant)
+
+ // TODO(gri) having a combined add-and-shift primitive
+ // could make this code significantly faster
+ switch {
+ case ex < ey:
+ if al {
+ t := nat(nil).shl(y.mant, uint(ey-ex))
+ z.mant = z.mant.add(x.mant, t)
+ } else {
+ z.mant = z.mant.shl(y.mant, uint(ey-ex))
+ z.mant = z.mant.add(x.mant, z.mant)
+ }
+ default:
+ // ex == ey, no shift needed
+ z.mant = z.mant.add(x.mant, y.mant)
+ case ex > ey:
+ if al {
+ t := nat(nil).shl(x.mant, uint(ex-ey))
+ z.mant = z.mant.add(t, y.mant)
+ } else {
+ z.mant = z.mant.shl(x.mant, uint(ex-ey))
+ z.mant = z.mant.add(z.mant, y.mant)
+ }
+ ex = ey
+ }
+ // len(z.mant) > 0
+
+ z.setExpAndRound(ex+int64(len(z.mant))*_W-fnorm(z.mant), 0)
+}
+
+// z = x - y for |x| > |y|, ignoring signs of x and y for the subtraction
+// but using the sign of z for rounding the result.
+// x and y must have a non-empty mantissa and valid exponent.
+func (z *Float) usub(x, y *Float) {
+ // This code is symmetric to uadd.
+ // We have not factored the common code out because
+ // eventually uadd (and usub) should be optimized
+ // by special-casing, and the code will diverge.
+
+ if debugFloat {
+ validateBinaryOperands(x, y)
+ }
+
+ ex := int64(x.exp) - int64(len(x.mant))*_W
+ ey := int64(y.exp) - int64(len(y.mant))*_W
+
+ al := alias(z.mant, x.mant) || alias(z.mant, y.mant)
+
+ switch {
+ case ex < ey:
+ if al {
+ t := nat(nil).shl(y.mant, uint(ey-ex))
+ z.mant = t.sub(x.mant, t)
+ } else {
+ z.mant = z.mant.shl(y.mant, uint(ey-ex))
+ z.mant = z.mant.sub(x.mant, z.mant)
+ }
+ default:
+ // ex == ey, no shift needed
+ z.mant = z.mant.sub(x.mant, y.mant)
+ case ex > ey:
+ if al {
+ t := nat(nil).shl(x.mant, uint(ex-ey))
+ z.mant = t.sub(t, y.mant)
+ } else {
+ z.mant = z.mant.shl(x.mant, uint(ex-ey))
+ z.mant = z.mant.sub(z.mant, y.mant)
+ }
+ ex = ey
+ }
+
+ // operands may have canceled each other out
+ if len(z.mant) == 0 {
+ z.acc = Exact
+ z.form = zero
+ z.neg = false
+ return
+ }
+ // len(z.mant) > 0
+
+ z.setExpAndRound(ex+int64(len(z.mant))*_W-fnorm(z.mant), 0)
+}
+
+// z = x * y, ignoring signs of x and y for the multiplication
+// but using the sign of z for rounding the result.
+// x and y must have a non-empty mantissa and valid exponent.
+func (z *Float) umul(x, y *Float) {
+ if debugFloat {
+ validateBinaryOperands(x, y)
+ }
+
+ // Note: This is doing too much work if the precision
+ // of z is less than the sum of the precisions of x
+ // and y which is often the case (e.g., if all floats
+ // have the same precision).
+ // TODO(gri) Optimize this for the common case.
+
+ e := int64(x.exp) + int64(y.exp)
+ if x == y {
+ z.mant = z.mant.sqr(x.mant)
+ } else {
+ z.mant = z.mant.mul(x.mant, y.mant)
+ }
+ z.setExpAndRound(e-fnorm(z.mant), 0)
+}
+
+// z = x / y, ignoring signs of x and y for the division
+// but using the sign of z for rounding the result.
+// x and y must have a non-empty mantissa and valid exponent.
+func (z *Float) uquo(x, y *Float) {
+ if debugFloat {
+ validateBinaryOperands(x, y)
+ }
+
+ // mantissa length in words for desired result precision + 1
+ // (at least one extra bit so we get the rounding bit after
+ // the division)
+ n := int(z.prec/_W) + 1
+
+ // compute adjusted x.mant such that we get enough result precision
+ xadj := x.mant
+ if d := n - len(x.mant) + len(y.mant); d > 0 {
+ // d extra words needed => add d "0 digits" to x
+ xadj = make(nat, len(x.mant)+d)
+ copy(xadj[d:], x.mant)
+ }
+ // TODO(gri): If we have too many digits (d < 0), we should be able
+ // to shorten x for faster division. But we must be extra careful
+ // with rounding in that case.
+
+ // Compute d before division since there may be aliasing of x.mant
+ // (via xadj) or y.mant with z.mant.
+ d := len(xadj) - len(y.mant)
+
+ // divide
+ var r nat
+ z.mant, r = z.mant.div(nil, xadj, y.mant)
+ e := int64(x.exp) - int64(y.exp) - int64(d-len(z.mant))*_W
+
+ // The result is long enough to include (at least) the rounding bit.
+ // If there's a non-zero remainder, the corresponding fractional part
+ // (if it were computed), would have a non-zero sticky bit (if it were
+ // zero, it couldn't have a non-zero remainder).
+ var sbit uint
+ if len(r) > 0 {
+ sbit = 1
+ }
+
+ z.setExpAndRound(e-fnorm(z.mant), sbit)
+}
+
+// ucmp returns -1, 0, or +1, depending on whether
+// |x| < |y|, |x| == |y|, or |x| > |y|.
+// x and y must have a non-empty mantissa and valid exponent.
+func (x *Float) ucmp(y *Float) int {
+ if debugFloat {
+ validateBinaryOperands(x, y)
+ }
+
+ switch {
+ case x.exp < y.exp:
+ return -1
+ case x.exp > y.exp:
+ return +1
+ }
+ // x.exp == y.exp
+
+ // compare mantissas
+ i := len(x.mant)
+ j := len(y.mant)
+ for i > 0 || j > 0 {
+ var xm, ym Word
+ if i > 0 {
+ i--
+ xm = x.mant[i]
+ }
+ if j > 0 {
+ j--
+ ym = y.mant[j]
+ }
+ switch {
+ case xm < ym:
+ return -1
+ case xm > ym:
+ return +1
+ }
+ }
+
+ return 0
+}
+
+// Handling of sign bit as defined by IEEE 754-2008, section 6.3:
+//
+// When neither the inputs nor result are NaN, the sign of a product or
+// quotient is the exclusive OR of the operands’ signs; the sign of a sum,
+// or of a difference x−y regarded as a sum x+(−y), differs from at most
+// one of the addends’ signs; and the sign of the result of conversions,
+// the quantize operation, the roundToIntegral operations, and the
+// roundToIntegralExact (see 5.3.1) is the sign of the first or only operand.
+// These rules shall apply even when operands or results are zero or infinite.
+//
+// When the sum of two operands with opposite signs (or the difference of
+// two operands with like signs) is exactly zero, the sign of that sum (or
+// difference) shall be +0 in all rounding-direction attributes except
+// roundTowardNegative; under that attribute, the sign of an exact zero
+// sum (or difference) shall be −0. However, x+x = x−(−x) retains the same
+// sign as x even when x is zero.
+//
+// See also: https://play.golang.org/p/RtH3UCt5IH
+
+// Add sets z to the rounded sum x+y and returns z. If z's precision is 0,
+// it is changed to the larger of x's or y's precision before the operation.
+// Rounding is performed according to z's precision and rounding mode; and
+// z's accuracy reports the result error relative to the exact (not rounded)
+// result. Add panics with ErrNaN if x and y are infinities with opposite
+// signs. The value of z is undefined in that case.
+func (z *Float) Add(x, y *Float) *Float {
+ if debugFloat {
+ x.validate()
+ y.validate()
+ }
+
+ if z.prec == 0 {
+ z.prec = umax32(x.prec, y.prec)
+ }
+
+ if x.form == finite && y.form == finite {
+ // x + y (common case)
+
+ // Below we set z.neg = x.neg, and when z aliases y this will
+ // change the y operand's sign. This is fine, because if an
+ // operand aliases the receiver it'll be overwritten, but we still
+ // want the original x.neg and y.neg values when we evaluate
+ // x.neg != y.neg, so we need to save y.neg before setting z.neg.
+ yneg := y.neg
+
+ z.neg = x.neg
+ if x.neg == yneg {
+ // x + y == x + y
+ // (-x) + (-y) == -(x + y)
+ z.uadd(x, y)
+ } else {
+ // x + (-y) == x - y == -(y - x)
+ // (-x) + y == y - x == -(x - y)
+ if x.ucmp(y) > 0 {
+ z.usub(x, y)
+ } else {
+ z.neg = !z.neg
+ z.usub(y, x)
+ }
+ }
+ if z.form == zero && z.mode == ToNegativeInf && z.acc == Exact {
+ z.neg = true
+ }
+ return z
+ }
+
+ if x.form == inf && y.form == inf && x.neg != y.neg {
+ // +Inf + -Inf
+ // -Inf + +Inf
+ // value of z is undefined but make sure it's valid
+ z.acc = Exact
+ z.form = zero
+ z.neg = false
+ panic(ErrNaN{"addition of infinities with opposite signs"})
+ }
+
+ if x.form == zero && y.form == zero {
+ // ±0 + ±0
+ z.acc = Exact
+ z.form = zero
+ z.neg = x.neg && y.neg // -0 + -0 == -0
+ return z
+ }
+
+ if x.form == inf || y.form == zero {
+ // ±Inf + y
+ // x + ±0
+ return z.Set(x)
+ }
+
+ // ±0 + y
+ // x + ±Inf
+ return z.Set(y)
+}
+
+// Sub sets z to the rounded difference x-y and returns z.
+// Precision, rounding, and accuracy reporting are as for Add.
+// Sub panics with ErrNaN if x and y are infinities with equal
+// signs. The value of z is undefined in that case.
+func (z *Float) Sub(x, y *Float) *Float {
+ if debugFloat {
+ x.validate()
+ y.validate()
+ }
+
+ if z.prec == 0 {
+ z.prec = umax32(x.prec, y.prec)
+ }
+
+ if x.form == finite && y.form == finite {
+ // x - y (common case)
+ yneg := y.neg
+ z.neg = x.neg
+ if x.neg != yneg {
+ // x - (-y) == x + y
+ // (-x) - y == -(x + y)
+ z.uadd(x, y)
+ } else {
+ // x - y == x - y == -(y - x)
+ // (-x) - (-y) == y - x == -(x - y)
+ if x.ucmp(y) > 0 {
+ z.usub(x, y)
+ } else {
+ z.neg = !z.neg
+ z.usub(y, x)
+ }
+ }
+ if z.form == zero && z.mode == ToNegativeInf && z.acc == Exact {
+ z.neg = true
+ }
+ return z
+ }
+
+ if x.form == inf && y.form == inf && x.neg == y.neg {
+ // +Inf - +Inf
+ // -Inf - -Inf
+ // value of z is undefined but make sure it's valid
+ z.acc = Exact
+ z.form = zero
+ z.neg = false
+ panic(ErrNaN{"subtraction of infinities with equal signs"})
+ }
+
+ if x.form == zero && y.form == zero {
+ // ±0 - ±0
+ z.acc = Exact
+ z.form = zero
+ z.neg = x.neg && !y.neg // -0 - +0 == -0
+ return z
+ }
+
+ if x.form == inf || y.form == zero {
+ // ±Inf - y
+ // x - ±0
+ return z.Set(x)
+ }
+
+ // ±0 - y
+ // x - ±Inf
+ return z.Neg(y)
+}
+
+// Mul sets z to the rounded product x*y and returns z.
+// Precision, rounding, and accuracy reporting are as for Add.
+// Mul panics with ErrNaN if one operand is zero and the other
+// operand an infinity. The value of z is undefined in that case.
+func (z *Float) Mul(x, y *Float) *Float {
+ if debugFloat {
+ x.validate()
+ y.validate()
+ }
+
+ if z.prec == 0 {
+ z.prec = umax32(x.prec, y.prec)
+ }
+
+ z.neg = x.neg != y.neg
+
+ if x.form == finite && y.form == finite {
+ // x * y (common case)
+ z.umul(x, y)
+ return z
+ }
+
+ z.acc = Exact
+ if x.form == zero && y.form == inf || x.form == inf && y.form == zero {
+ // ±0 * ±Inf
+ // ±Inf * ±0
+ // value of z is undefined but make sure it's valid
+ z.form = zero
+ z.neg = false
+ panic(ErrNaN{"multiplication of zero with infinity"})
+ }
+
+ if x.form == inf || y.form == inf {
+ // ±Inf * y
+ // x * ±Inf
+ z.form = inf
+ return z
+ }
+
+ // ±0 * y
+ // x * ±0
+ z.form = zero
+ return z
+}
+
+// Quo sets z to the rounded quotient x/y and returns z.
+// Precision, rounding, and accuracy reporting are as for Add.
+// Quo panics with ErrNaN if both operands are zero or infinities.
+// The value of z is undefined in that case.
+func (z *Float) Quo(x, y *Float) *Float {
+ if debugFloat {
+ x.validate()
+ y.validate()
+ }
+
+ if z.prec == 0 {
+ z.prec = umax32(x.prec, y.prec)
+ }
+
+ z.neg = x.neg != y.neg
+
+ if x.form == finite && y.form == finite {
+ // x / y (common case)
+ z.uquo(x, y)
+ return z
+ }
+
+ z.acc = Exact
+ if x.form == zero && y.form == zero || x.form == inf && y.form == inf {
+ // ±0 / ±0
+ // ±Inf / ±Inf
+ // value of z is undefined but make sure it's valid
+ z.form = zero
+ z.neg = false
+ panic(ErrNaN{"division of zero by zero or infinity by infinity"})
+ }
+
+ if x.form == zero || y.form == inf {
+ // ±0 / y
+ // x / ±Inf
+ z.form = zero
+ return z
+ }
+
+ // x / ±0
+ // ±Inf / y
+ z.form = inf
+ return z
+}
+
+// Cmp compares x and y and returns:
+//
+// -1 if x < y
+// 0 if x == y (incl. -0 == 0, -Inf == -Inf, and +Inf == +Inf)
+// +1 if x > y
+//
+func (x *Float) Cmp(y *Float) int {
+ if debugFloat {
+ x.validate()
+ y.validate()
+ }
+
+ mx := x.ord()
+ my := y.ord()
+ switch {
+ case mx < my:
+ return -1
+ case mx > my:
+ return +1
+ }
+ // mx == my
+
+ // only if |mx| == 1 we have to compare the mantissae
+ switch mx {
+ case -1:
+ return y.ucmp(x)
+ case +1:
+ return x.ucmp(y)
+ }
+
+ return 0
+}
+
+// ord classifies x and returns:
+//
+// -2 if -Inf == x
+// -1 if -Inf < x < 0
+// 0 if x == 0 (signed or unsigned)
+// +1 if 0 < x < +Inf
+// +2 if x == +Inf
+//
+func (x *Float) ord() int {
+ var m int
+ switch x.form {
+ case finite:
+ m = 1
+ case zero:
+ return 0
+ case inf:
+ m = 2
+ }
+ if x.neg {
+ m = -m
+ }
+ return m
+}
+
+func umax32(x, y uint32) uint32 {
+ if x > y {
+ return x
+ }
+ return y
+}
diff --git a/src/math/big/float_test.go b/src/math/big/float_test.go
new file mode 100644
index 0000000..7d6bf03
--- /dev/null
+++ b/src/math/big/float_test.go
@@ -0,0 +1,1858 @@
+// Copyright 2014 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "flag"
+ "fmt"
+ "math"
+ "strconv"
+ "strings"
+ "testing"
+)
+
+// Verify that ErrNaN implements the error interface.
+var _ error = ErrNaN{}
+
+func (x *Float) uint64() uint64 {
+ u, acc := x.Uint64()
+ if acc != Exact {
+ panic(fmt.Sprintf("%s is not a uint64", x.Text('g', 10)))
+ }
+ return u
+}
+
+func (x *Float) int64() int64 {
+ i, acc := x.Int64()
+ if acc != Exact {
+ panic(fmt.Sprintf("%s is not an int64", x.Text('g', 10)))
+ }
+ return i
+}
+
+func TestFloatZeroValue(t *testing.T) {
+ // zero (uninitialized) value is a ready-to-use 0.0
+ var x Float
+ if s := x.Text('f', 1); s != "0.0" {
+ t.Errorf("zero value = %s; want 0.0", s)
+ }
+
+ // zero value has precision 0
+ if prec := x.Prec(); prec != 0 {
+ t.Errorf("prec = %d; want 0", prec)
+ }
+
+ // zero value can be used in any and all positions of binary operations
+ make := func(x int) *Float {
+ var f Float
+ if x != 0 {
+ f.SetInt64(int64(x))
+ }
+ // x == 0 translates into the zero value
+ return &f
+ }
+ for _, test := range []struct {
+ z, x, y, want int
+ opname rune
+ op func(z, x, y *Float) *Float
+ }{
+ {0, 0, 0, 0, '+', (*Float).Add},
+ {0, 1, 2, 3, '+', (*Float).Add},
+ {1, 2, 0, 2, '+', (*Float).Add},
+ {2, 0, 1, 1, '+', (*Float).Add},
+
+ {0, 0, 0, 0, '-', (*Float).Sub},
+ {0, 1, 2, -1, '-', (*Float).Sub},
+ {1, 2, 0, 2, '-', (*Float).Sub},
+ {2, 0, 1, -1, '-', (*Float).Sub},
+
+ {0, 0, 0, 0, '*', (*Float).Mul},
+ {0, 1, 2, 2, '*', (*Float).Mul},
+ {1, 2, 0, 0, '*', (*Float).Mul},
+ {2, 0, 1, 0, '*', (*Float).Mul},
+
+ // {0, 0, 0, 0, '/', (*Float).Quo}, // panics
+ {0, 2, 1, 2, '/', (*Float).Quo},
+ {1, 2, 0, 0, '/', (*Float).Quo}, // = +Inf
+ {2, 0, 1, 0, '/', (*Float).Quo},
+ } {
+ z := make(test.z)
+ test.op(z, make(test.x), make(test.y))
+ got := 0
+ if !z.IsInf() {
+ got = int(z.int64())
+ }
+ if got != test.want {
+ t.Errorf("%d %c %d = %d; want %d", test.x, test.opname, test.y, got, test.want)
+ }
+ }
+
+ // TODO(gri) test how precision is set for zero value results
+}
+
+func makeFloat(s string) *Float {
+ x, _, err := ParseFloat(s, 0, 1000, ToNearestEven)
+ if err != nil {
+ panic(err)
+ }
+ return x
+}
+
+func TestFloatSetPrec(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ prec uint
+ want string
+ acc Accuracy
+ }{
+ // prec 0
+ {"0", 0, "0", Exact},
+ {"-0", 0, "-0", Exact},
+ {"-Inf", 0, "-Inf", Exact},
+ {"+Inf", 0, "+Inf", Exact},
+ {"123", 0, "0", Below},
+ {"-123", 0, "-0", Above},
+
+ // prec at upper limit
+ {"0", MaxPrec, "0", Exact},
+ {"-0", MaxPrec, "-0", Exact},
+ {"-Inf", MaxPrec, "-Inf", Exact},
+ {"+Inf", MaxPrec, "+Inf", Exact},
+
+ // just a few regular cases - general rounding is tested elsewhere
+ {"1.5", 1, "2", Above},
+ {"-1.5", 1, "-2", Below},
+ {"123", 1e6, "123", Exact},
+ {"-123", 1e6, "-123", Exact},
+ } {
+ x := makeFloat(test.x).SetPrec(test.prec)
+ prec := test.prec
+ if prec > MaxPrec {
+ prec = MaxPrec
+ }
+ if got := x.Prec(); got != prec {
+ t.Errorf("%s.SetPrec(%d).Prec() == %d; want %d", test.x, test.prec, got, prec)
+ }
+ if got, acc := x.String(), x.Acc(); got != test.want || acc != test.acc {
+ t.Errorf("%s.SetPrec(%d) = %s (%s); want %s (%s)", test.x, test.prec, got, acc, test.want, test.acc)
+ }
+ }
+}
+
+func TestFloatMinPrec(t *testing.T) {
+ const max = 100
+ for _, test := range []struct {
+ x string
+ want uint
+ }{
+ {"0", 0},
+ {"-0", 0},
+ {"+Inf", 0},
+ {"-Inf", 0},
+ {"1", 1},
+ {"2", 1},
+ {"3", 2},
+ {"0x8001", 16},
+ {"0x8001p-1000", 16},
+ {"0x8001p+1000", 16},
+ {"0.1", max},
+ } {
+ x := makeFloat(test.x).SetPrec(max)
+ if got := x.MinPrec(); got != test.want {
+ t.Errorf("%s.MinPrec() = %d; want %d", test.x, got, test.want)
+ }
+ }
+}
+
+func TestFloatSign(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ s int
+ }{
+ {"-Inf", -1},
+ {"-1", -1},
+ {"-0", 0},
+ {"+0", 0},
+ {"+1", +1},
+ {"+Inf", +1},
+ } {
+ x := makeFloat(test.x)
+ s := x.Sign()
+ if s != test.s {
+ t.Errorf("%s.Sign() = %d; want %d", test.x, s, test.s)
+ }
+ }
+}
+
+// alike(x, y) is like x.Cmp(y) == 0 but also considers the sign of 0 (0 != -0).
+func alike(x, y *Float) bool {
+ return x.Cmp(y) == 0 && x.Signbit() == y.Signbit()
+}
+
+func alike32(x, y float32) bool {
+ // we can ignore NaNs
+ return x == y && math.Signbit(float64(x)) == math.Signbit(float64(y))
+
+}
+
+func alike64(x, y float64) bool {
+ // we can ignore NaNs
+ return x == y && math.Signbit(x) == math.Signbit(y)
+
+}
+
+func TestFloatMantExp(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ mant string
+ exp int
+ }{
+ {"0", "0", 0},
+ {"+0", "0", 0},
+ {"-0", "-0", 0},
+ {"Inf", "+Inf", 0},
+ {"+Inf", "+Inf", 0},
+ {"-Inf", "-Inf", 0},
+ {"1.5", "0.75", 1},
+ {"1.024e3", "0.5", 11},
+ {"-0.125", "-0.5", -2},
+ } {
+ x := makeFloat(test.x)
+ mant := makeFloat(test.mant)
+ m := new(Float)
+ e := x.MantExp(m)
+ if !alike(m, mant) || e != test.exp {
+ t.Errorf("%s.MantExp() = %s, %d; want %s, %d", test.x, m.Text('g', 10), e, test.mant, test.exp)
+ }
+ }
+}
+
+func TestFloatMantExpAliasing(t *testing.T) {
+ x := makeFloat("0.5p10")
+ if e := x.MantExp(x); e != 10 {
+ t.Fatalf("Float.MantExp aliasing error: got %d; want 10", e)
+ }
+ if want := makeFloat("0.5"); !alike(x, want) {
+ t.Fatalf("Float.MantExp aliasing error: got %s; want %s", x.Text('g', 10), want.Text('g', 10))
+ }
+}
+
+func TestFloatSetMantExp(t *testing.T) {
+ for _, test := range []struct {
+ frac string
+ exp int
+ z string
+ }{
+ {"0", 0, "0"},
+ {"+0", 0, "0"},
+ {"-0", 0, "-0"},
+ {"Inf", 1234, "+Inf"},
+ {"+Inf", -1234, "+Inf"},
+ {"-Inf", -1234, "-Inf"},
+ {"0", MinExp, "0"},
+ {"0.25", MinExp, "+0"}, // exponent underflow
+ {"-0.25", MinExp, "-0"}, // exponent underflow
+ {"1", MaxExp, "+Inf"}, // exponent overflow
+ {"2", MaxExp - 1, "+Inf"}, // exponent overflow
+ {"0.75", 1, "1.5"},
+ {"0.5", 11, "1024"},
+ {"-0.5", -2, "-0.125"},
+ {"32", 5, "1024"},
+ {"1024", -10, "1"},
+ } {
+ frac := makeFloat(test.frac)
+ want := makeFloat(test.z)
+ var z Float
+ z.SetMantExp(frac, test.exp)
+ if !alike(&z, want) {
+ t.Errorf("SetMantExp(%s, %d) = %s; want %s", test.frac, test.exp, z.Text('g', 10), test.z)
+ }
+ // test inverse property
+ mant := new(Float)
+ if z.SetMantExp(mant, want.MantExp(mant)).Cmp(want) != 0 {
+ t.Errorf("Inverse property not satisfied: got %s; want %s", z.Text('g', 10), test.z)
+ }
+ }
+}
+
+func TestFloatPredicates(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ sign int
+ signbit, inf bool
+ }{
+ {x: "-Inf", sign: -1, signbit: true, inf: true},
+ {x: "-1", sign: -1, signbit: true},
+ {x: "-0", signbit: true},
+ {x: "0"},
+ {x: "1", sign: 1},
+ {x: "+Inf", sign: 1, inf: true},
+ } {
+ x := makeFloat(test.x)
+ if got := x.Signbit(); got != test.signbit {
+ t.Errorf("(%s).Signbit() = %v; want %v", test.x, got, test.signbit)
+ }
+ if got := x.Sign(); got != test.sign {
+ t.Errorf("(%s).Sign() = %d; want %d", test.x, got, test.sign)
+ }
+ if got := x.IsInf(); got != test.inf {
+ t.Errorf("(%s).IsInf() = %v; want %v", test.x, got, test.inf)
+ }
+ }
+}
+
+func TestFloatIsInt(t *testing.T) {
+ for _, test := range []string{
+ "0 int",
+ "-0 int",
+ "1 int",
+ "-1 int",
+ "0.5",
+ "1.23",
+ "1.23e1",
+ "1.23e2 int",
+ "0.000000001e+8",
+ "0.000000001e+9 int",
+ "1.2345e200 int",
+ "Inf",
+ "+Inf",
+ "-Inf",
+ } {
+ s := strings.TrimSuffix(test, " int")
+ want := s != test
+ if got := makeFloat(s).IsInt(); got != want {
+ t.Errorf("%s.IsInt() == %t", s, got)
+ }
+ }
+}
+
+func fromBinary(s string) int64 {
+ x, err := strconv.ParseInt(s, 2, 64)
+ if err != nil {
+ panic(err)
+ }
+ return x
+}
+
+func toBinary(x int64) string {
+ return strconv.FormatInt(x, 2)
+}
+
+func testFloatRound(t *testing.T, x, r int64, prec uint, mode RoundingMode) {
+ // verify test data
+ var ok bool
+ switch mode {
+ case ToNearestEven, ToNearestAway:
+ ok = true // nothing to do for now
+ case ToZero:
+ if x < 0 {
+ ok = r >= x
+ } else {
+ ok = r <= x
+ }
+ case AwayFromZero:
+ if x < 0 {
+ ok = r <= x
+ } else {
+ ok = r >= x
+ }
+ case ToNegativeInf:
+ ok = r <= x
+ case ToPositiveInf:
+ ok = r >= x
+ default:
+ panic("unreachable")
+ }
+ if !ok {
+ t.Fatalf("incorrect test data for prec = %d, %s: x = %s, r = %s", prec, mode, toBinary(x), toBinary(r))
+ }
+
+ // compute expected accuracy
+ a := Exact
+ switch {
+ case r < x:
+ a = Below
+ case r > x:
+ a = Above
+ }
+
+ // round
+ f := new(Float).SetMode(mode).SetInt64(x).SetPrec(prec)
+
+ // check result
+ r1 := f.int64()
+ p1 := f.Prec()
+ a1 := f.Acc()
+ if r1 != r || p1 != prec || a1 != a {
+ t.Errorf("round %s (%d bits, %s) incorrect: got %s (%d bits, %s); want %s (%d bits, %s)",
+ toBinary(x), prec, mode,
+ toBinary(r1), p1, a1,
+ toBinary(r), prec, a)
+ return
+ }
+
+ // g and f should be the same
+ // (rounding by SetPrec after SetInt64 using default precision
+ // should be the same as rounding by SetInt64 after setting the
+ // precision)
+ g := new(Float).SetMode(mode).SetPrec(prec).SetInt64(x)
+ if !alike(g, f) {
+ t.Errorf("round %s (%d bits, %s) not symmetric: got %s and %s; want %s",
+ toBinary(x), prec, mode,
+ toBinary(g.int64()),
+ toBinary(r1),
+ toBinary(r),
+ )
+ return
+ }
+
+ // h and f should be the same
+ // (repeated rounding should be idempotent)
+ h := new(Float).SetMode(mode).SetPrec(prec).Set(f)
+ if !alike(h, f) {
+ t.Errorf("round %s (%d bits, %s) not idempotent: got %s and %s; want %s",
+ toBinary(x), prec, mode,
+ toBinary(h.int64()),
+ toBinary(r1),
+ toBinary(r),
+ )
+ return
+ }
+}
+
+// TestFloatRound tests basic rounding.
+func TestFloatRound(t *testing.T) {
+ for _, test := range []struct {
+ prec uint
+ x, zero, neven, naway, away string // input, results rounded to prec bits
+ }{
+ {5, "1000", "1000", "1000", "1000", "1000"},
+ {5, "1001", "1001", "1001", "1001", "1001"},
+ {5, "1010", "1010", "1010", "1010", "1010"},
+ {5, "1011", "1011", "1011", "1011", "1011"},
+ {5, "1100", "1100", "1100", "1100", "1100"},
+ {5, "1101", "1101", "1101", "1101", "1101"},
+ {5, "1110", "1110", "1110", "1110", "1110"},
+ {5, "1111", "1111", "1111", "1111", "1111"},
+
+ {4, "1000", "1000", "1000", "1000", "1000"},
+ {4, "1001", "1001", "1001", "1001", "1001"},
+ {4, "1010", "1010", "1010", "1010", "1010"},
+ {4, "1011", "1011", "1011", "1011", "1011"},
+ {4, "1100", "1100", "1100", "1100", "1100"},
+ {4, "1101", "1101", "1101", "1101", "1101"},
+ {4, "1110", "1110", "1110", "1110", "1110"},
+ {4, "1111", "1111", "1111", "1111", "1111"},
+
+ {3, "1000", "1000", "1000", "1000", "1000"},
+ {3, "1001", "1000", "1000", "1010", "1010"},
+ {3, "1010", "1010", "1010", "1010", "1010"},
+ {3, "1011", "1010", "1100", "1100", "1100"},
+ {3, "1100", "1100", "1100", "1100", "1100"},
+ {3, "1101", "1100", "1100", "1110", "1110"},
+ {3, "1110", "1110", "1110", "1110", "1110"},
+ {3, "1111", "1110", "10000", "10000", "10000"},
+
+ {3, "1000001", "1000000", "1000000", "1000000", "1010000"},
+ {3, "1001001", "1000000", "1010000", "1010000", "1010000"},
+ {3, "1010001", "1010000", "1010000", "1010000", "1100000"},
+ {3, "1011001", "1010000", "1100000", "1100000", "1100000"},
+ {3, "1100001", "1100000", "1100000", "1100000", "1110000"},
+ {3, "1101001", "1100000", "1110000", "1110000", "1110000"},
+ {3, "1110001", "1110000", "1110000", "1110000", "10000000"},
+ {3, "1111001", "1110000", "10000000", "10000000", "10000000"},
+
+ {2, "1000", "1000", "1000", "1000", "1000"},
+ {2, "1001", "1000", "1000", "1000", "1100"},
+ {2, "1010", "1000", "1000", "1100", "1100"},
+ {2, "1011", "1000", "1100", "1100", "1100"},
+ {2, "1100", "1100", "1100", "1100", "1100"},
+ {2, "1101", "1100", "1100", "1100", "10000"},
+ {2, "1110", "1100", "10000", "10000", "10000"},
+ {2, "1111", "1100", "10000", "10000", "10000"},
+
+ {2, "1000001", "1000000", "1000000", "1000000", "1100000"},
+ {2, "1001001", "1000000", "1000000", "1000000", "1100000"},
+ {2, "1010001", "1000000", "1100000", "1100000", "1100000"},
+ {2, "1011001", "1000000", "1100000", "1100000", "1100000"},
+ {2, "1100001", "1100000", "1100000", "1100000", "10000000"},
+ {2, "1101001", "1100000", "1100000", "1100000", "10000000"},
+ {2, "1110001", "1100000", "10000000", "10000000", "10000000"},
+ {2, "1111001", "1100000", "10000000", "10000000", "10000000"},
+
+ {1, "1000", "1000", "1000", "1000", "1000"},
+ {1, "1001", "1000", "1000", "1000", "10000"},
+ {1, "1010", "1000", "1000", "1000", "10000"},
+ {1, "1011", "1000", "1000", "1000", "10000"},
+ {1, "1100", "1000", "10000", "10000", "10000"},
+ {1, "1101", "1000", "10000", "10000", "10000"},
+ {1, "1110", "1000", "10000", "10000", "10000"},
+ {1, "1111", "1000", "10000", "10000", "10000"},
+
+ {1, "1000001", "1000000", "1000000", "1000000", "10000000"},
+ {1, "1001001", "1000000", "1000000", "1000000", "10000000"},
+ {1, "1010001", "1000000", "1000000", "1000000", "10000000"},
+ {1, "1011001", "1000000", "1000000", "1000000", "10000000"},
+ {1, "1100001", "1000000", "10000000", "10000000", "10000000"},
+ {1, "1101001", "1000000", "10000000", "10000000", "10000000"},
+ {1, "1110001", "1000000", "10000000", "10000000", "10000000"},
+ {1, "1111001", "1000000", "10000000", "10000000", "10000000"},
+ } {
+ x := fromBinary(test.x)
+ z := fromBinary(test.zero)
+ e := fromBinary(test.neven)
+ n := fromBinary(test.naway)
+ a := fromBinary(test.away)
+ prec := test.prec
+
+ testFloatRound(t, x, z, prec, ToZero)
+ testFloatRound(t, x, e, prec, ToNearestEven)
+ testFloatRound(t, x, n, prec, ToNearestAway)
+ testFloatRound(t, x, a, prec, AwayFromZero)
+
+ testFloatRound(t, x, z, prec, ToNegativeInf)
+ testFloatRound(t, x, a, prec, ToPositiveInf)
+
+ testFloatRound(t, -x, -a, prec, ToNegativeInf)
+ testFloatRound(t, -x, -z, prec, ToPositiveInf)
+ }
+}
+
+// TestFloatRound24 tests that rounding a float64 to 24 bits
+// matches IEEE-754 rounding to nearest when converting a
+// float64 to a float32 (excluding denormal numbers).
+func TestFloatRound24(t *testing.T) {
+ const x0 = 1<<26 - 0x10 // 11...110000 (26 bits)
+ for d := 0; d <= 0x10; d++ {
+ x := float64(x0 + d)
+ f := new(Float).SetPrec(24).SetFloat64(x)
+ got, _ := f.Float32()
+ want := float32(x)
+ if got != want {
+ t.Errorf("Round(%g, 24) = %g; want %g", x, got, want)
+ }
+ }
+}
+
+func TestFloatSetUint64(t *testing.T) {
+ for _, want := range []uint64{
+ 0,
+ 1,
+ 2,
+ 10,
+ 100,
+ 1<<32 - 1,
+ 1 << 32,
+ 1<<64 - 1,
+ } {
+ var f Float
+ f.SetUint64(want)
+ if got := f.uint64(); got != want {
+ t.Errorf("got %#x (%s); want %#x", got, f.Text('p', 0), want)
+ }
+ }
+
+ // test basic rounding behavior (exhaustive rounding testing is done elsewhere)
+ const x uint64 = 0x8765432187654321 // 64 bits needed
+ for prec := uint(1); prec <= 64; prec++ {
+ f := new(Float).SetPrec(prec).SetMode(ToZero).SetUint64(x)
+ got := f.uint64()
+ want := x &^ (1<<(64-prec) - 1) // cut off (round to zero) low 64-prec bits
+ if got != want {
+ t.Errorf("got %#x (%s); want %#x", got, f.Text('p', 0), want)
+ }
+ }
+}
+
+func TestFloatSetInt64(t *testing.T) {
+ for _, want := range []int64{
+ 0,
+ 1,
+ 2,
+ 10,
+ 100,
+ 1<<32 - 1,
+ 1 << 32,
+ 1<<63 - 1,
+ } {
+ for i := range [2]int{} {
+ if i&1 != 0 {
+ want = -want
+ }
+ var f Float
+ f.SetInt64(want)
+ if got := f.int64(); got != want {
+ t.Errorf("got %#x (%s); want %#x", got, f.Text('p', 0), want)
+ }
+ }
+ }
+
+ // test basic rounding behavior (exhaustive rounding testing is done elsewhere)
+ const x int64 = 0x7654321076543210 // 63 bits needed
+ for prec := uint(1); prec <= 63; prec++ {
+ f := new(Float).SetPrec(prec).SetMode(ToZero).SetInt64(x)
+ got := f.int64()
+ want := x &^ (1<<(63-prec) - 1) // cut off (round to zero) low 63-prec bits
+ if got != want {
+ t.Errorf("got %#x (%s); want %#x", got, f.Text('p', 0), want)
+ }
+ }
+}
+
+func TestFloatSetFloat64(t *testing.T) {
+ for _, want := range []float64{
+ 0,
+ 1,
+ 2,
+ 12345,
+ 1e10,
+ 1e100,
+ 3.14159265e10,
+ 2.718281828e-123,
+ 1.0 / 3,
+ math.MaxFloat32,
+ math.MaxFloat64,
+ math.SmallestNonzeroFloat32,
+ math.SmallestNonzeroFloat64,
+ math.Inf(-1),
+ math.Inf(0),
+ -math.Inf(1),
+ } {
+ for i := range [2]int{} {
+ if i&1 != 0 {
+ want = -want
+ }
+ var f Float
+ f.SetFloat64(want)
+ if got, acc := f.Float64(); got != want || acc != Exact {
+ t.Errorf("got %g (%s, %s); want %g (Exact)", got, f.Text('p', 0), acc, want)
+ }
+ }
+ }
+
+ // test basic rounding behavior (exhaustive rounding testing is done elsewhere)
+ const x uint64 = 0x8765432143218 // 53 bits needed
+ for prec := uint(1); prec <= 52; prec++ {
+ f := new(Float).SetPrec(prec).SetMode(ToZero).SetFloat64(float64(x))
+ got, _ := f.Float64()
+ want := float64(x &^ (1<<(52-prec) - 1)) // cut off (round to zero) low 53-prec bits
+ if got != want {
+ t.Errorf("got %g (%s); want %g", got, f.Text('p', 0), want)
+ }
+ }
+
+ // test NaN
+ defer func() {
+ if p, ok := recover().(ErrNaN); !ok {
+ t.Errorf("got %v; want ErrNaN panic", p)
+ }
+ }()
+ var f Float
+ f.SetFloat64(math.NaN())
+ // should not reach here
+ t.Errorf("got %s; want ErrNaN panic", f.Text('p', 0))
+}
+
+func TestFloatSetInt(t *testing.T) {
+ for _, want := range []string{
+ "0",
+ "1",
+ "-1",
+ "1234567890",
+ "123456789012345678901234567890",
+ "123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890",
+ } {
+ var x Int
+ _, ok := x.SetString(want, 0)
+ if !ok {
+ t.Errorf("invalid integer %s", want)
+ continue
+ }
+ n := x.BitLen()
+
+ var f Float
+ f.SetInt(&x)
+
+ // check precision
+ if n < 64 {
+ n = 64
+ }
+ if prec := f.Prec(); prec != uint(n) {
+ t.Errorf("got prec = %d; want %d", prec, n)
+ }
+
+ // check value
+ got := f.Text('g', 100)
+ if got != want {
+ t.Errorf("got %s (%s); want %s", got, f.Text('p', 0), want)
+ }
+ }
+
+ // TODO(gri) test basic rounding behavior
+}
+
+func TestFloatSetRat(t *testing.T) {
+ for _, want := range []string{
+ "0",
+ "1",
+ "-1",
+ "1234567890",
+ "123456789012345678901234567890",
+ "123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890",
+ "1.2",
+ "3.14159265",
+ // TODO(gri) expand
+ } {
+ var x Rat
+ _, ok := x.SetString(want)
+ if !ok {
+ t.Errorf("invalid fraction %s", want)
+ continue
+ }
+ n := max(x.Num().BitLen(), x.Denom().BitLen())
+
+ var f1, f2 Float
+ f2.SetPrec(1000)
+ f1.SetRat(&x)
+ f2.SetRat(&x)
+
+ // check precision when set automatically
+ if n < 64 {
+ n = 64
+ }
+ if prec := f1.Prec(); prec != uint(n) {
+ t.Errorf("got prec = %d; want %d", prec, n)
+ }
+
+ got := f2.Text('g', 100)
+ if got != want {
+ t.Errorf("got %s (%s); want %s", got, f2.Text('p', 0), want)
+ }
+ }
+}
+
+func TestFloatSetInf(t *testing.T) {
+ var f Float
+ for _, test := range []struct {
+ signbit bool
+ prec uint
+ want string
+ }{
+ {false, 0, "+Inf"},
+ {true, 0, "-Inf"},
+ {false, 10, "+Inf"},
+ {true, 30, "-Inf"},
+ } {
+ x := f.SetPrec(test.prec).SetInf(test.signbit)
+ if got := x.String(); got != test.want || x.Prec() != test.prec {
+ t.Errorf("SetInf(%v) = %s (prec = %d); want %s (prec = %d)", test.signbit, got, x.Prec(), test.want, test.prec)
+ }
+ }
+}
+
+func TestFloatUint64(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ out uint64
+ acc Accuracy
+ }{
+ {"-Inf", 0, Above},
+ {"-1", 0, Above},
+ {"-1e-1000", 0, Above},
+ {"-0", 0, Exact},
+ {"0", 0, Exact},
+ {"1e-1000", 0, Below},
+ {"1", 1, Exact},
+ {"1.000000000000000000001", 1, Below},
+ {"12345.0", 12345, Exact},
+ {"12345.000000000000000000001", 12345, Below},
+ {"18446744073709551615", 18446744073709551615, Exact},
+ {"18446744073709551615.000000000000000000001", math.MaxUint64, Below},
+ {"18446744073709551616", math.MaxUint64, Below},
+ {"1e10000", math.MaxUint64, Below},
+ {"+Inf", math.MaxUint64, Below},
+ } {
+ x := makeFloat(test.x)
+ out, acc := x.Uint64()
+ if out != test.out || acc != test.acc {
+ t.Errorf("%s: got %d (%s); want %d (%s)", test.x, out, acc, test.out, test.acc)
+ }
+ }
+}
+
+func TestFloatInt64(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ out int64
+ acc Accuracy
+ }{
+ {"-Inf", math.MinInt64, Above},
+ {"-1e10000", math.MinInt64, Above},
+ {"-9223372036854775809", math.MinInt64, Above},
+ {"-9223372036854775808.000000000000000000001", math.MinInt64, Above},
+ {"-9223372036854775808", -9223372036854775808, Exact},
+ {"-9223372036854775807.000000000000000000001", -9223372036854775807, Above},
+ {"-9223372036854775807", -9223372036854775807, Exact},
+ {"-12345.000000000000000000001", -12345, Above},
+ {"-12345.0", -12345, Exact},
+ {"-1.000000000000000000001", -1, Above},
+ {"-1.5", -1, Above},
+ {"-1", -1, Exact},
+ {"-1e-1000", 0, Above},
+ {"0", 0, Exact},
+ {"1e-1000", 0, Below},
+ {"1", 1, Exact},
+ {"1.000000000000000000001", 1, Below},
+ {"1.5", 1, Below},
+ {"12345.0", 12345, Exact},
+ {"12345.000000000000000000001", 12345, Below},
+ {"9223372036854775807", 9223372036854775807, Exact},
+ {"9223372036854775807.000000000000000000001", math.MaxInt64, Below},
+ {"9223372036854775808", math.MaxInt64, Below},
+ {"1e10000", math.MaxInt64, Below},
+ {"+Inf", math.MaxInt64, Below},
+ } {
+ x := makeFloat(test.x)
+ out, acc := x.Int64()
+ if out != test.out || acc != test.acc {
+ t.Errorf("%s: got %d (%s); want %d (%s)", test.x, out, acc, test.out, test.acc)
+ }
+ }
+}
+
+func TestFloatFloat32(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ out float32
+ acc Accuracy
+ }{
+ {"0", 0, Exact},
+
+ // underflow to zero
+ {"1e-1000", 0, Below},
+ {"0x0.000002p-127", 0, Below},
+ {"0x.0000010p-126", 0, Below},
+
+ // denormals
+ {"1.401298464e-45", math.SmallestNonzeroFloat32, Above}, // rounded up to smallest denormal
+ {"0x.ffffff8p-149", math.SmallestNonzeroFloat32, Above}, // rounded up to smallest denormal
+ {"0x.0000018p-126", math.SmallestNonzeroFloat32, Above}, // rounded up to smallest denormal
+ {"0x.0000020p-126", math.SmallestNonzeroFloat32, Exact},
+ {"0x.8p-148", math.SmallestNonzeroFloat32, Exact},
+ {"1p-149", math.SmallestNonzeroFloat32, Exact},
+ {"0x.fffffep-126", math.Float32frombits(0x7fffff), Exact}, // largest denormal
+
+ // special denormal cases (see issues 14553, 14651)
+ {"0x0.0000001p-126", math.Float32frombits(0x00000000), Below}, // underflow to zero
+ {"0x0.0000008p-126", math.Float32frombits(0x00000000), Below}, // underflow to zero
+ {"0x0.0000010p-126", math.Float32frombits(0x00000000), Below}, // rounded down to even
+ {"0x0.0000011p-126", math.Float32frombits(0x00000001), Above}, // rounded up to smallest denormal
+ {"0x0.0000018p-126", math.Float32frombits(0x00000001), Above}, // rounded up to smallest denormal
+
+ {"0x1.0000000p-149", math.Float32frombits(0x00000001), Exact}, // smallest denormal
+ {"0x0.0000020p-126", math.Float32frombits(0x00000001), Exact}, // smallest denormal
+ {"0x0.fffffe0p-126", math.Float32frombits(0x007fffff), Exact}, // largest denormal
+ {"0x1.0000000p-126", math.Float32frombits(0x00800000), Exact}, // smallest normal
+
+ {"0x0.8p-149", math.Float32frombits(0x000000000), Below}, // rounded down to even
+ {"0x0.9p-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
+ {"0x0.ap-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
+ {"0x0.bp-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
+ {"0x0.cp-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
+
+ {"0x1.0p-149", math.Float32frombits(0x000000001), Exact}, // smallest denormal
+ {"0x1.7p-149", math.Float32frombits(0x000000001), Below},
+ {"0x1.8p-149", math.Float32frombits(0x000000002), Above},
+ {"0x1.9p-149", math.Float32frombits(0x000000002), Above},
+
+ {"0x2.0p-149", math.Float32frombits(0x000000002), Exact},
+ {"0x2.8p-149", math.Float32frombits(0x000000002), Below}, // rounded down to even
+ {"0x2.9p-149", math.Float32frombits(0x000000003), Above},
+
+ {"0x3.0p-149", math.Float32frombits(0x000000003), Exact},
+ {"0x3.7p-149", math.Float32frombits(0x000000003), Below},
+ {"0x3.8p-149", math.Float32frombits(0x000000004), Above}, // rounded up to even
+
+ {"0x4.0p-149", math.Float32frombits(0x000000004), Exact},
+ {"0x4.8p-149", math.Float32frombits(0x000000004), Below}, // rounded down to even
+ {"0x4.9p-149", math.Float32frombits(0x000000005), Above},
+
+ // specific case from issue 14553
+ {"0x7.7p-149", math.Float32frombits(0x000000007), Below},
+ {"0x7.8p-149", math.Float32frombits(0x000000008), Above},
+ {"0x7.9p-149", math.Float32frombits(0x000000008), Above},
+
+ // normals
+ {"0x.ffffffp-126", math.Float32frombits(0x00800000), Above}, // rounded up to smallest normal
+ {"1p-126", math.Float32frombits(0x00800000), Exact}, // smallest normal
+ {"0x1.fffffep-126", math.Float32frombits(0x00ffffff), Exact},
+ {"0x1.ffffffp-126", math.Float32frombits(0x01000000), Above}, // rounded up
+ {"1", 1, Exact},
+ {"1.000000000000000000001", 1, Below},
+ {"12345.0", 12345, Exact},
+ {"12345.000000000000000000001", 12345, Below},
+ {"0x1.fffffe0p127", math.MaxFloat32, Exact},
+ {"0x1.fffffe8p127", math.MaxFloat32, Below},
+
+ // overflow
+ {"0x1.ffffff0p127", float32(math.Inf(+1)), Above},
+ {"0x1p128", float32(math.Inf(+1)), Above},
+ {"1e10000", float32(math.Inf(+1)), Above},
+ {"0x1.ffffff0p2147483646", float32(math.Inf(+1)), Above}, // overflow in rounding
+
+ // inf
+ {"Inf", float32(math.Inf(+1)), Exact},
+ } {
+ for i := 0; i < 2; i++ {
+ // test both signs
+ tx, tout, tacc := test.x, test.out, test.acc
+ if i != 0 {
+ tx = "-" + tx
+ tout = -tout
+ tacc = -tacc
+ }
+
+ // conversion should match strconv where syntax is agreeable
+ if f, err := strconv.ParseFloat(tx, 32); err == nil && !alike32(float32(f), tout) {
+ t.Errorf("%s: got %g; want %g (incorrect test data)", tx, f, tout)
+ }
+
+ x := makeFloat(tx)
+ out, acc := x.Float32()
+ if !alike32(out, tout) || acc != tacc {
+ t.Errorf("%s: got %g (%#08x, %s); want %g (%#08x, %s)", tx, out, math.Float32bits(out), acc, test.out, math.Float32bits(test.out), tacc)
+ }
+
+ // test that x.SetFloat64(float64(f)).Float32() == f
+ var x2 Float
+ out2, acc2 := x2.SetFloat64(float64(out)).Float32()
+ if !alike32(out2, out) || acc2 != Exact {
+ t.Errorf("idempotency test: got %g (%s); want %g (Exact)", out2, acc2, out)
+ }
+ }
+ }
+}
+
+func TestFloatFloat64(t *testing.T) {
+ const smallestNormalFloat64 = 2.2250738585072014e-308 // 1p-1022
+ for _, test := range []struct {
+ x string
+ out float64
+ acc Accuracy
+ }{
+ {"0", 0, Exact},
+
+ // underflow to zero
+ {"1e-1000", 0, Below},
+ {"0x0.0000000000001p-1023", 0, Below},
+ {"0x0.00000000000008p-1022", 0, Below},
+
+ // denormals
+ {"0x0.0000000000000cp-1022", math.SmallestNonzeroFloat64, Above}, // rounded up to smallest denormal
+ {"0x0.00000000000010p-1022", math.SmallestNonzeroFloat64, Exact}, // smallest denormal
+ {"0x.8p-1073", math.SmallestNonzeroFloat64, Exact},
+ {"1p-1074", math.SmallestNonzeroFloat64, Exact},
+ {"0x.fffffffffffffp-1022", math.Float64frombits(0x000fffffffffffff), Exact}, // largest denormal
+
+ // special denormal cases (see issues 14553, 14651)
+ {"0x0.00000000000001p-1022", math.Float64frombits(0x00000000000000000), Below}, // underflow to zero
+ {"0x0.00000000000004p-1022", math.Float64frombits(0x00000000000000000), Below}, // underflow to zero
+ {"0x0.00000000000008p-1022", math.Float64frombits(0x00000000000000000), Below}, // rounded down to even
+ {"0x0.00000000000009p-1022", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+ {"0x0.0000000000000ap-1022", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+
+ {"0x0.8p-1074", math.Float64frombits(0x00000000000000000), Below}, // rounded down to even
+ {"0x0.9p-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+ {"0x0.ap-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+ {"0x0.bp-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+ {"0x0.cp-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+
+ {"0x1.0p-1074", math.Float64frombits(0x00000000000000001), Exact},
+ {"0x1.7p-1074", math.Float64frombits(0x00000000000000001), Below},
+ {"0x1.8p-1074", math.Float64frombits(0x00000000000000002), Above},
+ {"0x1.9p-1074", math.Float64frombits(0x00000000000000002), Above},
+
+ {"0x2.0p-1074", math.Float64frombits(0x00000000000000002), Exact},
+ {"0x2.8p-1074", math.Float64frombits(0x00000000000000002), Below}, // rounded down to even
+ {"0x2.9p-1074", math.Float64frombits(0x00000000000000003), Above},
+
+ {"0x3.0p-1074", math.Float64frombits(0x00000000000000003), Exact},
+ {"0x3.7p-1074", math.Float64frombits(0x00000000000000003), Below},
+ {"0x3.8p-1074", math.Float64frombits(0x00000000000000004), Above}, // rounded up to even
+
+ {"0x4.0p-1074", math.Float64frombits(0x00000000000000004), Exact},
+ {"0x4.8p-1074", math.Float64frombits(0x00000000000000004), Below}, // rounded down to even
+ {"0x4.9p-1074", math.Float64frombits(0x00000000000000005), Above},
+
+ // normals
+ {"0x.fffffffffffff8p-1022", math.Float64frombits(0x0010000000000000), Above}, // rounded up to smallest normal
+ {"1p-1022", math.Float64frombits(0x0010000000000000), Exact}, // smallest normal
+ {"1", 1, Exact},
+ {"1.000000000000000000001", 1, Below},
+ {"12345.0", 12345, Exact},
+ {"12345.000000000000000000001", 12345, Below},
+ {"0x1.fffffffffffff0p1023", math.MaxFloat64, Exact},
+ {"0x1.fffffffffffff4p1023", math.MaxFloat64, Below},
+
+ // overflow
+ {"0x1.fffffffffffff8p1023", math.Inf(+1), Above},
+ {"0x1p1024", math.Inf(+1), Above},
+ {"1e10000", math.Inf(+1), Above},
+ {"0x1.fffffffffffff8p2147483646", math.Inf(+1), Above}, // overflow in rounding
+ {"Inf", math.Inf(+1), Exact},
+
+ // selected denormalized values that were handled incorrectly in the past
+ {"0x.fffffffffffffp-1022", smallestNormalFloat64 - math.SmallestNonzeroFloat64, Exact},
+ {"4503599627370495p-1074", smallestNormalFloat64 - math.SmallestNonzeroFloat64, Exact},
+
+ // https://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
+ {"2.2250738585072011e-308", 2.225073858507201e-308, Below},
+ // https://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
+ {"2.2250738585072012e-308", 2.2250738585072014e-308, Above},
+ } {
+ for i := 0; i < 2; i++ {
+ // test both signs
+ tx, tout, tacc := test.x, test.out, test.acc
+ if i != 0 {
+ tx = "-" + tx
+ tout = -tout
+ tacc = -tacc
+ }
+
+ // conversion should match strconv where syntax is agreeable
+ if f, err := strconv.ParseFloat(tx, 64); err == nil && !alike64(f, tout) {
+ t.Errorf("%s: got %g; want %g (incorrect test data)", tx, f, tout)
+ }
+
+ x := makeFloat(tx)
+ out, acc := x.Float64()
+ if !alike64(out, tout) || acc != tacc {
+ t.Errorf("%s: got %g (%#016x, %s); want %g (%#016x, %s)", tx, out, math.Float64bits(out), acc, test.out, math.Float64bits(test.out), tacc)
+ }
+
+ // test that x.SetFloat64(f).Float64() == f
+ var x2 Float
+ out2, acc2 := x2.SetFloat64(out).Float64()
+ if !alike64(out2, out) || acc2 != Exact {
+ t.Errorf("idempotency test: got %g (%s); want %g (Exact)", out2, acc2, out)
+ }
+ }
+ }
+}
+
+func TestFloatInt(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ want string
+ acc Accuracy
+ }{
+ {"0", "0", Exact},
+ {"+0", "0", Exact},
+ {"-0", "0", Exact},
+ {"Inf", "nil", Below},
+ {"+Inf", "nil", Below},
+ {"-Inf", "nil", Above},
+ {"1", "1", Exact},
+ {"-1", "-1", Exact},
+ {"1.23", "1", Below},
+ {"-1.23", "-1", Above},
+ {"123e-2", "1", Below},
+ {"123e-3", "0", Below},
+ {"123e-4", "0", Below},
+ {"1e-1000", "0", Below},
+ {"-1e-1000", "0", Above},
+ {"1e+10", "10000000000", Exact},
+ {"1e+100", "10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", Exact},
+ } {
+ x := makeFloat(test.x)
+ res, acc := x.Int(nil)
+ got := "nil"
+ if res != nil {
+ got = res.String()
+ }
+ if got != test.want || acc != test.acc {
+ t.Errorf("%s: got %s (%s); want %s (%s)", test.x, got, acc, test.want, test.acc)
+ }
+ }
+
+ // check that supplied *Int is used
+ for _, f := range []string{"0", "1", "-1", "1234"} {
+ x := makeFloat(f)
+ i := new(Int)
+ if res, _ := x.Int(i); res != i {
+ t.Errorf("(%s).Int is not using supplied *Int", f)
+ }
+ }
+}
+
+func TestFloatRat(t *testing.T) {
+ for _, test := range []struct {
+ x, want string
+ acc Accuracy
+ }{
+ {"0", "0/1", Exact},
+ {"+0", "0/1", Exact},
+ {"-0", "0/1", Exact},
+ {"Inf", "nil", Below},
+ {"+Inf", "nil", Below},
+ {"-Inf", "nil", Above},
+ {"1", "1/1", Exact},
+ {"-1", "-1/1", Exact},
+ {"1.25", "5/4", Exact},
+ {"-1.25", "-5/4", Exact},
+ {"1e10", "10000000000/1", Exact},
+ {"1p10", "1024/1", Exact},
+ {"-1p-10", "-1/1024", Exact},
+ {"3.14159265", "7244019449799623199/2305843009213693952", Exact},
+ } {
+ x := makeFloat(test.x).SetPrec(64)
+ res, acc := x.Rat(nil)
+ got := "nil"
+ if res != nil {
+ got = res.String()
+ }
+ if got != test.want {
+ t.Errorf("%s: got %s; want %s", test.x, got, test.want)
+ continue
+ }
+ if acc != test.acc {
+ t.Errorf("%s: got %s; want %s", test.x, acc, test.acc)
+ continue
+ }
+
+ // inverse conversion
+ if res != nil {
+ got := new(Float).SetPrec(64).SetRat(res)
+ if got.Cmp(x) != 0 {
+ t.Errorf("%s: got %s; want %s", test.x, got, x)
+ }
+ }
+ }
+
+ // check that supplied *Rat is used
+ for _, f := range []string{"0", "1", "-1", "1234"} {
+ x := makeFloat(f)
+ r := new(Rat)
+ if res, _ := x.Rat(r); res != r {
+ t.Errorf("(%s).Rat is not using supplied *Rat", f)
+ }
+ }
+}
+
+func TestFloatAbs(t *testing.T) {
+ for _, test := range []string{
+ "0",
+ "1",
+ "1234",
+ "1.23e-2",
+ "1e-1000",
+ "1e1000",
+ "Inf",
+ } {
+ p := makeFloat(test)
+ a := new(Float).Abs(p)
+ if !alike(a, p) {
+ t.Errorf("%s: got %s; want %s", test, a.Text('g', 10), test)
+ }
+
+ n := makeFloat("-" + test)
+ a.Abs(n)
+ if !alike(a, p) {
+ t.Errorf("-%s: got %s; want %s", test, a.Text('g', 10), test)
+ }
+ }
+}
+
+func TestFloatNeg(t *testing.T) {
+ for _, test := range []string{
+ "0",
+ "1",
+ "1234",
+ "1.23e-2",
+ "1e-1000",
+ "1e1000",
+ "Inf",
+ } {
+ p1 := makeFloat(test)
+ n1 := makeFloat("-" + test)
+ n2 := new(Float).Neg(p1)
+ p2 := new(Float).Neg(n2)
+ if !alike(n2, n1) {
+ t.Errorf("%s: got %s; want %s", test, n2.Text('g', 10), n1.Text('g', 10))
+ }
+ if !alike(p2, p1) {
+ t.Errorf("%s: got %s; want %s", test, p2.Text('g', 10), p1.Text('g', 10))
+ }
+ }
+}
+
+func TestFloatInc(t *testing.T) {
+ const n = 10
+ for _, prec := range precList {
+ if 1<<prec < n {
+ continue // prec must be large enough to hold all numbers from 0 to n
+ }
+ var x, one Float
+ x.SetPrec(prec)
+ one.SetInt64(1)
+ for i := 0; i < n; i++ {
+ x.Add(&x, &one)
+ }
+ if x.Cmp(new(Float).SetInt64(n)) != 0 {
+ t.Errorf("prec = %d: got %s; want %d", prec, &x, n)
+ }
+ }
+}
+
+// Selected precisions with which to run various tests.
+var precList = [...]uint{1, 2, 5, 8, 10, 16, 23, 24, 32, 50, 53, 64, 100, 128, 500, 511, 512, 513, 1000, 10000}
+
+// Selected bits with which to run various tests.
+// Each entry is a list of bits representing a floating-point number (see fromBits).
+var bitsList = [...]Bits{
+ {}, // = 0
+ {0}, // = 1
+ {1}, // = 2
+ {-1}, // = 1/2
+ {10}, // = 2**10 == 1024
+ {-10}, // = 2**-10 == 1/1024
+ {100, 10, 1}, // = 2**100 + 2**10 + 2**1
+ {0, -1, -2, -10},
+ // TODO(gri) add more test cases
+}
+
+// TestFloatAdd tests Float.Add/Sub by comparing the result of a "manual"
+// addition/subtraction of arguments represented by Bits values with the
+// respective Float addition/subtraction for a variety of precisions
+// and rounding modes.
+func TestFloatAdd(t *testing.T) {
+ for _, xbits := range bitsList {
+ for _, ybits := range bitsList {
+ // exact values
+ x := xbits.Float()
+ y := ybits.Float()
+ zbits := xbits.add(ybits)
+ z := zbits.Float()
+
+ for i, mode := range [...]RoundingMode{ToZero, ToNearestEven, AwayFromZero} {
+ for _, prec := range precList {
+ got := new(Float).SetPrec(prec).SetMode(mode)
+ got.Add(x, y)
+ want := zbits.round(prec, mode)
+ if got.Cmp(want) != 0 {
+ t.Errorf("i = %d, prec = %d, %s:\n\t %s %v\n\t+ %s %v\n\t= %s\n\twant %s",
+ i, prec, mode, x, xbits, y, ybits, got, want)
+ }
+
+ got.Sub(z, x)
+ want = ybits.round(prec, mode)
+ if got.Cmp(want) != 0 {
+ t.Errorf("i = %d, prec = %d, %s:\n\t %s %v\n\t- %s %v\n\t= %s\n\twant %s",
+ i, prec, mode, z, zbits, x, xbits, got, want)
+ }
+ }
+ }
+ }
+ }
+}
+
+// TestFloatAddRoundZero tests Float.Add/Sub rounding when the result is exactly zero.
+// x + (-x) or x - x for non-zero x should be +0 in all cases except when
+// the rounding mode is ToNegativeInf in which case it should be -0.
+func TestFloatAddRoundZero(t *testing.T) {
+ for _, mode := range [...]RoundingMode{ToNearestEven, ToNearestAway, ToZero, AwayFromZero, ToPositiveInf, ToNegativeInf} {
+ x := NewFloat(5.0)
+ y := new(Float).Neg(x)
+ want := NewFloat(0.0)
+ if mode == ToNegativeInf {
+ want.Neg(want)
+ }
+ got := new(Float).SetMode(mode)
+ got.Add(x, y)
+ if got.Cmp(want) != 0 || got.neg != (mode == ToNegativeInf) {
+ t.Errorf("%s:\n\t %v\n\t+ %v\n\t= %v\n\twant %v",
+ mode, x, y, got, want)
+ }
+ got.Sub(x, x)
+ if got.Cmp(want) != 0 || got.neg != (mode == ToNegativeInf) {
+ t.Errorf("%v:\n\t %v\n\t- %v\n\t= %v\n\twant %v",
+ mode, x, x, got, want)
+ }
+ }
+}
+
+// TestFloatAdd32 tests that Float.Add/Sub of numbers with
+// 24bit mantissa behaves like float32 addition/subtraction
+// (excluding denormal numbers).
+func TestFloatAdd32(t *testing.T) {
+ // chose base such that we cross the mantissa precision limit
+ const base = 1<<26 - 0x10 // 11...110000 (26 bits)
+ for d := 0; d <= 0x10; d++ {
+ for i := range [2]int{} {
+ x0, y0 := float64(base), float64(d)
+ if i&1 != 0 {
+ x0, y0 = y0, x0
+ }
+
+ x := NewFloat(x0)
+ y := NewFloat(y0)
+ z := new(Float).SetPrec(24)
+
+ z.Add(x, y)
+ got, acc := z.Float32()
+ want := float32(y0) + float32(x0)
+ if got != want || acc != Exact {
+ t.Errorf("d = %d: %g + %g = %g (%s); want %g (Exact)", d, x0, y0, got, acc, want)
+ }
+
+ z.Sub(z, y)
+ got, acc = z.Float32()
+ want = float32(want) - float32(y0)
+ if got != want || acc != Exact {
+ t.Errorf("d = %d: %g - %g = %g (%s); want %g (Exact)", d, x0+y0, y0, got, acc, want)
+ }
+ }
+ }
+}
+
+// TestFloatAdd64 tests that Float.Add/Sub of numbers with
+// 53bit mantissa behaves like float64 addition/subtraction.
+func TestFloatAdd64(t *testing.T) {
+ // chose base such that we cross the mantissa precision limit
+ const base = 1<<55 - 0x10 // 11...110000 (55 bits)
+ for d := 0; d <= 0x10; d++ {
+ for i := range [2]int{} {
+ x0, y0 := float64(base), float64(d)
+ if i&1 != 0 {
+ x0, y0 = y0, x0
+ }
+
+ x := NewFloat(x0)
+ y := NewFloat(y0)
+ z := new(Float).SetPrec(53)
+
+ z.Add(x, y)
+ got, acc := z.Float64()
+ want := x0 + y0
+ if got != want || acc != Exact {
+ t.Errorf("d = %d: %g + %g = %g (%s); want %g (Exact)", d, x0, y0, got, acc, want)
+ }
+
+ z.Sub(z, y)
+ got, acc = z.Float64()
+ want -= y0
+ if got != want || acc != Exact {
+ t.Errorf("d = %d: %g - %g = %g (%s); want %g (Exact)", d, x0+y0, y0, got, acc, want)
+ }
+ }
+ }
+}
+
+func TestIssue20490(t *testing.T) {
+ var tests = []struct {
+ a, b float64
+ }{
+ {4, 1},
+ {-4, 1},
+ {4, -1},
+ {-4, -1},
+ }
+
+ for _, test := range tests {
+ a, b := NewFloat(test.a), NewFloat(test.b)
+ diff := new(Float).Sub(a, b)
+ b.Sub(a, b)
+ if b.Cmp(diff) != 0 {
+ t.Errorf("got %g - %g = %g; want %g\n", a, NewFloat(test.b), b, diff)
+ }
+
+ b = NewFloat(test.b)
+ sum := new(Float).Add(a, b)
+ b.Add(a, b)
+ if b.Cmp(sum) != 0 {
+ t.Errorf("got %g + %g = %g; want %g\n", a, NewFloat(test.b), b, sum)
+ }
+
+ }
+}
+
+// TestFloatMul tests Float.Mul/Quo by comparing the result of a "manual"
+// multiplication/division of arguments represented by Bits values with the
+// respective Float multiplication/division for a variety of precisions
+// and rounding modes.
+func TestFloatMul(t *testing.T) {
+ for _, xbits := range bitsList {
+ for _, ybits := range bitsList {
+ // exact values
+ x := xbits.Float()
+ y := ybits.Float()
+ zbits := xbits.mul(ybits)
+ z := zbits.Float()
+
+ for i, mode := range [...]RoundingMode{ToZero, ToNearestEven, AwayFromZero} {
+ for _, prec := range precList {
+ got := new(Float).SetPrec(prec).SetMode(mode)
+ got.Mul(x, y)
+ want := zbits.round(prec, mode)
+ if got.Cmp(want) != 0 {
+ t.Errorf("i = %d, prec = %d, %s:\n\t %v %v\n\t* %v %v\n\t= %v\n\twant %v",
+ i, prec, mode, x, xbits, y, ybits, got, want)
+ }
+
+ if x.Sign() == 0 {
+ continue // ignore div-0 case (not invertable)
+ }
+ got.Quo(z, x)
+ want = ybits.round(prec, mode)
+ if got.Cmp(want) != 0 {
+ t.Errorf("i = %d, prec = %d, %s:\n\t %v %v\n\t/ %v %v\n\t= %v\n\twant %v",
+ i, prec, mode, z, zbits, x, xbits, got, want)
+ }
+ }
+ }
+ }
+ }
+}
+
+// TestFloatMul64 tests that Float.Mul/Quo of numbers with
+// 53bit mantissa behaves like float64 multiplication/division.
+func TestFloatMul64(t *testing.T) {
+ for _, test := range []struct {
+ x, y float64
+ }{
+ {0, 0},
+ {0, 1},
+ {1, 1},
+ {1, 1.5},
+ {1.234, 0.5678},
+ {2.718281828, 3.14159265358979},
+ {2.718281828e10, 3.14159265358979e-32},
+ {1.0 / 3, 1e200},
+ } {
+ for i := range [8]int{} {
+ x0, y0 := test.x, test.y
+ if i&1 != 0 {
+ x0 = -x0
+ }
+ if i&2 != 0 {
+ y0 = -y0
+ }
+ if i&4 != 0 {
+ x0, y0 = y0, x0
+ }
+
+ x := NewFloat(x0)
+ y := NewFloat(y0)
+ z := new(Float).SetPrec(53)
+
+ z.Mul(x, y)
+ got, _ := z.Float64()
+ want := x0 * y0
+ if got != want {
+ t.Errorf("%g * %g = %g; want %g", x0, y0, got, want)
+ }
+
+ if y0 == 0 {
+ continue // avoid division-by-zero
+ }
+ z.Quo(z, y)
+ got, _ = z.Float64()
+ want /= y0
+ if got != want {
+ t.Errorf("%g / %g = %g; want %g", x0*y0, y0, got, want)
+ }
+ }
+ }
+}
+
+func TestIssue6866(t *testing.T) {
+ for _, prec := range precList {
+ two := new(Float).SetPrec(prec).SetInt64(2)
+ one := new(Float).SetPrec(prec).SetInt64(1)
+ three := new(Float).SetPrec(prec).SetInt64(3)
+ msix := new(Float).SetPrec(prec).SetInt64(-6)
+ psix := new(Float).SetPrec(prec).SetInt64(+6)
+
+ p := new(Float).SetPrec(prec)
+ z1 := new(Float).SetPrec(prec)
+ z2 := new(Float).SetPrec(prec)
+
+ // z1 = 2 + 1.0/3*-6
+ p.Quo(one, three)
+ p.Mul(p, msix)
+ z1.Add(two, p)
+
+ // z2 = 2 - 1.0/3*+6
+ p.Quo(one, three)
+ p.Mul(p, psix)
+ z2.Sub(two, p)
+
+ if z1.Cmp(z2) != 0 {
+ t.Fatalf("prec %d: got z1 = %v != z2 = %v; want z1 == z2\n", prec, z1, z2)
+ }
+ if z1.Sign() != 0 {
+ t.Errorf("prec %d: got z1 = %v; want 0", prec, z1)
+ }
+ if z2.Sign() != 0 {
+ t.Errorf("prec %d: got z2 = %v; want 0", prec, z2)
+ }
+ }
+}
+
+func TestFloatQuo(t *testing.T) {
+ // TODO(gri) make the test vary these precisions
+ preci := 200 // precision of integer part
+ precf := 20 // precision of fractional part
+
+ for i := 0; i < 8; i++ {
+ // compute accurate (not rounded) result z
+ bits := Bits{preci - 1}
+ if i&3 != 0 {
+ bits = append(bits, 0)
+ }
+ if i&2 != 0 {
+ bits = append(bits, -1)
+ }
+ if i&1 != 0 {
+ bits = append(bits, -precf)
+ }
+ z := bits.Float()
+
+ // compute accurate x as z*y
+ y := NewFloat(3.14159265358979323e123)
+
+ x := new(Float).SetPrec(z.Prec() + y.Prec()).SetMode(ToZero)
+ x.Mul(z, y)
+
+ // leave for debugging
+ // fmt.Printf("x = %s\ny = %s\nz = %s\n", x, y, z)
+
+ if got := x.Acc(); got != Exact {
+ t.Errorf("got acc = %s; want exact", got)
+ }
+
+ // round accurate z for a variety of precisions and
+ // modes and compare against result of x / y.
+ for _, mode := range [...]RoundingMode{ToZero, ToNearestEven, AwayFromZero} {
+ for d := -5; d < 5; d++ {
+ prec := uint(preci + d)
+ got := new(Float).SetPrec(prec).SetMode(mode).Quo(x, y)
+ want := bits.round(prec, mode)
+ if got.Cmp(want) != 0 {
+ t.Errorf("i = %d, prec = %d, %s:\n\t %s\n\t/ %s\n\t= %s\n\twant %s",
+ i, prec, mode, x, y, got, want)
+ }
+ }
+ }
+ }
+}
+
+var long = flag.Bool("long", false, "run very long tests")
+
+// TestFloatQuoSmoke tests all divisions x/y for values x, y in the range [-n, +n];
+// it serves as a smoke test for basic correctness of division.
+func TestFloatQuoSmoke(t *testing.T) {
+ n := 10
+ if *long {
+ n = 1000
+ }
+
+ const dprec = 3 // max. precision variation
+ const prec = 10 + dprec // enough bits to hold n precisely
+ for x := -n; x <= n; x++ {
+ for y := -n; y < n; y++ {
+ if y == 0 {
+ continue
+ }
+
+ a := float64(x)
+ b := float64(y)
+ c := a / b
+
+ // vary operand precision (only ok as long as a, b can be represented correctly)
+ for ad := -dprec; ad <= dprec; ad++ {
+ for bd := -dprec; bd <= dprec; bd++ {
+ A := new(Float).SetPrec(uint(prec + ad)).SetFloat64(a)
+ B := new(Float).SetPrec(uint(prec + bd)).SetFloat64(b)
+ C := new(Float).SetPrec(53).Quo(A, B) // C has float64 mantissa width
+
+ cc, acc := C.Float64()
+ if cc != c {
+ t.Errorf("%g/%g = %s; want %.5g\n", a, b, C.Text('g', 5), c)
+ continue
+ }
+ if acc != Exact {
+ t.Errorf("%g/%g got %s result; want exact result", a, b, acc)
+ }
+ }
+ }
+ }
+ }
+}
+
+// TestFloatArithmeticSpecialValues tests that Float operations produce the
+// correct results for combinations of zero (±0), finite (±1 and ±2.71828),
+// and infinite (±Inf) operands.
+func TestFloatArithmeticSpecialValues(t *testing.T) {
+ zero := 0.0
+ args := []float64{math.Inf(-1), -2.71828, -1, -zero, zero, 1, 2.71828, math.Inf(1)}
+ xx := new(Float)
+ yy := new(Float)
+ got := new(Float)
+ want := new(Float)
+ for i := 0; i < 4; i++ {
+ for _, x := range args {
+ xx.SetFloat64(x)
+ // check conversion is correct
+ // (no need to do this for y, since we see exactly the
+ // same values there)
+ if got, acc := xx.Float64(); got != x || acc != Exact {
+ t.Errorf("Float(%g) == %g (%s)", x, got, acc)
+ }
+ for _, y := range args {
+ yy.SetFloat64(y)
+ var (
+ op string
+ z float64
+ f func(z, x, y *Float) *Float
+ )
+ switch i {
+ case 0:
+ op = "+"
+ z = x + y
+ f = (*Float).Add
+ case 1:
+ op = "-"
+ z = x - y
+ f = (*Float).Sub
+ case 2:
+ op = "*"
+ z = x * y
+ f = (*Float).Mul
+ case 3:
+ op = "/"
+ z = x / y
+ f = (*Float).Quo
+ default:
+ panic("unreachable")
+ }
+ var errnan bool // set if execution of f panicked with ErrNaN
+ // protect execution of f
+ func() {
+ defer func() {
+ if p := recover(); p != nil {
+ _ = p.(ErrNaN) // re-panic if not ErrNaN
+ errnan = true
+ }
+ }()
+ f(got, xx, yy)
+ }()
+ if math.IsNaN(z) {
+ if !errnan {
+ t.Errorf("%5g %s %5g = %5s; want ErrNaN panic", x, op, y, got)
+ }
+ continue
+ }
+ if errnan {
+ t.Errorf("%5g %s %5g panicked with ErrNan; want %5s", x, op, y, want)
+ continue
+ }
+ want.SetFloat64(z)
+ if !alike(got, want) {
+ t.Errorf("%5g %s %5g = %5s; want %5s", x, op, y, got, want)
+ }
+ }
+ }
+ }
+}
+
+func TestFloatArithmeticOverflow(t *testing.T) {
+ for _, test := range []struct {
+ prec uint
+ mode RoundingMode
+ op byte
+ x, y, want string
+ acc Accuracy
+ }{
+ {4, ToNearestEven, '+', "0", "0", "0", Exact}, // smoke test
+ {4, ToNearestEven, '+', "0x.8p+0", "0x.8p+0", "0x.8p+1", Exact}, // smoke test
+
+ {4, ToNearestEven, '+', "0", "0x.8p2147483647", "0x.8p+2147483647", Exact},
+ {4, ToNearestEven, '+', "0x.8p2147483500", "0x.8p2147483647", "0x.8p+2147483647", Below}, // rounded to zero
+ {4, ToNearestEven, '+', "0x.8p2147483647", "0x.8p2147483647", "+Inf", Above}, // exponent overflow in +
+ {4, ToNearestEven, '+', "-0x.8p2147483647", "-0x.8p2147483647", "-Inf", Below}, // exponent overflow in +
+ {4, ToNearestEven, '-', "-0x.8p2147483647", "0x.8p2147483647", "-Inf", Below}, // exponent overflow in -
+
+ {4, ToZero, '+', "0x.fp2147483647", "0x.8p2147483643", "0x.fp+2147483647", Below}, // rounded to zero
+ {4, ToNearestEven, '+', "0x.fp2147483647", "0x.8p2147483643", "+Inf", Above}, // exponent overflow in rounding
+ {4, AwayFromZero, '+', "0x.fp2147483647", "0x.8p2147483643", "+Inf", Above}, // exponent overflow in rounding
+
+ {4, AwayFromZero, '-', "-0x.fp2147483647", "0x.8p2147483644", "-Inf", Below}, // exponent overflow in rounding
+ {4, ToNearestEven, '-', "-0x.fp2147483647", "0x.8p2147483643", "-Inf", Below}, // exponent overflow in rounding
+ {4, ToZero, '-', "-0x.fp2147483647", "0x.8p2147483643", "-0x.fp+2147483647", Above}, // rounded to zero
+
+ {4, ToNearestEven, '+', "0", "0x.8p-2147483648", "0x.8p-2147483648", Exact},
+ {4, ToNearestEven, '+', "0x.8p-2147483648", "0x.8p-2147483648", "0x.8p-2147483647", Exact},
+
+ {4, ToNearestEven, '*', "1", "0x.8p2147483647", "0x.8p+2147483647", Exact},
+ {4, ToNearestEven, '*', "2", "0x.8p2147483647", "+Inf", Above}, // exponent overflow in *
+ {4, ToNearestEven, '*', "-2", "0x.8p2147483647", "-Inf", Below}, // exponent overflow in *
+
+ {4, ToNearestEven, '/', "0.5", "0x.8p2147483647", "0x.8p-2147483646", Exact},
+ {4, ToNearestEven, '/', "0x.8p+0", "0x.8p2147483647", "0x.8p-2147483646", Exact},
+ {4, ToNearestEven, '/', "0x.8p-1", "0x.8p2147483647", "0x.8p-2147483647", Exact},
+ {4, ToNearestEven, '/', "0x.8p-2", "0x.8p2147483647", "0x.8p-2147483648", Exact},
+ {4, ToNearestEven, '/', "0x.8p-3", "0x.8p2147483647", "0", Below}, // exponent underflow in /
+ } {
+ x := makeFloat(test.x)
+ y := makeFloat(test.y)
+ z := new(Float).SetPrec(test.prec).SetMode(test.mode)
+ switch test.op {
+ case '+':
+ z.Add(x, y)
+ case '-':
+ z.Sub(x, y)
+ case '*':
+ z.Mul(x, y)
+ case '/':
+ z.Quo(x, y)
+ default:
+ panic("unreachable")
+ }
+ if got := z.Text('p', 0); got != test.want || z.Acc() != test.acc {
+ t.Errorf(
+ "prec = %d (%s): %s %c %s = %s (%s); want %s (%s)",
+ test.prec, test.mode, x.Text('p', 0), test.op, y.Text('p', 0), got, z.Acc(), test.want, test.acc,
+ )
+ }
+ }
+}
+
+// TODO(gri) Add tests that check correctness in the presence of aliasing.
+
+// For rounding modes ToNegativeInf and ToPositiveInf, rounding is affected
+// by the sign of the value to be rounded. Test that rounding happens after
+// the sign of a result has been set.
+// This test uses specific values that are known to fail if rounding is
+// "factored" out before setting the result sign.
+func TestFloatArithmeticRounding(t *testing.T) {
+ for _, test := range []struct {
+ mode RoundingMode
+ prec uint
+ x, y, want int64
+ op byte
+ }{
+ {ToZero, 3, -0x8, -0x1, -0x8, '+'},
+ {AwayFromZero, 3, -0x8, -0x1, -0xa, '+'},
+ {ToNegativeInf, 3, -0x8, -0x1, -0xa, '+'},
+
+ {ToZero, 3, -0x8, 0x1, -0x8, '-'},
+ {AwayFromZero, 3, -0x8, 0x1, -0xa, '-'},
+ {ToNegativeInf, 3, -0x8, 0x1, -0xa, '-'},
+
+ {ToZero, 3, -0x9, 0x1, -0x8, '*'},
+ {AwayFromZero, 3, -0x9, 0x1, -0xa, '*'},
+ {ToNegativeInf, 3, -0x9, 0x1, -0xa, '*'},
+
+ {ToZero, 3, -0x9, 0x1, -0x8, '/'},
+ {AwayFromZero, 3, -0x9, 0x1, -0xa, '/'},
+ {ToNegativeInf, 3, -0x9, 0x1, -0xa, '/'},
+ } {
+ var x, y, z Float
+ x.SetInt64(test.x)
+ y.SetInt64(test.y)
+ z.SetPrec(test.prec).SetMode(test.mode)
+ switch test.op {
+ case '+':
+ z.Add(&x, &y)
+ case '-':
+ z.Sub(&x, &y)
+ case '*':
+ z.Mul(&x, &y)
+ case '/':
+ z.Quo(&x, &y)
+ default:
+ panic("unreachable")
+ }
+ if got, acc := z.Int64(); got != test.want || acc != Exact {
+ t.Errorf("%s, %d bits: %d %c %d = %d (%s); want %d (Exact)",
+ test.mode, test.prec, test.x, test.op, test.y, got, acc, test.want,
+ )
+ }
+ }
+}
+
+// TestFloatCmpSpecialValues tests that Cmp produces the correct results for
+// combinations of zero (±0), finite (±1 and ±2.71828), and infinite (±Inf)
+// operands.
+func TestFloatCmpSpecialValues(t *testing.T) {
+ zero := 0.0
+ args := []float64{math.Inf(-1), -2.71828, -1, -zero, zero, 1, 2.71828, math.Inf(1)}
+ xx := new(Float)
+ yy := new(Float)
+ for i := 0; i < 4; i++ {
+ for _, x := range args {
+ xx.SetFloat64(x)
+ // check conversion is correct
+ // (no need to do this for y, since we see exactly the
+ // same values there)
+ if got, acc := xx.Float64(); got != x || acc != Exact {
+ t.Errorf("Float(%g) == %g (%s)", x, got, acc)
+ }
+ for _, y := range args {
+ yy.SetFloat64(y)
+ got := xx.Cmp(yy)
+ want := 0
+ switch {
+ case x < y:
+ want = -1
+ case x > y:
+ want = +1
+ }
+ if got != want {
+ t.Errorf("(%g).Cmp(%g) = %v; want %v", x, y, got, want)
+ }
+ }
+ }
+ }
+}
+
+func BenchmarkFloatAdd(b *testing.B) {
+ x := new(Float)
+ y := new(Float)
+ z := new(Float)
+
+ for _, prec := range []uint{10, 1e2, 1e3, 1e4, 1e5} {
+ x.SetPrec(prec).SetRat(NewRat(1, 3))
+ y.SetPrec(prec).SetRat(NewRat(1, 6))
+ z.SetPrec(prec)
+
+ b.Run(fmt.Sprintf("%v", prec), func(b *testing.B) {
+ b.ReportAllocs()
+ for i := 0; i < b.N; i++ {
+ z.Add(x, y)
+ }
+ })
+ }
+}
+
+func BenchmarkFloatSub(b *testing.B) {
+ x := new(Float)
+ y := new(Float)
+ z := new(Float)
+
+ for _, prec := range []uint{10, 1e2, 1e3, 1e4, 1e5} {
+ x.SetPrec(prec).SetRat(NewRat(1, 3))
+ y.SetPrec(prec).SetRat(NewRat(1, 6))
+ z.SetPrec(prec)
+
+ b.Run(fmt.Sprintf("%v", prec), func(b *testing.B) {
+ b.ReportAllocs()
+ for i := 0; i < b.N; i++ {
+ z.Sub(x, y)
+ }
+ })
+ }
+}
diff --git a/src/math/big/floatconv.go b/src/math/big/floatconv.go
new file mode 100644
index 0000000..57b7df3
--- /dev/null
+++ b/src/math/big/floatconv.go
@@ -0,0 +1,304 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements string-to-Float conversion functions.
+
+package big
+
+import (
+ "fmt"
+ "io"
+ "strings"
+)
+
+var floatZero Float
+
+// SetString sets z to the value of s and returns z and a boolean indicating
+// success. s must be a floating-point number of the same format as accepted
+// by Parse, with base argument 0. The entire string (not just a prefix) must
+// be valid for success. If the operation failed, the value of z is undefined
+// but the returned value is nil.
+func (z *Float) SetString(s string) (*Float, bool) {
+ if f, _, err := z.Parse(s, 0); err == nil {
+ return f, true
+ }
+ return nil, false
+}
+
+// scan is like Parse but reads the longest possible prefix representing a valid
+// floating point number from an io.ByteScanner rather than a string. It serves
+// as the implementation of Parse. It does not recognize ±Inf and does not expect
+// EOF at the end.
+func (z *Float) scan(r io.ByteScanner, base int) (f *Float, b int, err error) {
+ prec := z.prec
+ if prec == 0 {
+ prec = 64
+ }
+
+ // A reasonable value in case of an error.
+ z.form = zero
+
+ // sign
+ z.neg, err = scanSign(r)
+ if err != nil {
+ return
+ }
+
+ // mantissa
+ var fcount int // fractional digit count; valid if <= 0
+ z.mant, b, fcount, err = z.mant.scan(r, base, true)
+ if err != nil {
+ return
+ }
+
+ // exponent
+ var exp int64
+ var ebase int
+ exp, ebase, err = scanExponent(r, true, base == 0)
+ if err != nil {
+ return
+ }
+
+ // special-case 0
+ if len(z.mant) == 0 {
+ z.prec = prec
+ z.acc = Exact
+ z.form = zero
+ f = z
+ return
+ }
+ // len(z.mant) > 0
+
+ // The mantissa may have a radix point (fcount <= 0) and there
+ // may be a nonzero exponent exp. The radix point amounts to a
+ // division by b**(-fcount). An exponent means multiplication by
+ // ebase**exp. Finally, mantissa normalization (shift left) requires
+ // a correcting multiplication by 2**(-shiftcount). Multiplications
+ // are commutative, so we can apply them in any order as long as there
+ // is no loss of precision. We only have powers of 2 and 10, and
+ // we split powers of 10 into the product of the same powers of
+ // 2 and 5. This reduces the size of the multiplication factor
+ // needed for base-10 exponents.
+
+ // normalize mantissa and determine initial exponent contributions
+ exp2 := int64(len(z.mant))*_W - fnorm(z.mant)
+ exp5 := int64(0)
+
+ // determine binary or decimal exponent contribution of radix point
+ if fcount < 0 {
+ // The mantissa has a radix point ddd.dddd; and
+ // -fcount is the number of digits to the right
+ // of '.'. Adjust relevant exponent accordingly.
+ d := int64(fcount)
+ switch b {
+ case 10:
+ exp5 = d
+ fallthrough // 10**e == 5**e * 2**e
+ case 2:
+ exp2 += d
+ case 8:
+ exp2 += d * 3 // octal digits are 3 bits each
+ case 16:
+ exp2 += d * 4 // hexadecimal digits are 4 bits each
+ default:
+ panic("unexpected mantissa base")
+ }
+ // fcount consumed - not needed anymore
+ }
+
+ // take actual exponent into account
+ switch ebase {
+ case 10:
+ exp5 += exp
+ fallthrough // see fallthrough above
+ case 2:
+ exp2 += exp
+ default:
+ panic("unexpected exponent base")
+ }
+ // exp consumed - not needed anymore
+
+ // apply 2**exp2
+ if MinExp <= exp2 && exp2 <= MaxExp {
+ z.prec = prec
+ z.form = finite
+ z.exp = int32(exp2)
+ f = z
+ } else {
+ err = fmt.Errorf("exponent overflow")
+ return
+ }
+
+ if exp5 == 0 {
+ // no decimal exponent contribution
+ z.round(0)
+ return
+ }
+ // exp5 != 0
+
+ // apply 5**exp5
+ p := new(Float).SetPrec(z.Prec() + 64) // use more bits for p -- TODO(gri) what is the right number?
+ if exp5 < 0 {
+ z.Quo(z, p.pow5(uint64(-exp5)))
+ } else {
+ z.Mul(z, p.pow5(uint64(exp5)))
+ }
+
+ return
+}
+
+// These powers of 5 fit into a uint64.
+//
+// for p, q := uint64(0), uint64(1); p < q; p, q = q, q*5 {
+// fmt.Println(q)
+// }
+//
+var pow5tab = [...]uint64{
+ 1,
+ 5,
+ 25,
+ 125,
+ 625,
+ 3125,
+ 15625,
+ 78125,
+ 390625,
+ 1953125,
+ 9765625,
+ 48828125,
+ 244140625,
+ 1220703125,
+ 6103515625,
+ 30517578125,
+ 152587890625,
+ 762939453125,
+ 3814697265625,
+ 19073486328125,
+ 95367431640625,
+ 476837158203125,
+ 2384185791015625,
+ 11920928955078125,
+ 59604644775390625,
+ 298023223876953125,
+ 1490116119384765625,
+ 7450580596923828125,
+}
+
+// pow5 sets z to 5**n and returns z.
+// n must not be negative.
+func (z *Float) pow5(n uint64) *Float {
+ const m = uint64(len(pow5tab) - 1)
+ if n <= m {
+ return z.SetUint64(pow5tab[n])
+ }
+ // n > m
+
+ z.SetUint64(pow5tab[m])
+ n -= m
+
+ // use more bits for f than for z
+ // TODO(gri) what is the right number?
+ f := new(Float).SetPrec(z.Prec() + 64).SetUint64(5)
+
+ for n > 0 {
+ if n&1 != 0 {
+ z.Mul(z, f)
+ }
+ f.Mul(f, f)
+ n >>= 1
+ }
+
+ return z
+}
+
+// Parse parses s which must contain a text representation of a floating-
+// point number with a mantissa in the given conversion base (the exponent
+// is always a decimal number), or a string representing an infinite value.
+//
+// For base 0, an underscore character ``_'' may appear between a base
+// prefix and an adjacent digit, and between successive digits; such
+// underscores do not change the value of the number, or the returned
+// digit count. Incorrect placement of underscores is reported as an
+// error if there are no other errors. If base != 0, underscores are
+// not recognized and thus terminate scanning like any other character
+// that is not a valid radix point or digit.
+//
+// It sets z to the (possibly rounded) value of the corresponding floating-
+// point value, and returns z, the actual base b, and an error err, if any.
+// The entire string (not just a prefix) must be consumed for success.
+// If z's precision is 0, it is changed to 64 before rounding takes effect.
+// The number must be of the form:
+//
+// number = [ sign ] ( float | "inf" | "Inf" ) .
+// sign = "+" | "-" .
+// float = ( mantissa | prefix pmantissa ) [ exponent ] .
+// prefix = "0" [ "b" | "B" | "o" | "O" | "x" | "X" ] .
+// mantissa = digits "." [ digits ] | digits | "." digits .
+// pmantissa = [ "_" ] digits "." [ digits ] | [ "_" ] digits | "." digits .
+// exponent = ( "e" | "E" | "p" | "P" ) [ sign ] digits .
+// digits = digit { [ "_" ] digit } .
+// digit = "0" ... "9" | "a" ... "z" | "A" ... "Z" .
+//
+// The base argument must be 0, 2, 8, 10, or 16. Providing an invalid base
+// argument will lead to a run-time panic.
+//
+// For base 0, the number prefix determines the actual base: A prefix of
+// ``0b'' or ``0B'' selects base 2, ``0o'' or ``0O'' selects base 8, and
+// ``0x'' or ``0X'' selects base 16. Otherwise, the actual base is 10 and
+// no prefix is accepted. The octal prefix "0" is not supported (a leading
+// "0" is simply considered a "0").
+//
+// A "p" or "P" exponent indicates a base 2 (rather then base 10) exponent;
+// for instance, "0x1.fffffffffffffp1023" (using base 0) represents the
+// maximum float64 value. For hexadecimal mantissae, the exponent character
+// must be one of 'p' or 'P', if present (an "e" or "E" exponent indicator
+// cannot be distinguished from a mantissa digit).
+//
+// The returned *Float f is nil and the value of z is valid but not
+// defined if an error is reported.
+//
+func (z *Float) Parse(s string, base int) (f *Float, b int, err error) {
+ // scan doesn't handle ±Inf
+ if len(s) == 3 && (s == "Inf" || s == "inf") {
+ f = z.SetInf(false)
+ return
+ }
+ if len(s) == 4 && (s[0] == '+' || s[0] == '-') && (s[1:] == "Inf" || s[1:] == "inf") {
+ f = z.SetInf(s[0] == '-')
+ return
+ }
+
+ r := strings.NewReader(s)
+ if f, b, err = z.scan(r, base); err != nil {
+ return
+ }
+
+ // entire string must have been consumed
+ if ch, err2 := r.ReadByte(); err2 == nil {
+ err = fmt.Errorf("expected end of string, found %q", ch)
+ } else if err2 != io.EOF {
+ err = err2
+ }
+
+ return
+}
+
+// ParseFloat is like f.Parse(s, base) with f set to the given precision
+// and rounding mode.
+func ParseFloat(s string, base int, prec uint, mode RoundingMode) (f *Float, b int, err error) {
+ return new(Float).SetPrec(prec).SetMode(mode).Parse(s, base)
+}
+
+var _ fmt.Scanner = (*Float)(nil) // *Float must implement fmt.Scanner
+
+// Scan is a support routine for fmt.Scanner; it sets z to the value of
+// the scanned number. It accepts formats whose verbs are supported by
+// fmt.Scan for floating point values, which are:
+// 'b' (binary), 'e', 'E', 'f', 'F', 'g' and 'G'.
+// Scan doesn't handle ±Inf.
+func (z *Float) Scan(s fmt.ScanState, ch rune) error {
+ s.SkipSpace()
+ _, _, err := z.scan(byteReader{s}, 0)
+ return err
+}
diff --git a/src/math/big/floatconv_test.go b/src/math/big/floatconv_test.go
new file mode 100644
index 0000000..3aa6834
--- /dev/null
+++ b/src/math/big/floatconv_test.go
@@ -0,0 +1,825 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "fmt"
+ "math"
+ "math/bits"
+ "strconv"
+ "testing"
+)
+
+var zero_ float64
+
+func TestFloatSetFloat64String(t *testing.T) {
+ inf := math.Inf(0)
+ nan := math.NaN()
+
+ for _, test := range []struct {
+ s string
+ x float64 // NaNs represent invalid inputs
+ }{
+ // basics
+ {"0", 0},
+ {"-0", -zero_},
+ {"+0", 0},
+ {"1", 1},
+ {"-1", -1},
+ {"+1", 1},
+ {"1.234", 1.234},
+ {"-1.234", -1.234},
+ {"+1.234", 1.234},
+ {".1", 0.1},
+ {"1.", 1},
+ {"+1.", 1},
+
+ // various zeros
+ {"0e100", 0},
+ {"-0e+100", -zero_},
+ {"+0e-100", 0},
+ {"0E100", 0},
+ {"-0E+100", -zero_},
+ {"+0E-100", 0},
+
+ // various decimal exponent formats
+ {"1.e10", 1e10},
+ {"1e+10", 1e10},
+ {"+1e-10", 1e-10},
+ {"1E10", 1e10},
+ {"1.E+10", 1e10},
+ {"+1E-10", 1e-10},
+
+ // infinities
+ {"Inf", inf},
+ {"+Inf", inf},
+ {"-Inf", -inf},
+ {"inf", inf},
+ {"+inf", inf},
+ {"-inf", -inf},
+
+ // invalid numbers
+ {"", nan},
+ {"-", nan},
+ {"0x", nan},
+ {"0e", nan},
+ {"1.2ef", nan},
+ {"2..3", nan},
+ {"123..", nan},
+ {"infinity", nan},
+ {"foobar", nan},
+
+ // invalid underscores
+ {"_", nan},
+ {"0_", nan},
+ {"1__0", nan},
+ {"123_.", nan},
+ {"123._", nan},
+ {"123._4", nan},
+ {"1_2.3_4_", nan},
+ {"_.123", nan},
+ {"_123.456", nan},
+ {"10._0", nan},
+ {"10.0e_0", nan},
+ {"10.0e0_", nan},
+ {"0P-0__0", nan},
+
+ // misc decimal values
+ {"3.14159265", 3.14159265},
+ {"-687436.79457e-245", -687436.79457e-245},
+ {"-687436.79457E245", -687436.79457e245},
+ {".0000000000000000000000000000000000000001", 1e-40},
+ {"+10000000000000000000000000000000000000000e-0", 1e40},
+
+ // decimal mantissa, binary exponent
+ {"0p0", 0},
+ {"-0p0", -zero_},
+ {"1p10", 1 << 10},
+ {"1p+10", 1 << 10},
+ {"+1p-10", 1.0 / (1 << 10)},
+ {"1024p-12", 0.25},
+ {"-1p10", -1024},
+ {"1.5p1", 3},
+
+ // binary mantissa, decimal exponent
+ {"0b0", 0},
+ {"-0b0", -zero_},
+ {"0b0e+10", 0},
+ {"-0b0e-10", -zero_},
+ {"0b1010", 10},
+ {"0B1010E2", 1000},
+ {"0b.1", 0.5},
+ {"0b.001", 0.125},
+ {"0b.001e3", 125},
+
+ // binary mantissa, binary exponent
+ {"0b0p+10", 0},
+ {"-0b0p-10", -zero_},
+ {"0b.1010p4", 10},
+ {"0b1p-1", 0.5},
+ {"0b001p-3", 0.125},
+ {"0b.001p3", 1},
+ {"0b0.01p2", 1},
+ {"0b0.01P+2", 1},
+
+ // octal mantissa, decimal exponent
+ {"0o0", 0},
+ {"-0o0", -zero_},
+ {"0o0e+10", 0},
+ {"-0o0e-10", -zero_},
+ {"0o12", 10},
+ {"0O12E2", 1000},
+ {"0o.4", 0.5},
+ {"0o.01", 0.015625},
+ {"0o.01e3", 15.625},
+
+ // octal mantissa, binary exponent
+ {"0o0p+10", 0},
+ {"-0o0p-10", -zero_},
+ {"0o.12p6", 10},
+ {"0o4p-3", 0.5},
+ {"0o0014p-6", 0.1875},
+ {"0o.001p9", 1},
+ {"0o0.01p7", 2},
+ {"0O0.01P+2", 0.0625},
+
+ // hexadecimal mantissa and exponent
+ {"0x0", 0},
+ {"-0x0", -zero_},
+ {"0x0p+10", 0},
+ {"-0x0p-10", -zero_},
+ {"0xff", 255},
+ {"0X.8p1", 1},
+ {"-0X0.00008p16", -0.5},
+ {"-0X0.00008P+16", -0.5},
+ {"0x0.0000000000001p-1022", math.SmallestNonzeroFloat64},
+ {"0x1.fffffffffffffp1023", math.MaxFloat64},
+
+ // underscores
+ {"0_0", 0},
+ {"1_000.", 1000},
+ {"1_2_3.4_5_6", 123.456},
+ {"1.0e0_0", 1},
+ {"1p+1_0", 1024},
+ {"0b_1000", 0x8},
+ {"0b_1011_1101", 0xbd},
+ {"0x_f0_0d_1eP+0_8", 0xf00d1e00},
+ } {
+ var x Float
+ x.SetPrec(53)
+ _, ok := x.SetString(test.s)
+ if math.IsNaN(test.x) {
+ // test.s is invalid
+ if ok {
+ t.Errorf("%s: want parse error", test.s)
+ }
+ continue
+ }
+ // test.s is valid
+ if !ok {
+ t.Errorf("%s: got parse error", test.s)
+ continue
+ }
+ f, _ := x.Float64()
+ want := new(Float).SetFloat64(test.x)
+ if x.Cmp(want) != 0 || x.Signbit() != want.Signbit() {
+ t.Errorf("%s: got %v (%v); want %v", test.s, &x, f, test.x)
+ }
+ }
+}
+
+func fdiv(a, b float64) float64 { return a / b }
+
+const (
+ below1e23 = 99999999999999974834176
+ above1e23 = 100000000000000008388608
+)
+
+func TestFloat64Text(t *testing.T) {
+ for _, test := range []struct {
+ x float64
+ format byte
+ prec int
+ want string
+ }{
+ {0, 'f', 0, "0"},
+ {math.Copysign(0, -1), 'f', 0, "-0"},
+ {1, 'f', 0, "1"},
+ {-1, 'f', 0, "-1"},
+
+ {0.001, 'e', 0, "1e-03"},
+ {0.459, 'e', 0, "5e-01"},
+ {1.459, 'e', 0, "1e+00"},
+ {2.459, 'e', 1, "2.5e+00"},
+ {3.459, 'e', 2, "3.46e+00"},
+ {4.459, 'e', 3, "4.459e+00"},
+ {5.459, 'e', 4, "5.4590e+00"},
+
+ {0.001, 'f', 0, "0"},
+ {0.459, 'f', 0, "0"},
+ {1.459, 'f', 0, "1"},
+ {2.459, 'f', 1, "2.5"},
+ {3.459, 'f', 2, "3.46"},
+ {4.459, 'f', 3, "4.459"},
+ {5.459, 'f', 4, "5.4590"},
+
+ {0, 'b', 0, "0"},
+ {math.Copysign(0, -1), 'b', 0, "-0"},
+ {1.0, 'b', 0, "4503599627370496p-52"},
+ {-1.0, 'b', 0, "-4503599627370496p-52"},
+ {4503599627370496, 'b', 0, "4503599627370496p+0"},
+
+ {0, 'p', 0, "0"},
+ {math.Copysign(0, -1), 'p', 0, "-0"},
+ {1024.0, 'p', 0, "0x.8p+11"},
+ {-1024.0, 'p', 0, "-0x.8p+11"},
+
+ // all test cases below from strconv/ftoa_test.go
+ {1, 'e', 5, "1.00000e+00"},
+ {1, 'f', 5, "1.00000"},
+ {1, 'g', 5, "1"},
+ {1, 'g', -1, "1"},
+ {20, 'g', -1, "20"},
+ {1234567.8, 'g', -1, "1.2345678e+06"},
+ {200000, 'g', -1, "200000"},
+ {2000000, 'g', -1, "2e+06"},
+
+ // g conversion and zero suppression
+ {400, 'g', 2, "4e+02"},
+ {40, 'g', 2, "40"},
+ {4, 'g', 2, "4"},
+ {.4, 'g', 2, "0.4"},
+ {.04, 'g', 2, "0.04"},
+ {.004, 'g', 2, "0.004"},
+ {.0004, 'g', 2, "0.0004"},
+ {.00004, 'g', 2, "4e-05"},
+ {.000004, 'g', 2, "4e-06"},
+
+ {0, 'e', 5, "0.00000e+00"},
+ {0, 'f', 5, "0.00000"},
+ {0, 'g', 5, "0"},
+ {0, 'g', -1, "0"},
+
+ {-1, 'e', 5, "-1.00000e+00"},
+ {-1, 'f', 5, "-1.00000"},
+ {-1, 'g', 5, "-1"},
+ {-1, 'g', -1, "-1"},
+
+ {12, 'e', 5, "1.20000e+01"},
+ {12, 'f', 5, "12.00000"},
+ {12, 'g', 5, "12"},
+ {12, 'g', -1, "12"},
+
+ {123456700, 'e', 5, "1.23457e+08"},
+ {123456700, 'f', 5, "123456700.00000"},
+ {123456700, 'g', 5, "1.2346e+08"},
+ {123456700, 'g', -1, "1.234567e+08"},
+
+ {1.2345e6, 'e', 5, "1.23450e+06"},
+ {1.2345e6, 'f', 5, "1234500.00000"},
+ {1.2345e6, 'g', 5, "1.2345e+06"},
+
+ {1e23, 'e', 17, "9.99999999999999916e+22"},
+ {1e23, 'f', 17, "99999999999999991611392.00000000000000000"},
+ {1e23, 'g', 17, "9.9999999999999992e+22"},
+
+ {1e23, 'e', -1, "1e+23"},
+ {1e23, 'f', -1, "100000000000000000000000"},
+ {1e23, 'g', -1, "1e+23"},
+
+ {below1e23, 'e', 17, "9.99999999999999748e+22"},
+ {below1e23, 'f', 17, "99999999999999974834176.00000000000000000"},
+ {below1e23, 'g', 17, "9.9999999999999975e+22"},
+
+ {below1e23, 'e', -1, "9.999999999999997e+22"},
+ {below1e23, 'f', -1, "99999999999999970000000"},
+ {below1e23, 'g', -1, "9.999999999999997e+22"},
+
+ {above1e23, 'e', 17, "1.00000000000000008e+23"},
+ {above1e23, 'f', 17, "100000000000000008388608.00000000000000000"},
+ {above1e23, 'g', 17, "1.0000000000000001e+23"},
+
+ {above1e23, 'e', -1, "1.0000000000000001e+23"},
+ {above1e23, 'f', -1, "100000000000000010000000"},
+ {above1e23, 'g', -1, "1.0000000000000001e+23"},
+
+ {5e-304 / 1e20, 'g', -1, "5e-324"},
+ {-5e-304 / 1e20, 'g', -1, "-5e-324"},
+ {fdiv(5e-304, 1e20), 'g', -1, "5e-324"}, // avoid constant arithmetic
+ {fdiv(-5e-304, 1e20), 'g', -1, "-5e-324"}, // avoid constant arithmetic
+
+ {32, 'g', -1, "32"},
+ {32, 'g', 0, "3e+01"},
+
+ {100, 'x', -1, "0x1.9p+06"},
+
+ // {math.NaN(), 'g', -1, "NaN"}, // Float doesn't support NaNs
+ // {-math.NaN(), 'g', -1, "NaN"}, // Float doesn't support NaNs
+ {math.Inf(0), 'g', -1, "+Inf"},
+ {math.Inf(-1), 'g', -1, "-Inf"},
+ {-math.Inf(0), 'g', -1, "-Inf"},
+
+ {-1, 'b', -1, "-4503599627370496p-52"},
+
+ // fixed bugs
+ {0.9, 'f', 1, "0.9"},
+ {0.09, 'f', 1, "0.1"},
+ {0.0999, 'f', 1, "0.1"},
+ {0.05, 'f', 1, "0.1"},
+ {0.05, 'f', 0, "0"},
+ {0.5, 'f', 1, "0.5"},
+ {0.5, 'f', 0, "0"},
+ {1.5, 'f', 0, "2"},
+
+ // https://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
+ {2.2250738585072012e-308, 'g', -1, "2.2250738585072014e-308"},
+ // https://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
+ {2.2250738585072011e-308, 'g', -1, "2.225073858507201e-308"},
+
+ // Issue 2625.
+ {383260575764816448, 'f', 0, "383260575764816448"},
+ {383260575764816448, 'g', -1, "3.8326057576481645e+17"},
+
+ // Issue 15918.
+ {1, 'f', -10, "1"},
+ {1, 'f', -11, "1"},
+ {1, 'f', -12, "1"},
+ } {
+ // The test cases are from the strconv package which tests float64 values.
+ // When formatting values with prec = -1 (shortest representation),
+ // the actually available mantissa precision matters.
+ // For denormalized values, that precision is < 53 (SetFloat64 default).
+ // Compute and set the actual precision explicitly.
+ f := new(Float).SetPrec(actualPrec(test.x)).SetFloat64(test.x)
+ got := f.Text(test.format, test.prec)
+ if got != test.want {
+ t.Errorf("%v: got %s; want %s", test, got, test.want)
+ continue
+ }
+
+ if test.format == 'b' && test.x == 0 {
+ continue // 'b' format in strconv.Float requires knowledge of bias for 0.0
+ }
+ if test.format == 'p' {
+ continue // 'p' format not supported in strconv.Format
+ }
+
+ // verify that Float format matches strconv format
+ want := strconv.FormatFloat(test.x, test.format, test.prec, 64)
+ if got != want {
+ t.Errorf("%v: got %s; want %s (strconv)", test, got, want)
+ }
+ }
+}
+
+// actualPrec returns the number of actually used mantissa bits.
+func actualPrec(x float64) uint {
+ if mant := math.Float64bits(x); x != 0 && mant&(0x7ff<<52) == 0 {
+ // x is denormalized
+ return 64 - uint(bits.LeadingZeros64(mant&(1<<52-1)))
+ }
+ return 53
+}
+
+func TestFloatText(t *testing.T) {
+ const defaultRound = ^RoundingMode(0)
+
+ for _, test := range []struct {
+ x string
+ round RoundingMode
+ prec uint
+ format byte
+ digits int
+ want string
+ }{
+ {"0", defaultRound, 10, 'f', 0, "0"},
+ {"-0", defaultRound, 10, 'f', 0, "-0"},
+ {"1", defaultRound, 10, 'f', 0, "1"},
+ {"-1", defaultRound, 10, 'f', 0, "-1"},
+
+ {"1.459", defaultRound, 100, 'e', 0, "1e+00"},
+ {"2.459", defaultRound, 100, 'e', 1, "2.5e+00"},
+ {"3.459", defaultRound, 100, 'e', 2, "3.46e+00"},
+ {"4.459", defaultRound, 100, 'e', 3, "4.459e+00"},
+ {"5.459", defaultRound, 100, 'e', 4, "5.4590e+00"},
+
+ {"1.459", defaultRound, 100, 'E', 0, "1E+00"},
+ {"2.459", defaultRound, 100, 'E', 1, "2.5E+00"},
+ {"3.459", defaultRound, 100, 'E', 2, "3.46E+00"},
+ {"4.459", defaultRound, 100, 'E', 3, "4.459E+00"},
+ {"5.459", defaultRound, 100, 'E', 4, "5.4590E+00"},
+
+ {"1.459", defaultRound, 100, 'f', 0, "1"},
+ {"2.459", defaultRound, 100, 'f', 1, "2.5"},
+ {"3.459", defaultRound, 100, 'f', 2, "3.46"},
+ {"4.459", defaultRound, 100, 'f', 3, "4.459"},
+ {"5.459", defaultRound, 100, 'f', 4, "5.4590"},
+
+ {"1.459", defaultRound, 100, 'g', 0, "1"},
+ {"2.459", defaultRound, 100, 'g', 1, "2"},
+ {"3.459", defaultRound, 100, 'g', 2, "3.5"},
+ {"4.459", defaultRound, 100, 'g', 3, "4.46"},
+ {"5.459", defaultRound, 100, 'g', 4, "5.459"},
+
+ {"1459", defaultRound, 53, 'g', 0, "1e+03"},
+ {"2459", defaultRound, 53, 'g', 1, "2e+03"},
+ {"3459", defaultRound, 53, 'g', 2, "3.5e+03"},
+ {"4459", defaultRound, 53, 'g', 3, "4.46e+03"},
+ {"5459", defaultRound, 53, 'g', 4, "5459"},
+
+ {"1459", defaultRound, 53, 'G', 0, "1E+03"},
+ {"2459", defaultRound, 53, 'G', 1, "2E+03"},
+ {"3459", defaultRound, 53, 'G', 2, "3.5E+03"},
+ {"4459", defaultRound, 53, 'G', 3, "4.46E+03"},
+ {"5459", defaultRound, 53, 'G', 4, "5459"},
+
+ {"3", defaultRound, 10, 'e', 40, "3.0000000000000000000000000000000000000000e+00"},
+ {"3", defaultRound, 10, 'f', 40, "3.0000000000000000000000000000000000000000"},
+ {"3", defaultRound, 10, 'g', 40, "3"},
+
+ {"3e40", defaultRound, 100, 'e', 40, "3.0000000000000000000000000000000000000000e+40"},
+ {"3e40", defaultRound, 100, 'f', 4, "30000000000000000000000000000000000000000.0000"},
+ {"3e40", defaultRound, 100, 'g', 40, "3e+40"},
+
+ // make sure "stupid" exponents don't stall the machine
+ {"1e1000000", defaultRound, 64, 'p', 0, "0x.88b3a28a05eade3ap+3321929"},
+ {"1e646456992", defaultRound, 64, 'p', 0, "0x.e883a0c5c8c7c42ap+2147483644"},
+ {"1e646456993", defaultRound, 64, 'p', 0, "+Inf"},
+ {"1e1000000000", defaultRound, 64, 'p', 0, "+Inf"},
+ {"1e-1000000", defaultRound, 64, 'p', 0, "0x.efb4542cc8ca418ap-3321928"},
+ {"1e-646456993", defaultRound, 64, 'p', 0, "0x.e17c8956983d9d59p-2147483647"},
+ {"1e-646456994", defaultRound, 64, 'p', 0, "0"},
+ {"1e-1000000000", defaultRound, 64, 'p', 0, "0"},
+
+ // minimum and maximum values
+ {"1p2147483646", defaultRound, 64, 'p', 0, "0x.8p+2147483647"},
+ {"0x.8p2147483647", defaultRound, 64, 'p', 0, "0x.8p+2147483647"},
+ {"0x.8p-2147483647", defaultRound, 64, 'p', 0, "0x.8p-2147483647"},
+ {"1p-2147483649", defaultRound, 64, 'p', 0, "0x.8p-2147483648"},
+
+ // TODO(gri) need tests for actual large Floats
+
+ {"0", defaultRound, 53, 'b', 0, "0"},
+ {"-0", defaultRound, 53, 'b', 0, "-0"},
+ {"1.0", defaultRound, 53, 'b', 0, "4503599627370496p-52"},
+ {"-1.0", defaultRound, 53, 'b', 0, "-4503599627370496p-52"},
+ {"4503599627370496", defaultRound, 53, 'b', 0, "4503599627370496p+0"},
+
+ // issue 9939
+ {"3", defaultRound, 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+ {"03", defaultRound, 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+ {"3.", defaultRound, 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+ {"3.0", defaultRound, 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+ {"3.00", defaultRound, 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+ {"3.000", defaultRound, 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+
+ {"3", defaultRound, 350, 'p', 0, "0x.cp+2"},
+ {"03", defaultRound, 350, 'p', 0, "0x.cp+2"},
+ {"3.", defaultRound, 350, 'p', 0, "0x.cp+2"},
+ {"3.0", defaultRound, 350, 'p', 0, "0x.cp+2"},
+ {"3.00", defaultRound, 350, 'p', 0, "0x.cp+2"},
+ {"3.000", defaultRound, 350, 'p', 0, "0x.cp+2"},
+
+ {"0", defaultRound, 64, 'p', 0, "0"},
+ {"-0", defaultRound, 64, 'p', 0, "-0"},
+ {"1024.0", defaultRound, 64, 'p', 0, "0x.8p+11"},
+ {"-1024.0", defaultRound, 64, 'p', 0, "-0x.8p+11"},
+
+ {"0", defaultRound, 64, 'x', -1, "0x0p+00"},
+ {"0", defaultRound, 64, 'x', 0, "0x0p+00"},
+ {"0", defaultRound, 64, 'x', 1, "0x0.0p+00"},
+ {"0", defaultRound, 64, 'x', 5, "0x0.00000p+00"},
+ {"3.25", defaultRound, 64, 'x', 0, "0x1p+02"},
+ {"-3.25", defaultRound, 64, 'x', 0, "-0x1p+02"},
+ {"3.25", defaultRound, 64, 'x', 1, "0x1.ap+01"},
+ {"-3.25", defaultRound, 64, 'x', 1, "-0x1.ap+01"},
+ {"3.25", defaultRound, 64, 'x', -1, "0x1.ap+01"},
+ {"-3.25", defaultRound, 64, 'x', -1, "-0x1.ap+01"},
+ {"1024.0", defaultRound, 64, 'x', 0, "0x1p+10"},
+ {"-1024.0", defaultRound, 64, 'x', 0, "-0x1p+10"},
+ {"1024.0", defaultRound, 64, 'x', 5, "0x1.00000p+10"},
+ {"8191.0", defaultRound, 53, 'x', -1, "0x1.fffp+12"},
+ {"8191.5", defaultRound, 53, 'x', -1, "0x1.fff8p+12"},
+ {"8191.53125", defaultRound, 53, 'x', -1, "0x1.fff88p+12"},
+ {"8191.53125", defaultRound, 53, 'x', 4, "0x1.fff8p+12"},
+ {"8191.53125", defaultRound, 53, 'x', 3, "0x1.000p+13"},
+ {"8191.53125", defaultRound, 53, 'x', 0, "0x1p+13"},
+ {"8191.533203125", defaultRound, 53, 'x', -1, "0x1.fff888p+12"},
+ {"8191.533203125", defaultRound, 53, 'x', 5, "0x1.fff88p+12"},
+ {"8191.533203125", defaultRound, 53, 'x', 4, "0x1.fff9p+12"},
+
+ {"8191.53125", defaultRound, 53, 'x', -1, "0x1.fff88p+12"},
+ {"8191.53125", ToNearestEven, 53, 'x', 5, "0x1.fff88p+12"},
+ {"8191.53125", ToNearestAway, 53, 'x', 5, "0x1.fff88p+12"},
+ {"8191.53125", ToZero, 53, 'x', 5, "0x1.fff88p+12"},
+ {"8191.53125", AwayFromZero, 53, 'x', 5, "0x1.fff88p+12"},
+ {"8191.53125", ToNegativeInf, 53, 'x', 5, "0x1.fff88p+12"},
+ {"8191.53125", ToPositiveInf, 53, 'x', 5, "0x1.fff88p+12"},
+
+ {"8191.53125", defaultRound, 53, 'x', 4, "0x1.fff8p+12"},
+ {"8191.53125", defaultRound, 53, 'x', 3, "0x1.000p+13"},
+ {"8191.53125", defaultRound, 53, 'x', 0, "0x1p+13"},
+ {"8191.533203125", defaultRound, 53, 'x', -1, "0x1.fff888p+12"},
+ {"8191.533203125", defaultRound, 53, 'x', 6, "0x1.fff888p+12"},
+ {"8191.533203125", defaultRound, 53, 'x', 5, "0x1.fff88p+12"},
+ {"8191.533203125", defaultRound, 53, 'x', 4, "0x1.fff9p+12"},
+
+ {"8191.53125", ToNearestEven, 53, 'x', 4, "0x1.fff8p+12"},
+ {"8191.53125", ToNearestAway, 53, 'x', 4, "0x1.fff9p+12"},
+ {"8191.53125", ToZero, 53, 'x', 4, "0x1.fff8p+12"},
+ {"8191.53125", ToZero, 53, 'x', 2, "0x1.ffp+12"},
+ {"8191.53125", AwayFromZero, 53, 'x', 4, "0x1.fff9p+12"},
+ {"8191.53125", ToNegativeInf, 53, 'x', 4, "0x1.fff8p+12"},
+ {"-8191.53125", ToNegativeInf, 53, 'x', 4, "-0x1.fff9p+12"},
+ {"8191.53125", ToPositiveInf, 53, 'x', 4, "0x1.fff9p+12"},
+ {"-8191.53125", ToPositiveInf, 53, 'x', 4, "-0x1.fff8p+12"},
+
+ // issue 34343
+ {"0x.8p-2147483648", ToNearestEven, 4, 'p', -1, "0x.8p-2147483648"},
+ {"0x.8p-2147483648", ToNearestEven, 4, 'x', -1, "0x1p-2147483649"},
+ } {
+ f, _, err := ParseFloat(test.x, 0, test.prec, ToNearestEven)
+ if err != nil {
+ t.Errorf("%v: %s", test, err)
+ continue
+ }
+ if test.round != defaultRound {
+ f.SetMode(test.round)
+ }
+
+ got := f.Text(test.format, test.digits)
+ if got != test.want {
+ t.Errorf("%v: got %s; want %s", test, got, test.want)
+ }
+
+ // compare with strconv.FormatFloat output if possible
+ // ('p' format is not supported by strconv.FormatFloat,
+ // and its output for 0.0 prints a biased exponent value
+ // as in 0p-1074 which makes no sense to emulate here)
+ if test.prec == 53 && test.format != 'p' && f.Sign() != 0 && (test.round == ToNearestEven || test.round == defaultRound) {
+ f64, acc := f.Float64()
+ if acc != Exact {
+ t.Errorf("%v: expected exact conversion to float64", test)
+ continue
+ }
+ got := strconv.FormatFloat(f64, test.format, test.digits, 64)
+ if got != test.want {
+ t.Errorf("%v: got %s; want %s", test, got, test.want)
+ }
+ }
+ }
+}
+
+func TestFloatFormat(t *testing.T) {
+ for _, test := range []struct {
+ format string
+ value interface{} // float32, float64, or string (== 512bit *Float)
+ want string
+ }{
+ // from fmt/fmt_test.go
+ {"%+.3e", 0.0, "+0.000e+00"},
+ {"%+.3e", 1.0, "+1.000e+00"},
+ {"%+.3f", -1.0, "-1.000"},
+ {"%+.3F", -1.0, "-1.000"},
+ {"%+.3F", float32(-1.0), "-1.000"},
+ {"%+07.2f", 1.0, "+001.00"},
+ {"%+07.2f", -1.0, "-001.00"},
+ {"%+10.2f", +1.0, " +1.00"},
+ {"%+10.2f", -1.0, " -1.00"},
+ {"% .3E", -1.0, "-1.000E+00"},
+ {"% .3e", 1.0, " 1.000e+00"},
+ {"%+.3g", 0.0, "+0"},
+ {"%+.3g", 1.0, "+1"},
+ {"%+.3g", -1.0, "-1"},
+ {"% .3g", -1.0, "-1"},
+ {"% .3g", 1.0, " 1"},
+ {"%b", float32(1.0), "8388608p-23"},
+ {"%b", 1.0, "4503599627370496p-52"},
+
+ // from fmt/fmt_test.go: old test/fmt_test.go
+ {"%e", 1.0, "1.000000e+00"},
+ {"%e", 1234.5678e3, "1.234568e+06"},
+ {"%e", 1234.5678e-8, "1.234568e-05"},
+ {"%e", -7.0, "-7.000000e+00"},
+ {"%e", -1e-9, "-1.000000e-09"},
+ {"%f", 1234.5678e3, "1234567.800000"},
+ {"%f", 1234.5678e-8, "0.000012"},
+ {"%f", -7.0, "-7.000000"},
+ {"%f", -1e-9, "-0.000000"},
+ {"%g", 1234.5678e3, "1.2345678e+06"},
+ {"%g", float32(1234.5678e3), "1.2345678e+06"},
+ {"%g", 1234.5678e-8, "1.2345678e-05"},
+ {"%g", -7.0, "-7"},
+ {"%g", -1e-9, "-1e-09"},
+ {"%g", float32(-1e-9), "-1e-09"},
+ {"%E", 1.0, "1.000000E+00"},
+ {"%E", 1234.5678e3, "1.234568E+06"},
+ {"%E", 1234.5678e-8, "1.234568E-05"},
+ {"%E", -7.0, "-7.000000E+00"},
+ {"%E", -1e-9, "-1.000000E-09"},
+ {"%G", 1234.5678e3, "1.2345678E+06"},
+ {"%G", float32(1234.5678e3), "1.2345678E+06"},
+ {"%G", 1234.5678e-8, "1.2345678E-05"},
+ {"%G", -7.0, "-7"},
+ {"%G", -1e-9, "-1E-09"},
+ {"%G", float32(-1e-9), "-1E-09"},
+
+ {"%20.6e", 1.2345e3, " 1.234500e+03"},
+ {"%20.6e", 1.2345e-3, " 1.234500e-03"},
+ {"%20e", 1.2345e3, " 1.234500e+03"},
+ {"%20e", 1.2345e-3, " 1.234500e-03"},
+ {"%20.8e", 1.2345e3, " 1.23450000e+03"},
+ {"%20f", 1.23456789e3, " 1234.567890"},
+ {"%20f", 1.23456789e-3, " 0.001235"},
+ {"%20f", 12345678901.23456789, " 12345678901.234568"},
+ {"%-20f", 1.23456789e3, "1234.567890 "},
+ {"%20.8f", 1.23456789e3, " 1234.56789000"},
+ {"%20.8f", 1.23456789e-3, " 0.00123457"},
+ {"%g", 1.23456789e3, "1234.56789"},
+ {"%g", 1.23456789e-3, "0.00123456789"},
+ {"%g", 1.23456789e20, "1.23456789e+20"},
+ {"%20e", math.Inf(1), " +Inf"},
+ {"%-20f", math.Inf(-1), "-Inf "},
+
+ // from fmt/fmt_test.go: comparison of padding rules with C printf
+ {"%.2f", 1.0, "1.00"},
+ {"%.2f", -1.0, "-1.00"},
+ {"% .2f", 1.0, " 1.00"},
+ {"% .2f", -1.0, "-1.00"},
+ {"%+.2f", 1.0, "+1.00"},
+ {"%+.2f", -1.0, "-1.00"},
+ {"%7.2f", 1.0, " 1.00"},
+ {"%7.2f", -1.0, " -1.00"},
+ {"% 7.2f", 1.0, " 1.00"},
+ {"% 7.2f", -1.0, " -1.00"},
+ {"%+7.2f", 1.0, " +1.00"},
+ {"%+7.2f", -1.0, " -1.00"},
+ {"%07.2f", 1.0, "0001.00"},
+ {"%07.2f", -1.0, "-001.00"},
+ {"% 07.2f", 1.0, " 001.00"},
+ {"% 07.2f", -1.0, "-001.00"},
+ {"%+07.2f", 1.0, "+001.00"},
+ {"%+07.2f", -1.0, "-001.00"},
+
+ // from fmt/fmt_test.go: zero padding does not apply to infinities
+ {"%020f", math.Inf(-1), " -Inf"},
+ {"%020f", math.Inf(+1), " +Inf"},
+ {"% 020f", math.Inf(-1), " -Inf"},
+ {"% 020f", math.Inf(+1), " Inf"},
+ {"%+020f", math.Inf(-1), " -Inf"},
+ {"%+020f", math.Inf(+1), " +Inf"},
+ {"%20f", -1.0, " -1.000000"},
+
+ // handle %v like %g
+ {"%v", 0.0, "0"},
+ {"%v", -7.0, "-7"},
+ {"%v", -1e-9, "-1e-09"},
+ {"%v", float32(-1e-9), "-1e-09"},
+ {"%010v", 0.0, "0000000000"},
+
+ // *Float cases
+ {"%.20f", "1e-20", "0.00000000000000000001"},
+ {"%.20f", "-1e-20", "-0.00000000000000000001"},
+ {"%30.20f", "-1e-20", " -0.00000000000000000001"},
+ {"%030.20f", "-1e-20", "-00000000.00000000000000000001"},
+ {"%030.20f", "+1e-20", "000000000.00000000000000000001"},
+ {"% 030.20f", "+1e-20", " 00000000.00000000000000000001"},
+
+ // erroneous formats
+ {"%s", 1.0, "%!s(*big.Float=1)"},
+ } {
+ value := new(Float)
+ switch v := test.value.(type) {
+ case float32:
+ value.SetPrec(24).SetFloat64(float64(v))
+ case float64:
+ value.SetPrec(53).SetFloat64(v)
+ case string:
+ value.SetPrec(512).Parse(v, 0)
+ default:
+ t.Fatalf("unsupported test value: %v (%T)", v, v)
+ }
+
+ if got := fmt.Sprintf(test.format, value); got != test.want {
+ t.Errorf("%v: got %q; want %q", test, got, test.want)
+ }
+ }
+}
+
+func BenchmarkParseFloatSmallExp(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ for _, s := range []string{
+ "1e0",
+ "1e-1",
+ "1e-2",
+ "1e-3",
+ "1e-4",
+ "1e-5",
+ "1e-10",
+ "1e-20",
+ "1e-50",
+ "1e1",
+ "1e2",
+ "1e3",
+ "1e4",
+ "1e5",
+ "1e10",
+ "1e20",
+ "1e50",
+ } {
+ var x Float
+ _, _, err := x.Parse(s, 0)
+ if err != nil {
+ b.Fatalf("%s: %v", s, err)
+ }
+ }
+ }
+}
+
+func BenchmarkParseFloatLargeExp(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ for _, s := range []string{
+ "1e0",
+ "1e-10",
+ "1e-20",
+ "1e-30",
+ "1e-40",
+ "1e-50",
+ "1e-100",
+ "1e-500",
+ "1e-1000",
+ "1e-5000",
+ "1e-10000",
+ "1e10",
+ "1e20",
+ "1e30",
+ "1e40",
+ "1e50",
+ "1e100",
+ "1e500",
+ "1e1000",
+ "1e5000",
+ "1e10000",
+ } {
+ var x Float
+ _, _, err := x.Parse(s, 0)
+ if err != nil {
+ b.Fatalf("%s: %v", s, err)
+ }
+ }
+ }
+}
+
+func TestFloatScan(t *testing.T) {
+ var floatScanTests = []struct {
+ input string
+ format string
+ output string
+ remaining int
+ wantErr bool
+ }{
+ 0: {"10.0", "%f", "10", 0, false},
+ 1: {"23.98+2.0", "%v", "23.98", 4, false},
+ 2: {"-1+1", "%v", "-1", 2, false},
+ 3: {" 00000", "%v", "0", 0, false},
+ 4: {"-123456p-78", "%b", "-4.084816388e-19", 0, false},
+ 5: {"+123", "%b", "123", 0, false},
+ 6: {"-1.234e+56", "%e", "-1.234e+56", 0, false},
+ 7: {"-1.234E-56", "%E", "-1.234e-56", 0, false},
+ 8: {"-1.234e+567", "%g", "-1.234e+567", 0, false},
+ 9: {"+1234567891011.234", "%G", "1.234567891e+12", 0, false},
+
+ // Scan doesn't handle ±Inf.
+ 10: {"Inf", "%v", "", 3, true},
+ 11: {"-Inf", "%v", "", 3, true},
+ 12: {"-Inf", "%v", "", 3, true},
+ }
+
+ var buf bytes.Buffer
+ for i, test := range floatScanTests {
+ x := new(Float)
+ buf.Reset()
+ buf.WriteString(test.input)
+ _, err := fmt.Fscanf(&buf, test.format, x)
+ if test.wantErr {
+ if err == nil {
+ t.Errorf("#%d want non-nil err", i)
+ }
+ continue
+ }
+
+ if err != nil {
+ t.Errorf("#%d error: %s", i, err)
+ }
+
+ if x.String() != test.output {
+ t.Errorf("#%d got %s; want %s", i, x.String(), test.output)
+ }
+ if buf.Len() != test.remaining {
+ t.Errorf("#%d got %d bytes remaining; want %d", i, buf.Len(), test.remaining)
+ }
+ }
+}
diff --git a/src/math/big/floatexample_test.go b/src/math/big/floatexample_test.go
new file mode 100644
index 0000000..0c6668c
--- /dev/null
+++ b/src/math/big/floatexample_test.go
@@ -0,0 +1,141 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big_test
+
+import (
+ "fmt"
+ "math"
+ "math/big"
+)
+
+func ExampleFloat_Add() {
+ // Operate on numbers of different precision.
+ var x, y, z big.Float
+ x.SetInt64(1000) // x is automatically set to 64bit precision
+ y.SetFloat64(2.718281828) // y is automatically set to 53bit precision
+ z.SetPrec(32)
+ z.Add(&x, &y)
+ fmt.Printf("x = %.10g (%s, prec = %d, acc = %s)\n", &x, x.Text('p', 0), x.Prec(), x.Acc())
+ fmt.Printf("y = %.10g (%s, prec = %d, acc = %s)\n", &y, y.Text('p', 0), y.Prec(), y.Acc())
+ fmt.Printf("z = %.10g (%s, prec = %d, acc = %s)\n", &z, z.Text('p', 0), z.Prec(), z.Acc())
+ // Output:
+ // x = 1000 (0x.fap+10, prec = 64, acc = Exact)
+ // y = 2.718281828 (0x.adf85458248cd8p+2, prec = 53, acc = Exact)
+ // z = 1002.718282 (0x.faadf854p+10, prec = 32, acc = Below)
+}
+
+func ExampleFloat_shift() {
+ // Implement Float "shift" by modifying the (binary) exponents directly.
+ for s := -5; s <= 5; s++ {
+ x := big.NewFloat(0.5)
+ x.SetMantExp(x, x.MantExp(nil)+s) // shift x by s
+ fmt.Println(x)
+ }
+ // Output:
+ // 0.015625
+ // 0.03125
+ // 0.0625
+ // 0.125
+ // 0.25
+ // 0.5
+ // 1
+ // 2
+ // 4
+ // 8
+ // 16
+}
+
+func ExampleFloat_Cmp() {
+ inf := math.Inf(1)
+ zero := 0.0
+
+ operands := []float64{-inf, -1.2, -zero, 0, +1.2, +inf}
+
+ fmt.Println(" x y cmp")
+ fmt.Println("---------------")
+ for _, x64 := range operands {
+ x := big.NewFloat(x64)
+ for _, y64 := range operands {
+ y := big.NewFloat(y64)
+ fmt.Printf("%4g %4g %3d\n", x, y, x.Cmp(y))
+ }
+ fmt.Println()
+ }
+
+ // Output:
+ // x y cmp
+ // ---------------
+ // -Inf -Inf 0
+ // -Inf -1.2 -1
+ // -Inf -0 -1
+ // -Inf 0 -1
+ // -Inf 1.2 -1
+ // -Inf +Inf -1
+ //
+ // -1.2 -Inf 1
+ // -1.2 -1.2 0
+ // -1.2 -0 -1
+ // -1.2 0 -1
+ // -1.2 1.2 -1
+ // -1.2 +Inf -1
+ //
+ // -0 -Inf 1
+ // -0 -1.2 1
+ // -0 -0 0
+ // -0 0 0
+ // -0 1.2 -1
+ // -0 +Inf -1
+ //
+ // 0 -Inf 1
+ // 0 -1.2 1
+ // 0 -0 0
+ // 0 0 0
+ // 0 1.2 -1
+ // 0 +Inf -1
+ //
+ // 1.2 -Inf 1
+ // 1.2 -1.2 1
+ // 1.2 -0 1
+ // 1.2 0 1
+ // 1.2 1.2 0
+ // 1.2 +Inf -1
+ //
+ // +Inf -Inf 1
+ // +Inf -1.2 1
+ // +Inf -0 1
+ // +Inf 0 1
+ // +Inf 1.2 1
+ // +Inf +Inf 0
+}
+
+func ExampleRoundingMode() {
+ operands := []float64{2.6, 2.5, 2.1, -2.1, -2.5, -2.6}
+
+ fmt.Print(" x")
+ for mode := big.ToNearestEven; mode <= big.ToPositiveInf; mode++ {
+ fmt.Printf(" %s", mode)
+ }
+ fmt.Println()
+
+ for _, f64 := range operands {
+ fmt.Printf("%4g", f64)
+ for mode := big.ToNearestEven; mode <= big.ToPositiveInf; mode++ {
+ // sample operands above require 2 bits to represent mantissa
+ // set binary precision to 2 to round them to integer values
+ f := new(big.Float).SetPrec(2).SetMode(mode).SetFloat64(f64)
+ fmt.Printf(" %*g", len(mode.String()), f)
+ }
+ fmt.Println()
+ }
+
+ // Output:
+ // x ToNearestEven ToNearestAway ToZero AwayFromZero ToNegativeInf ToPositiveInf
+ // 2.6 3 3 2 3 2 3
+ // 2.5 2 3 2 3 2 3
+ // 2.1 2 2 2 3 2 3
+ // -2.1 -2 -2 -2 -3 -3 -2
+ // -2.5 -2 -3 -2 -3 -3 -2
+ // -2.6 -3 -3 -2 -3 -3 -2
+}
diff --git a/src/math/big/floatmarsh.go b/src/math/big/floatmarsh.go
new file mode 100644
index 0000000..d1c1dab
--- /dev/null
+++ b/src/math/big/floatmarsh.go
@@ -0,0 +1,120 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements encoding/decoding of Floats.
+
+package big
+
+import (
+ "encoding/binary"
+ "fmt"
+)
+
+// Gob codec version. Permits backward-compatible changes to the encoding.
+const floatGobVersion byte = 1
+
+// GobEncode implements the gob.GobEncoder interface.
+// The Float value and all its attributes (precision,
+// rounding mode, accuracy) are marshaled.
+func (x *Float) GobEncode() ([]byte, error) {
+ if x == nil {
+ return nil, nil
+ }
+
+ // determine max. space (bytes) required for encoding
+ sz := 1 + 1 + 4 // version + mode|acc|form|neg (3+2+2+1bit) + prec
+ n := 0 // number of mantissa words
+ if x.form == finite {
+ // add space for mantissa and exponent
+ n = int((x.prec + (_W - 1)) / _W) // required mantissa length in words for given precision
+ // actual mantissa slice could be shorter (trailing 0's) or longer (unused bits):
+ // - if shorter, only encode the words present
+ // - if longer, cut off unused words when encoding in bytes
+ // (in practice, this should never happen since rounding
+ // takes care of it, but be safe and do it always)
+ if len(x.mant) < n {
+ n = len(x.mant)
+ }
+ // len(x.mant) >= n
+ sz += 4 + n*_S // exp + mant
+ }
+ buf := make([]byte, sz)
+
+ buf[0] = floatGobVersion
+ b := byte(x.mode&7)<<5 | byte((x.acc+1)&3)<<3 | byte(x.form&3)<<1
+ if x.neg {
+ b |= 1
+ }
+ buf[1] = b
+ binary.BigEndian.PutUint32(buf[2:], x.prec)
+
+ if x.form == finite {
+ binary.BigEndian.PutUint32(buf[6:], uint32(x.exp))
+ x.mant[len(x.mant)-n:].bytes(buf[10:]) // cut off unused trailing words
+ }
+
+ return buf, nil
+}
+
+// GobDecode implements the gob.GobDecoder interface.
+// The result is rounded per the precision and rounding mode of
+// z unless z's precision is 0, in which case z is set exactly
+// to the decoded value.
+func (z *Float) GobDecode(buf []byte) error {
+ if len(buf) == 0 {
+ // Other side sent a nil or default value.
+ *z = Float{}
+ return nil
+ }
+
+ if buf[0] != floatGobVersion {
+ return fmt.Errorf("Float.GobDecode: encoding version %d not supported", buf[0])
+ }
+
+ oldPrec := z.prec
+ oldMode := z.mode
+
+ b := buf[1]
+ z.mode = RoundingMode((b >> 5) & 7)
+ z.acc = Accuracy((b>>3)&3) - 1
+ z.form = form((b >> 1) & 3)
+ z.neg = b&1 != 0
+ z.prec = binary.BigEndian.Uint32(buf[2:])
+
+ if z.form == finite {
+ z.exp = int32(binary.BigEndian.Uint32(buf[6:]))
+ z.mant = z.mant.setBytes(buf[10:])
+ }
+
+ if oldPrec != 0 {
+ z.mode = oldMode
+ z.SetPrec(uint(oldPrec))
+ }
+
+ return nil
+}
+
+// MarshalText implements the encoding.TextMarshaler interface.
+// Only the Float value is marshaled (in full precision), other
+// attributes such as precision or accuracy are ignored.
+func (x *Float) MarshalText() (text []byte, err error) {
+ if x == nil {
+ return []byte("<nil>"), nil
+ }
+ var buf []byte
+ return x.Append(buf, 'g', -1), nil
+}
+
+// UnmarshalText implements the encoding.TextUnmarshaler interface.
+// The result is rounded per the precision and rounding mode of z.
+// If z's precision is 0, it is changed to 64 before rounding takes
+// effect.
+func (z *Float) UnmarshalText(text []byte) error {
+ // TODO(gri): get rid of the []byte/string conversion
+ _, _, err := z.Parse(string(text), 0)
+ if err != nil {
+ err = fmt.Errorf("math/big: cannot unmarshal %q into a *big.Float (%v)", text, err)
+ }
+ return err
+}
diff --git a/src/math/big/floatmarsh_test.go b/src/math/big/floatmarsh_test.go
new file mode 100644
index 0000000..c056d78
--- /dev/null
+++ b/src/math/big/floatmarsh_test.go
@@ -0,0 +1,139 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "encoding/gob"
+ "encoding/json"
+ "io"
+ "testing"
+)
+
+var floatVals = []string{
+ "0",
+ "1",
+ "0.1",
+ "2.71828",
+ "1234567890",
+ "3.14e1234",
+ "3.14e-1234",
+ "0.738957395793475734757349579759957975985497e100",
+ "0.73895739579347546656564656573475734957975995797598589749859834759476745986795497e100",
+ "inf",
+ "Inf",
+}
+
+func TestFloatGobEncoding(t *testing.T) {
+ var medium bytes.Buffer
+ enc := gob.NewEncoder(&medium)
+ dec := gob.NewDecoder(&medium)
+ for _, test := range floatVals {
+ for _, sign := range []string{"", "+", "-"} {
+ for _, prec := range []uint{0, 1, 2, 10, 53, 64, 100, 1000} {
+ for _, mode := range []RoundingMode{ToNearestEven, ToNearestAway, ToZero, AwayFromZero, ToNegativeInf, ToPositiveInf} {
+ medium.Reset() // empty buffer for each test case (in case of failures)
+ x := sign + test
+
+ var tx Float
+ _, _, err := tx.SetPrec(prec).SetMode(mode).Parse(x, 0)
+ if err != nil {
+ t.Errorf("parsing of %s (%dbits, %v) failed (invalid test case): %v", x, prec, mode, err)
+ continue
+ }
+
+ // If tx was set to prec == 0, tx.Parse(x, 0) assumes precision 64. Correct it.
+ if prec == 0 {
+ tx.SetPrec(0)
+ }
+
+ if err := enc.Encode(&tx); err != nil {
+ t.Errorf("encoding of %v (%dbits, %v) failed: %v", &tx, prec, mode, err)
+ continue
+ }
+
+ var rx Float
+ if err := dec.Decode(&rx); err != nil {
+ t.Errorf("decoding of %v (%dbits, %v) failed: %v", &tx, prec, mode, err)
+ continue
+ }
+
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("transmission of %s failed: got %s want %s", x, rx.String(), tx.String())
+ continue
+ }
+
+ if rx.Prec() != prec {
+ t.Errorf("transmission of %s's prec failed: got %d want %d", x, rx.Prec(), prec)
+ }
+
+ if rx.Mode() != mode {
+ t.Errorf("transmission of %s's mode failed: got %s want %s", x, rx.Mode(), mode)
+ }
+
+ if rx.Acc() != tx.Acc() {
+ t.Errorf("transmission of %s's accuracy failed: got %s want %s", x, rx.Acc(), tx.Acc())
+ }
+ }
+ }
+ }
+ }
+}
+
+func TestFloatCorruptGob(t *testing.T) {
+ var buf bytes.Buffer
+ tx := NewFloat(4 / 3).SetPrec(1000).SetMode(ToPositiveInf)
+ if err := gob.NewEncoder(&buf).Encode(tx); err != nil {
+ t.Fatal(err)
+ }
+ b := buf.Bytes()
+
+ var rx Float
+ if err := gob.NewDecoder(bytes.NewReader(b)).Decode(&rx); err != nil {
+ t.Fatal(err)
+ }
+
+ if err := gob.NewDecoder(bytes.NewReader(b[:10])).Decode(&rx); err != io.ErrUnexpectedEOF {
+ t.Errorf("got %v want EOF", err)
+ }
+
+ b[1] = 0
+ if err := gob.NewDecoder(bytes.NewReader(b)).Decode(&rx); err == nil {
+ t.Fatal("got nil want version error")
+ }
+}
+
+func TestFloatJSONEncoding(t *testing.T) {
+ for _, test := range floatVals {
+ for _, sign := range []string{"", "+", "-"} {
+ for _, prec := range []uint{0, 1, 2, 10, 53, 64, 100, 1000} {
+ if prec > 53 && testing.Short() {
+ continue
+ }
+ x := sign + test
+ var tx Float
+ _, _, err := tx.SetPrec(prec).Parse(x, 0)
+ if err != nil {
+ t.Errorf("parsing of %s (prec = %d) failed (invalid test case): %v", x, prec, err)
+ continue
+ }
+ b, err := json.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %v (prec = %d) failed: %v", &tx, prec, err)
+ continue
+ }
+ var rx Float
+ rx.SetPrec(prec)
+ if err := json.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %v (prec = %d) failed: %v", &tx, prec, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("JSON encoding of %v (prec = %d) failed: got %v want %v", &tx, prec, &rx, &tx)
+ }
+ }
+ }
+ }
+}
diff --git a/src/math/big/ftoa.go b/src/math/big/ftoa.go
new file mode 100644
index 0000000..5506e6e
--- /dev/null
+++ b/src/math/big/ftoa.go
@@ -0,0 +1,536 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements Float-to-string conversion functions.
+// It is closely following the corresponding implementation
+// in strconv/ftoa.go, but modified and simplified for Float.
+
+package big
+
+import (
+ "bytes"
+ "fmt"
+ "strconv"
+)
+
+// Text converts the floating-point number x to a string according
+// to the given format and precision prec. The format is one of:
+//
+// 'e' -d.dddde±dd, decimal exponent, at least two (possibly 0) exponent digits
+// 'E' -d.ddddE±dd, decimal exponent, at least two (possibly 0) exponent digits
+// 'f' -ddddd.dddd, no exponent
+// 'g' like 'e' for large exponents, like 'f' otherwise
+// 'G' like 'E' for large exponents, like 'f' otherwise
+// 'x' -0xd.dddddp±dd, hexadecimal mantissa, decimal power of two exponent
+// 'p' -0x.dddp±dd, hexadecimal mantissa, decimal power of two exponent (non-standard)
+// 'b' -ddddddp±dd, decimal mantissa, decimal power of two exponent (non-standard)
+//
+// For the power-of-two exponent formats, the mantissa is printed in normalized form:
+//
+// 'x' hexadecimal mantissa in [1, 2), or 0
+// 'p' hexadecimal mantissa in [½, 1), or 0
+// 'b' decimal integer mantissa using x.Prec() bits, or 0
+//
+// Note that the 'x' form is the one used by most other languages and libraries.
+//
+// If format is a different character, Text returns a "%" followed by the
+// unrecognized format character.
+//
+// The precision prec controls the number of digits (excluding the exponent)
+// printed by the 'e', 'E', 'f', 'g', 'G', and 'x' formats.
+// For 'e', 'E', 'f', and 'x', it is the number of digits after the decimal point.
+// For 'g' and 'G' it is the total number of digits. A negative precision selects
+// the smallest number of decimal digits necessary to identify the value x uniquely
+// using x.Prec() mantissa bits.
+// The prec value is ignored for the 'b' and 'p' formats.
+func (x *Float) Text(format byte, prec int) string {
+ cap := 10 // TODO(gri) determine a good/better value here
+ if prec > 0 {
+ cap += prec
+ }
+ return string(x.Append(make([]byte, 0, cap), format, prec))
+}
+
+// String formats x like x.Text('g', 10).
+// (String must be called explicitly, Float.Format does not support %s verb.)
+func (x *Float) String() string {
+ return x.Text('g', 10)
+}
+
+// Append appends to buf the string form of the floating-point number x,
+// as generated by x.Text, and returns the extended buffer.
+func (x *Float) Append(buf []byte, fmt byte, prec int) []byte {
+ // sign
+ if x.neg {
+ buf = append(buf, '-')
+ }
+
+ // Inf
+ if x.form == inf {
+ if !x.neg {
+ buf = append(buf, '+')
+ }
+ return append(buf, "Inf"...)
+ }
+
+ // pick off easy formats
+ switch fmt {
+ case 'b':
+ return x.fmtB(buf)
+ case 'p':
+ return x.fmtP(buf)
+ case 'x':
+ return x.fmtX(buf, prec)
+ }
+
+ // Algorithm:
+ // 1) convert Float to multiprecision decimal
+ // 2) round to desired precision
+ // 3) read digits out and format
+
+ // 1) convert Float to multiprecision decimal
+ var d decimal // == 0.0
+ if x.form == finite {
+ // x != 0
+ d.init(x.mant, int(x.exp)-x.mant.bitLen())
+ }
+
+ // 2) round to desired precision
+ shortest := false
+ if prec < 0 {
+ shortest = true
+ roundShortest(&d, x)
+ // Precision for shortest representation mode.
+ switch fmt {
+ case 'e', 'E':
+ prec = len(d.mant) - 1
+ case 'f':
+ prec = max(len(d.mant)-d.exp, 0)
+ case 'g', 'G':
+ prec = len(d.mant)
+ }
+ } else {
+ // round appropriately
+ switch fmt {
+ case 'e', 'E':
+ // one digit before and number of digits after decimal point
+ d.round(1 + prec)
+ case 'f':
+ // number of digits before and after decimal point
+ d.round(d.exp + prec)
+ case 'g', 'G':
+ if prec == 0 {
+ prec = 1
+ }
+ d.round(prec)
+ }
+ }
+
+ // 3) read digits out and format
+ switch fmt {
+ case 'e', 'E':
+ return fmtE(buf, fmt, prec, d)
+ case 'f':
+ return fmtF(buf, prec, d)
+ case 'g', 'G':
+ // trim trailing fractional zeros in %e format
+ eprec := prec
+ if eprec > len(d.mant) && len(d.mant) >= d.exp {
+ eprec = len(d.mant)
+ }
+ // %e is used if the exponent from the conversion
+ // is less than -4 or greater than or equal to the precision.
+ // If precision was the shortest possible, use eprec = 6 for
+ // this decision.
+ if shortest {
+ eprec = 6
+ }
+ exp := d.exp - 1
+ if exp < -4 || exp >= eprec {
+ if prec > len(d.mant) {
+ prec = len(d.mant)
+ }
+ return fmtE(buf, fmt+'e'-'g', prec-1, d)
+ }
+ if prec > d.exp {
+ prec = len(d.mant)
+ }
+ return fmtF(buf, max(prec-d.exp, 0), d)
+ }
+
+ // unknown format
+ if x.neg {
+ buf = buf[:len(buf)-1] // sign was added prematurely - remove it again
+ }
+ return append(buf, '%', fmt)
+}
+
+func roundShortest(d *decimal, x *Float) {
+ // if the mantissa is zero, the number is zero - stop now
+ if len(d.mant) == 0 {
+ return
+ }
+
+ // Approach: All numbers in the interval [x - 1/2ulp, x + 1/2ulp]
+ // (possibly exclusive) round to x for the given precision of x.
+ // Compute the lower and upper bound in decimal form and find the
+ // shortest decimal number d such that lower <= d <= upper.
+
+ // TODO(gri) strconv/ftoa.do describes a shortcut in some cases.
+ // See if we can use it (in adjusted form) here as well.
+
+ // 1) Compute normalized mantissa mant and exponent exp for x such
+ // that the lsb of mant corresponds to 1/2 ulp for the precision of
+ // x (i.e., for mant we want x.prec + 1 bits).
+ mant := nat(nil).set(x.mant)
+ exp := int(x.exp) - mant.bitLen()
+ s := mant.bitLen() - int(x.prec+1)
+ switch {
+ case s < 0:
+ mant = mant.shl(mant, uint(-s))
+ case s > 0:
+ mant = mant.shr(mant, uint(+s))
+ }
+ exp += s
+ // x = mant * 2**exp with lsb(mant) == 1/2 ulp of x.prec
+
+ // 2) Compute lower bound by subtracting 1/2 ulp.
+ var lower decimal
+ var tmp nat
+ lower.init(tmp.sub(mant, natOne), exp)
+
+ // 3) Compute upper bound by adding 1/2 ulp.
+ var upper decimal
+ upper.init(tmp.add(mant, natOne), exp)
+
+ // The upper and lower bounds are possible outputs only if
+ // the original mantissa is even, so that ToNearestEven rounding
+ // would round to the original mantissa and not the neighbors.
+ inclusive := mant[0]&2 == 0 // test bit 1 since original mantissa was shifted by 1
+
+ // Now we can figure out the minimum number of digits required.
+ // Walk along until d has distinguished itself from upper and lower.
+ for i, m := range d.mant {
+ l := lower.at(i)
+ u := upper.at(i)
+
+ // Okay to round down (truncate) if lower has a different digit
+ // or if lower is inclusive and is exactly the result of rounding
+ // down (i.e., and we have reached the final digit of lower).
+ okdown := l != m || inclusive && i+1 == len(lower.mant)
+
+ // Okay to round up if upper has a different digit and either upper
+ // is inclusive or upper is bigger than the result of rounding up.
+ okup := m != u && (inclusive || m+1 < u || i+1 < len(upper.mant))
+
+ // If it's okay to do either, then round to the nearest one.
+ // If it's okay to do only one, do it.
+ switch {
+ case okdown && okup:
+ d.round(i + 1)
+ return
+ case okdown:
+ d.roundDown(i + 1)
+ return
+ case okup:
+ d.roundUp(i + 1)
+ return
+ }
+ }
+}
+
+// %e: d.ddddde±dd
+func fmtE(buf []byte, fmt byte, prec int, d decimal) []byte {
+ // first digit
+ ch := byte('0')
+ if len(d.mant) > 0 {
+ ch = d.mant[0]
+ }
+ buf = append(buf, ch)
+
+ // .moredigits
+ if prec > 0 {
+ buf = append(buf, '.')
+ i := 1
+ m := min(len(d.mant), prec+1)
+ if i < m {
+ buf = append(buf, d.mant[i:m]...)
+ i = m
+ }
+ for ; i <= prec; i++ {
+ buf = append(buf, '0')
+ }
+ }
+
+ // e±
+ buf = append(buf, fmt)
+ var exp int64
+ if len(d.mant) > 0 {
+ exp = int64(d.exp) - 1 // -1 because first digit was printed before '.'
+ }
+ if exp < 0 {
+ ch = '-'
+ exp = -exp
+ } else {
+ ch = '+'
+ }
+ buf = append(buf, ch)
+
+ // dd...d
+ if exp < 10 {
+ buf = append(buf, '0') // at least 2 exponent digits
+ }
+ return strconv.AppendInt(buf, exp, 10)
+}
+
+// %f: ddddddd.ddddd
+func fmtF(buf []byte, prec int, d decimal) []byte {
+ // integer, padded with zeros as needed
+ if d.exp > 0 {
+ m := min(len(d.mant), d.exp)
+ buf = append(buf, d.mant[:m]...)
+ for ; m < d.exp; m++ {
+ buf = append(buf, '0')
+ }
+ } else {
+ buf = append(buf, '0')
+ }
+
+ // fraction
+ if prec > 0 {
+ buf = append(buf, '.')
+ for i := 0; i < prec; i++ {
+ buf = append(buf, d.at(d.exp+i))
+ }
+ }
+
+ return buf
+}
+
+// fmtB appends the string of x in the format mantissa "p" exponent
+// with a decimal mantissa and a binary exponent, or 0" if x is zero,
+// and returns the extended buffer.
+// The mantissa is normalized such that is uses x.Prec() bits in binary
+// representation.
+// The sign of x is ignored, and x must not be an Inf.
+// (The caller handles Inf before invoking fmtB.)
+func (x *Float) fmtB(buf []byte) []byte {
+ if x.form == zero {
+ return append(buf, '0')
+ }
+
+ if debugFloat && x.form != finite {
+ panic("non-finite float")
+ }
+ // x != 0
+
+ // adjust mantissa to use exactly x.prec bits
+ m := x.mant
+ switch w := uint32(len(x.mant)) * _W; {
+ case w < x.prec:
+ m = nat(nil).shl(m, uint(x.prec-w))
+ case w > x.prec:
+ m = nat(nil).shr(m, uint(w-x.prec))
+ }
+
+ buf = append(buf, m.utoa(10)...)
+ buf = append(buf, 'p')
+ e := int64(x.exp) - int64(x.prec)
+ if e >= 0 {
+ buf = append(buf, '+')
+ }
+ return strconv.AppendInt(buf, e, 10)
+}
+
+// fmtX appends the string of x in the format "0x1." mantissa "p" exponent
+// with a hexadecimal mantissa and a binary exponent, or "0x0p0" if x is zero,
+// and returns the extended buffer.
+// A non-zero mantissa is normalized such that 1.0 <= mantissa < 2.0.
+// The sign of x is ignored, and x must not be an Inf.
+// (The caller handles Inf before invoking fmtX.)
+func (x *Float) fmtX(buf []byte, prec int) []byte {
+ if x.form == zero {
+ buf = append(buf, "0x0"...)
+ if prec > 0 {
+ buf = append(buf, '.')
+ for i := 0; i < prec; i++ {
+ buf = append(buf, '0')
+ }
+ }
+ buf = append(buf, "p+00"...)
+ return buf
+ }
+
+ if debugFloat && x.form != finite {
+ panic("non-finite float")
+ }
+
+ // round mantissa to n bits
+ var n uint
+ if prec < 0 {
+ n = 1 + (x.MinPrec()-1+3)/4*4 // round MinPrec up to 1 mod 4
+ } else {
+ n = 1 + 4*uint(prec)
+ }
+ // n%4 == 1
+ x = new(Float).SetPrec(n).SetMode(x.mode).Set(x)
+
+ // adjust mantissa to use exactly n bits
+ m := x.mant
+ switch w := uint(len(x.mant)) * _W; {
+ case w < n:
+ m = nat(nil).shl(m, n-w)
+ case w > n:
+ m = nat(nil).shr(m, w-n)
+ }
+ exp64 := int64(x.exp) - 1 // avoid wrap-around
+
+ hm := m.utoa(16)
+ if debugFloat && hm[0] != '1' {
+ panic("incorrect mantissa: " + string(hm))
+ }
+ buf = append(buf, "0x1"...)
+ if len(hm) > 1 {
+ buf = append(buf, '.')
+ buf = append(buf, hm[1:]...)
+ }
+
+ buf = append(buf, 'p')
+ if exp64 >= 0 {
+ buf = append(buf, '+')
+ } else {
+ exp64 = -exp64
+ buf = append(buf, '-')
+ }
+ // Force at least two exponent digits, to match fmt.
+ if exp64 < 10 {
+ buf = append(buf, '0')
+ }
+ return strconv.AppendInt(buf, exp64, 10)
+}
+
+// fmtP appends the string of x in the format "0x." mantissa "p" exponent
+// with a hexadecimal mantissa and a binary exponent, or "0" if x is zero,
+// and returns the extended buffer.
+// The mantissa is normalized such that 0.5 <= 0.mantissa < 1.0.
+// The sign of x is ignored, and x must not be an Inf.
+// (The caller handles Inf before invoking fmtP.)
+func (x *Float) fmtP(buf []byte) []byte {
+ if x.form == zero {
+ return append(buf, '0')
+ }
+
+ if debugFloat && x.form != finite {
+ panic("non-finite float")
+ }
+ // x != 0
+
+ // remove trailing 0 words early
+ // (no need to convert to hex 0's and trim later)
+ m := x.mant
+ i := 0
+ for i < len(m) && m[i] == 0 {
+ i++
+ }
+ m = m[i:]
+
+ buf = append(buf, "0x."...)
+ buf = append(buf, bytes.TrimRight(m.utoa(16), "0")...)
+ buf = append(buf, 'p')
+ if x.exp >= 0 {
+ buf = append(buf, '+')
+ }
+ return strconv.AppendInt(buf, int64(x.exp), 10)
+}
+
+func min(x, y int) int {
+ if x < y {
+ return x
+ }
+ return y
+}
+
+var _ fmt.Formatter = &floatZero // *Float must implement fmt.Formatter
+
+// Format implements fmt.Formatter. It accepts all the regular
+// formats for floating-point numbers ('b', 'e', 'E', 'f', 'F',
+// 'g', 'G', 'x') as well as 'p' and 'v'. See (*Float).Text for the
+// interpretation of 'p'. The 'v' format is handled like 'g'.
+// Format also supports specification of the minimum precision
+// in digits, the output field width, as well as the format flags
+// '+' and ' ' for sign control, '0' for space or zero padding,
+// and '-' for left or right justification. See the fmt package
+// for details.
+func (x *Float) Format(s fmt.State, format rune) {
+ prec, hasPrec := s.Precision()
+ if !hasPrec {
+ prec = 6 // default precision for 'e', 'f'
+ }
+
+ switch format {
+ case 'e', 'E', 'f', 'b', 'p', 'x':
+ // nothing to do
+ case 'F':
+ // (*Float).Text doesn't support 'F'; handle like 'f'
+ format = 'f'
+ case 'v':
+ // handle like 'g'
+ format = 'g'
+ fallthrough
+ case 'g', 'G':
+ if !hasPrec {
+ prec = -1 // default precision for 'g', 'G'
+ }
+ default:
+ fmt.Fprintf(s, "%%!%c(*big.Float=%s)", format, x.String())
+ return
+ }
+ var buf []byte
+ buf = x.Append(buf, byte(format), prec)
+ if len(buf) == 0 {
+ buf = []byte("?") // should never happen, but don't crash
+ }
+ // len(buf) > 0
+
+ var sign string
+ switch {
+ case buf[0] == '-':
+ sign = "-"
+ buf = buf[1:]
+ case buf[0] == '+':
+ // +Inf
+ sign = "+"
+ if s.Flag(' ') {
+ sign = " "
+ }
+ buf = buf[1:]
+ case s.Flag('+'):
+ sign = "+"
+ case s.Flag(' '):
+ sign = " "
+ }
+
+ var padding int
+ if width, hasWidth := s.Width(); hasWidth && width > len(sign)+len(buf) {
+ padding = width - len(sign) - len(buf)
+ }
+
+ switch {
+ case s.Flag('0') && !x.IsInf():
+ // 0-padding on left
+ writeMultiple(s, sign, 1)
+ writeMultiple(s, "0", padding)
+ s.Write(buf)
+ case s.Flag('-'):
+ // padding on right
+ writeMultiple(s, sign, 1)
+ s.Write(buf)
+ writeMultiple(s, " ", padding)
+ default:
+ // padding on left
+ writeMultiple(s, " ", padding)
+ writeMultiple(s, sign, 1)
+ s.Write(buf)
+ }
+}
diff --git a/src/math/big/gcd_test.go b/src/math/big/gcd_test.go
new file mode 100644
index 0000000..3cca2ec
--- /dev/null
+++ b/src/math/big/gcd_test.go
@@ -0,0 +1,64 @@
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements a GCD benchmark.
+// Usage: go test math/big -test.bench GCD
+
+package big
+
+import (
+ "math/rand"
+ "testing"
+)
+
+// randInt returns a pseudo-random Int in the range [1<<(size-1), (1<<size) - 1]
+func randInt(r *rand.Rand, size uint) *Int {
+ n := new(Int).Lsh(intOne, size-1)
+ x := new(Int).Rand(r, n)
+ return x.Add(x, n) // make sure result > 1<<(size-1)
+}
+
+func runGCD(b *testing.B, aSize, bSize uint) {
+ if isRaceBuilder && (aSize > 1000 || bSize > 1000) {
+ b.Skip("skipping on race builder")
+ }
+ b.Run("WithoutXY", func(b *testing.B) {
+ runGCDExt(b, aSize, bSize, false)
+ })
+ b.Run("WithXY", func(b *testing.B) {
+ runGCDExt(b, aSize, bSize, true)
+ })
+}
+
+func runGCDExt(b *testing.B, aSize, bSize uint, calcXY bool) {
+ b.StopTimer()
+ var r = rand.New(rand.NewSource(1234))
+ aa := randInt(r, aSize)
+ bb := randInt(r, bSize)
+ var x, y *Int
+ if calcXY {
+ x = new(Int)
+ y = new(Int)
+ }
+ b.StartTimer()
+ for i := 0; i < b.N; i++ {
+ new(Int).GCD(x, y, aa, bb)
+ }
+}
+
+func BenchmarkGCD10x10(b *testing.B) { runGCD(b, 10, 10) }
+func BenchmarkGCD10x100(b *testing.B) { runGCD(b, 10, 100) }
+func BenchmarkGCD10x1000(b *testing.B) { runGCD(b, 10, 1000) }
+func BenchmarkGCD10x10000(b *testing.B) { runGCD(b, 10, 10000) }
+func BenchmarkGCD10x100000(b *testing.B) { runGCD(b, 10, 100000) }
+func BenchmarkGCD100x100(b *testing.B) { runGCD(b, 100, 100) }
+func BenchmarkGCD100x1000(b *testing.B) { runGCD(b, 100, 1000) }
+func BenchmarkGCD100x10000(b *testing.B) { runGCD(b, 100, 10000) }
+func BenchmarkGCD100x100000(b *testing.B) { runGCD(b, 100, 100000) }
+func BenchmarkGCD1000x1000(b *testing.B) { runGCD(b, 1000, 1000) }
+func BenchmarkGCD1000x10000(b *testing.B) { runGCD(b, 1000, 10000) }
+func BenchmarkGCD1000x100000(b *testing.B) { runGCD(b, 1000, 100000) }
+func BenchmarkGCD10000x10000(b *testing.B) { runGCD(b, 10000, 10000) }
+func BenchmarkGCD10000x100000(b *testing.B) { runGCD(b, 10000, 100000) }
+func BenchmarkGCD100000x100000(b *testing.B) { runGCD(b, 100000, 100000) }
diff --git a/src/math/big/hilbert_test.go b/src/math/big/hilbert_test.go
new file mode 100644
index 0000000..1a84341
--- /dev/null
+++ b/src/math/big/hilbert_test.go
@@ -0,0 +1,160 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// A little test program and benchmark for rational arithmetics.
+// Computes a Hilbert matrix, its inverse, multiplies them
+// and verifies that the product is the identity matrix.
+
+package big
+
+import (
+ "fmt"
+ "testing"
+)
+
+type matrix struct {
+ n, m int
+ a []*Rat
+}
+
+func (a *matrix) at(i, j int) *Rat {
+ if !(0 <= i && i < a.n && 0 <= j && j < a.m) {
+ panic("index out of range")
+ }
+ return a.a[i*a.m+j]
+}
+
+func (a *matrix) set(i, j int, x *Rat) {
+ if !(0 <= i && i < a.n && 0 <= j && j < a.m) {
+ panic("index out of range")
+ }
+ a.a[i*a.m+j] = x
+}
+
+func newMatrix(n, m int) *matrix {
+ if !(0 <= n && 0 <= m) {
+ panic("illegal matrix")
+ }
+ a := new(matrix)
+ a.n = n
+ a.m = m
+ a.a = make([]*Rat, n*m)
+ return a
+}
+
+func newUnit(n int) *matrix {
+ a := newMatrix(n, n)
+ for i := 0; i < n; i++ {
+ for j := 0; j < n; j++ {
+ x := NewRat(0, 1)
+ if i == j {
+ x.SetInt64(1)
+ }
+ a.set(i, j, x)
+ }
+ }
+ return a
+}
+
+func newHilbert(n int) *matrix {
+ a := newMatrix(n, n)
+ for i := 0; i < n; i++ {
+ for j := 0; j < n; j++ {
+ a.set(i, j, NewRat(1, int64(i+j+1)))
+ }
+ }
+ return a
+}
+
+func newInverseHilbert(n int) *matrix {
+ a := newMatrix(n, n)
+ for i := 0; i < n; i++ {
+ for j := 0; j < n; j++ {
+ x1 := new(Rat).SetInt64(int64(i + j + 1))
+ x2 := new(Rat).SetInt(new(Int).Binomial(int64(n+i), int64(n-j-1)))
+ x3 := new(Rat).SetInt(new(Int).Binomial(int64(n+j), int64(n-i-1)))
+ x4 := new(Rat).SetInt(new(Int).Binomial(int64(i+j), int64(i)))
+
+ x1.Mul(x1, x2)
+ x1.Mul(x1, x3)
+ x1.Mul(x1, x4)
+ x1.Mul(x1, x4)
+
+ if (i+j)&1 != 0 {
+ x1.Neg(x1)
+ }
+
+ a.set(i, j, x1)
+ }
+ }
+ return a
+}
+
+func (a *matrix) mul(b *matrix) *matrix {
+ if a.m != b.n {
+ panic("illegal matrix multiply")
+ }
+ c := newMatrix(a.n, b.m)
+ for i := 0; i < c.n; i++ {
+ for j := 0; j < c.m; j++ {
+ x := NewRat(0, 1)
+ for k := 0; k < a.m; k++ {
+ x.Add(x, new(Rat).Mul(a.at(i, k), b.at(k, j)))
+ }
+ c.set(i, j, x)
+ }
+ }
+ return c
+}
+
+func (a *matrix) eql(b *matrix) bool {
+ if a.n != b.n || a.m != b.m {
+ return false
+ }
+ for i := 0; i < a.n; i++ {
+ for j := 0; j < a.m; j++ {
+ if a.at(i, j).Cmp(b.at(i, j)) != 0 {
+ return false
+ }
+ }
+ }
+ return true
+}
+
+func (a *matrix) String() string {
+ s := ""
+ for i := 0; i < a.n; i++ {
+ for j := 0; j < a.m; j++ {
+ s += fmt.Sprintf("\t%s", a.at(i, j))
+ }
+ s += "\n"
+ }
+ return s
+}
+
+func doHilbert(t *testing.T, n int) {
+ a := newHilbert(n)
+ b := newInverseHilbert(n)
+ I := newUnit(n)
+ ab := a.mul(b)
+ if !ab.eql(I) {
+ if t == nil {
+ panic("Hilbert failed")
+ }
+ t.Errorf("a = %s\n", a)
+ t.Errorf("b = %s\n", b)
+ t.Errorf("a*b = %s\n", ab)
+ t.Errorf("I = %s\n", I)
+ }
+}
+
+func TestHilbert(t *testing.T) {
+ doHilbert(t, 10)
+}
+
+func BenchmarkHilbert(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ doHilbert(nil, 10)
+ }
+}
diff --git a/src/math/big/int.go b/src/math/big/int.go
new file mode 100644
index 0000000..65f3248
--- /dev/null
+++ b/src/math/big/int.go
@@ -0,0 +1,1218 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements signed multi-precision integers.
+
+package big
+
+import (
+ "fmt"
+ "io"
+ "math/rand"
+ "strings"
+)
+
+// An Int represents a signed multi-precision integer.
+// The zero value for an Int represents the value 0.
+//
+// Operations always take pointer arguments (*Int) rather
+// than Int values, and each unique Int value requires
+// its own unique *Int pointer. To "copy" an Int value,
+// an existing (or newly allocated) Int must be set to
+// a new value using the Int.Set method; shallow copies
+// of Ints are not supported and may lead to errors.
+type Int struct {
+ neg bool // sign
+ abs nat // absolute value of the integer
+}
+
+var intOne = &Int{false, natOne}
+
+// Sign returns:
+//
+// -1 if x < 0
+// 0 if x == 0
+// +1 if x > 0
+//
+func (x *Int) Sign() int {
+ if len(x.abs) == 0 {
+ return 0
+ }
+ if x.neg {
+ return -1
+ }
+ return 1
+}
+
+// SetInt64 sets z to x and returns z.
+func (z *Int) SetInt64(x int64) *Int {
+ neg := false
+ if x < 0 {
+ neg = true
+ x = -x
+ }
+ z.abs = z.abs.setUint64(uint64(x))
+ z.neg = neg
+ return z
+}
+
+// SetUint64 sets z to x and returns z.
+func (z *Int) SetUint64(x uint64) *Int {
+ z.abs = z.abs.setUint64(x)
+ z.neg = false
+ return z
+}
+
+// NewInt allocates and returns a new Int set to x.
+func NewInt(x int64) *Int {
+ return new(Int).SetInt64(x)
+}
+
+// Set sets z to x and returns z.
+func (z *Int) Set(x *Int) *Int {
+ if z != x {
+ z.abs = z.abs.set(x.abs)
+ z.neg = x.neg
+ }
+ return z
+}
+
+// Bits provides raw (unchecked but fast) access to x by returning its
+// absolute value as a little-endian Word slice. The result and x share
+// the same underlying array.
+// Bits is intended to support implementation of missing low-level Int
+// functionality outside this package; it should be avoided otherwise.
+func (x *Int) Bits() []Word {
+ return x.abs
+}
+
+// SetBits provides raw (unchecked but fast) access to z by setting its
+// value to abs, interpreted as a little-endian Word slice, and returning
+// z. The result and abs share the same underlying array.
+// SetBits is intended to support implementation of missing low-level Int
+// functionality outside this package; it should be avoided otherwise.
+func (z *Int) SetBits(abs []Word) *Int {
+ z.abs = nat(abs).norm()
+ z.neg = false
+ return z
+}
+
+// Abs sets z to |x| (the absolute value of x) and returns z.
+func (z *Int) Abs(x *Int) *Int {
+ z.Set(x)
+ z.neg = false
+ return z
+}
+
+// Neg sets z to -x and returns z.
+func (z *Int) Neg(x *Int) *Int {
+ z.Set(x)
+ z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
+ return z
+}
+
+// Add sets z to the sum x+y and returns z.
+func (z *Int) Add(x, y *Int) *Int {
+ neg := x.neg
+ if x.neg == y.neg {
+ // x + y == x + y
+ // (-x) + (-y) == -(x + y)
+ z.abs = z.abs.add(x.abs, y.abs)
+ } else {
+ // x + (-y) == x - y == -(y - x)
+ // (-x) + y == y - x == -(x - y)
+ if x.abs.cmp(y.abs) >= 0 {
+ z.abs = z.abs.sub(x.abs, y.abs)
+ } else {
+ neg = !neg
+ z.abs = z.abs.sub(y.abs, x.abs)
+ }
+ }
+ z.neg = len(z.abs) > 0 && neg // 0 has no sign
+ return z
+}
+
+// Sub sets z to the difference x-y and returns z.
+func (z *Int) Sub(x, y *Int) *Int {
+ neg := x.neg
+ if x.neg != y.neg {
+ // x - (-y) == x + y
+ // (-x) - y == -(x + y)
+ z.abs = z.abs.add(x.abs, y.abs)
+ } else {
+ // x - y == x - y == -(y - x)
+ // (-x) - (-y) == y - x == -(x - y)
+ if x.abs.cmp(y.abs) >= 0 {
+ z.abs = z.abs.sub(x.abs, y.abs)
+ } else {
+ neg = !neg
+ z.abs = z.abs.sub(y.abs, x.abs)
+ }
+ }
+ z.neg = len(z.abs) > 0 && neg // 0 has no sign
+ return z
+}
+
+// Mul sets z to the product x*y and returns z.
+func (z *Int) Mul(x, y *Int) *Int {
+ // x * y == x * y
+ // x * (-y) == -(x * y)
+ // (-x) * y == -(x * y)
+ // (-x) * (-y) == x * y
+ if x == y {
+ z.abs = z.abs.sqr(x.abs)
+ z.neg = false
+ return z
+ }
+ z.abs = z.abs.mul(x.abs, y.abs)
+ z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
+ return z
+}
+
+// MulRange sets z to the product of all integers
+// in the range [a, b] inclusively and returns z.
+// If a > b (empty range), the result is 1.
+func (z *Int) MulRange(a, b int64) *Int {
+ switch {
+ case a > b:
+ return z.SetInt64(1) // empty range
+ case a <= 0 && b >= 0:
+ return z.SetInt64(0) // range includes 0
+ }
+ // a <= b && (b < 0 || a > 0)
+
+ neg := false
+ if a < 0 {
+ neg = (b-a)&1 == 0
+ a, b = -b, -a
+ }
+
+ z.abs = z.abs.mulRange(uint64(a), uint64(b))
+ z.neg = neg
+ return z
+}
+
+// Binomial sets z to the binomial coefficient of (n, k) and returns z.
+func (z *Int) Binomial(n, k int64) *Int {
+ // reduce the number of multiplications by reducing k
+ if n/2 < k && k <= n {
+ k = n - k // Binomial(n, k) == Binomial(n, n-k)
+ }
+ var a, b Int
+ a.MulRange(n-k+1, n)
+ b.MulRange(1, k)
+ return z.Quo(&a, &b)
+}
+
+// Quo sets z to the quotient x/y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Quo implements truncated division (like Go); see QuoRem for more details.
+func (z *Int) Quo(x, y *Int) *Int {
+ z.abs, _ = z.abs.div(nil, x.abs, y.abs)
+ z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
+ return z
+}
+
+// Rem sets z to the remainder x%y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Rem implements truncated modulus (like Go); see QuoRem for more details.
+func (z *Int) Rem(x, y *Int) *Int {
+ _, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
+ z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
+ return z
+}
+
+// QuoRem sets z to the quotient x/y and r to the remainder x%y
+// and returns the pair (z, r) for y != 0.
+// If y == 0, a division-by-zero run-time panic occurs.
+//
+// QuoRem implements T-division and modulus (like Go):
+//
+// q = x/y with the result truncated to zero
+// r = x - y*q
+//
+// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
+// See DivMod for Euclidean division and modulus (unlike Go).
+//
+func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
+ z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
+ z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
+ return z, r
+}
+
+// Div sets z to the quotient x/y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Div implements Euclidean division (unlike Go); see DivMod for more details.
+func (z *Int) Div(x, y *Int) *Int {
+ y_neg := y.neg // z may be an alias for y
+ var r Int
+ z.QuoRem(x, y, &r)
+ if r.neg {
+ if y_neg {
+ z.Add(z, intOne)
+ } else {
+ z.Sub(z, intOne)
+ }
+ }
+ return z
+}
+
+// Mod sets z to the modulus x%y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
+func (z *Int) Mod(x, y *Int) *Int {
+ y0 := y // save y
+ if z == y || alias(z.abs, y.abs) {
+ y0 = new(Int).Set(y)
+ }
+ var q Int
+ q.QuoRem(x, y, z)
+ if z.neg {
+ if y0.neg {
+ z.Sub(z, y0)
+ } else {
+ z.Add(z, y0)
+ }
+ }
+ return z
+}
+
+// DivMod sets z to the quotient x div y and m to the modulus x mod y
+// and returns the pair (z, m) for y != 0.
+// If y == 0, a division-by-zero run-time panic occurs.
+//
+// DivMod implements Euclidean division and modulus (unlike Go):
+//
+// q = x div y such that
+// m = x - y*q with 0 <= m < |y|
+//
+// (See Raymond T. Boute, ``The Euclidean definition of the functions
+// div and mod''. ACM Transactions on Programming Languages and
+// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
+// ACM press.)
+// See QuoRem for T-division and modulus (like Go).
+//
+func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
+ y0 := y // save y
+ if z == y || alias(z.abs, y.abs) {
+ y0 = new(Int).Set(y)
+ }
+ z.QuoRem(x, y, m)
+ if m.neg {
+ if y0.neg {
+ z.Add(z, intOne)
+ m.Sub(m, y0)
+ } else {
+ z.Sub(z, intOne)
+ m.Add(m, y0)
+ }
+ }
+ return z, m
+}
+
+// Cmp compares x and y and returns:
+//
+// -1 if x < y
+// 0 if x == y
+// +1 if x > y
+//
+func (x *Int) Cmp(y *Int) (r int) {
+ // x cmp y == x cmp y
+ // x cmp (-y) == x
+ // (-x) cmp y == y
+ // (-x) cmp (-y) == -(x cmp y)
+ switch {
+ case x == y:
+ // nothing to do
+ case x.neg == y.neg:
+ r = x.abs.cmp(y.abs)
+ if x.neg {
+ r = -r
+ }
+ case x.neg:
+ r = -1
+ default:
+ r = 1
+ }
+ return
+}
+
+// CmpAbs compares the absolute values of x and y and returns:
+//
+// -1 if |x| < |y|
+// 0 if |x| == |y|
+// +1 if |x| > |y|
+//
+func (x *Int) CmpAbs(y *Int) int {
+ return x.abs.cmp(y.abs)
+}
+
+// low32 returns the least significant 32 bits of x.
+func low32(x nat) uint32 {
+ if len(x) == 0 {
+ return 0
+ }
+ return uint32(x[0])
+}
+
+// low64 returns the least significant 64 bits of x.
+func low64(x nat) uint64 {
+ if len(x) == 0 {
+ return 0
+ }
+ v := uint64(x[0])
+ if _W == 32 && len(x) > 1 {
+ return uint64(x[1])<<32 | v
+ }
+ return v
+}
+
+// Int64 returns the int64 representation of x.
+// If x cannot be represented in an int64, the result is undefined.
+func (x *Int) Int64() int64 {
+ v := int64(low64(x.abs))
+ if x.neg {
+ v = -v
+ }
+ return v
+}
+
+// Uint64 returns the uint64 representation of x.
+// If x cannot be represented in a uint64, the result is undefined.
+func (x *Int) Uint64() uint64 {
+ return low64(x.abs)
+}
+
+// IsInt64 reports whether x can be represented as an int64.
+func (x *Int) IsInt64() bool {
+ if len(x.abs) <= 64/_W {
+ w := int64(low64(x.abs))
+ return w >= 0 || x.neg && w == -w
+ }
+ return false
+}
+
+// IsUint64 reports whether x can be represented as a uint64.
+func (x *Int) IsUint64() bool {
+ return !x.neg && len(x.abs) <= 64/_W
+}
+
+// SetString sets z to the value of s, interpreted in the given base,
+// and returns z and a boolean indicating success. The entire string
+// (not just a prefix) must be valid for success. If SetString fails,
+// the value of z is undefined but the returned value is nil.
+//
+// The base argument must be 0 or a value between 2 and MaxBase.
+// For base 0, the number prefix determines the actual base: A prefix of
+// ``0b'' or ``0B'' selects base 2, ``0'', ``0o'' or ``0O'' selects base 8,
+// and ``0x'' or ``0X'' selects base 16. Otherwise, the selected base is 10
+// and no prefix is accepted.
+//
+// For bases <= 36, lower and upper case letters are considered the same:
+// The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
+// For bases > 36, the upper case letters 'A' to 'Z' represent the digit
+// values 36 to 61.
+//
+// For base 0, an underscore character ``_'' may appear between a base
+// prefix and an adjacent digit, and between successive digits; such
+// underscores do not change the value of the number.
+// Incorrect placement of underscores is reported as an error if there
+// are no other errors. If base != 0, underscores are not recognized
+// and act like any other character that is not a valid digit.
+//
+func (z *Int) SetString(s string, base int) (*Int, bool) {
+ return z.setFromScanner(strings.NewReader(s), base)
+}
+
+// setFromScanner implements SetString given an io.BytesScanner.
+// For documentation see comments of SetString.
+func (z *Int) setFromScanner(r io.ByteScanner, base int) (*Int, bool) {
+ if _, _, err := z.scan(r, base); err != nil {
+ return nil, false
+ }
+ // entire content must have been consumed
+ if _, err := r.ReadByte(); err != io.EOF {
+ return nil, false
+ }
+ return z, true // err == io.EOF => scan consumed all content of r
+}
+
+// SetBytes interprets buf as the bytes of a big-endian unsigned
+// integer, sets z to that value, and returns z.
+func (z *Int) SetBytes(buf []byte) *Int {
+ z.abs = z.abs.setBytes(buf)
+ z.neg = false
+ return z
+}
+
+// Bytes returns the absolute value of x as a big-endian byte slice.
+//
+// To use a fixed length slice, or a preallocated one, use FillBytes.
+func (x *Int) Bytes() []byte {
+ buf := make([]byte, len(x.abs)*_S)
+ return buf[x.abs.bytes(buf):]
+}
+
+// FillBytes sets buf to the absolute value of x, storing it as a zero-extended
+// big-endian byte slice, and returns buf.
+//
+// If the absolute value of x doesn't fit in buf, FillBytes will panic.
+func (x *Int) FillBytes(buf []byte) []byte {
+ // Clear whole buffer. (This gets optimized into a memclr.)
+ for i := range buf {
+ buf[i] = 0
+ }
+ x.abs.bytes(buf)
+ return buf
+}
+
+// BitLen returns the length of the absolute value of x in bits.
+// The bit length of 0 is 0.
+func (x *Int) BitLen() int {
+ return x.abs.bitLen()
+}
+
+// TrailingZeroBits returns the number of consecutive least significant zero
+// bits of |x|.
+func (x *Int) TrailingZeroBits() uint {
+ return x.abs.trailingZeroBits()
+}
+
+// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
+// If m == nil or m == 0, z = x**y unless y <= 0 then z = 1. If m != 0, y < 0,
+// and x and m are not relatively prime, z is unchanged and nil is returned.
+//
+// Modular exponentiation of inputs of a particular size is not a
+// cryptographically constant-time operation.
+func (z *Int) Exp(x, y, m *Int) *Int {
+ // See Knuth, volume 2, section 4.6.3.
+ xWords := x.abs
+ if y.neg {
+ if m == nil || len(m.abs) == 0 {
+ return z.SetInt64(1)
+ }
+ // for y < 0: x**y mod m == (x**(-1))**|y| mod m
+ inverse := new(Int).ModInverse(x, m)
+ if inverse == nil {
+ return nil
+ }
+ xWords = inverse.abs
+ }
+ yWords := y.abs
+
+ var mWords nat
+ if m != nil {
+ mWords = m.abs // m.abs may be nil for m == 0
+ }
+
+ z.abs = z.abs.expNN(xWords, yWords, mWords)
+ z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
+ if z.neg && len(mWords) > 0 {
+ // make modulus result positive
+ z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
+ z.neg = false
+ }
+
+ return z
+}
+
+// GCD sets z to the greatest common divisor of a and b and returns z.
+// If x or y are not nil, GCD sets their value such that z = a*x + b*y.
+//
+// a and b may be positive, zero or negative. (Before Go 1.14 both had
+// to be > 0.) Regardless of the signs of a and b, z is always >= 0.
+//
+// If a == b == 0, GCD sets z = x = y = 0.
+//
+// If a == 0 and b != 0, GCD sets z = |b|, x = 0, y = sign(b) * 1.
+//
+// If a != 0 and b == 0, GCD sets z = |a|, x = sign(a) * 1, y = 0.
+func (z *Int) GCD(x, y, a, b *Int) *Int {
+ if len(a.abs) == 0 || len(b.abs) == 0 {
+ lenA, lenB, negA, negB := len(a.abs), len(b.abs), a.neg, b.neg
+ if lenA == 0 {
+ z.Set(b)
+ } else {
+ z.Set(a)
+ }
+ z.neg = false
+ if x != nil {
+ if lenA == 0 {
+ x.SetUint64(0)
+ } else {
+ x.SetUint64(1)
+ x.neg = negA
+ }
+ }
+ if y != nil {
+ if lenB == 0 {
+ y.SetUint64(0)
+ } else {
+ y.SetUint64(1)
+ y.neg = negB
+ }
+ }
+ return z
+ }
+
+ return z.lehmerGCD(x, y, a, b)
+}
+
+// lehmerSimulate attempts to simulate several Euclidean update steps
+// using the leading digits of A and B. It returns u0, u1, v0, v1
+// such that A and B can be updated as:
+// A = u0*A + v0*B
+// B = u1*A + v1*B
+// Requirements: A >= B and len(B.abs) >= 2
+// Since we are calculating with full words to avoid overflow,
+// we use 'even' to track the sign of the cosequences.
+// For even iterations: u0, v1 >= 0 && u1, v0 <= 0
+// For odd iterations: u0, v1 <= 0 && u1, v0 >= 0
+func lehmerSimulate(A, B *Int) (u0, u1, v0, v1 Word, even bool) {
+ // initialize the digits
+ var a1, a2, u2, v2 Word
+
+ m := len(B.abs) // m >= 2
+ n := len(A.abs) // n >= m >= 2
+
+ // extract the top Word of bits from A and B
+ h := nlz(A.abs[n-1])
+ a1 = A.abs[n-1]<<h | A.abs[n-2]>>(_W-h)
+ // B may have implicit zero words in the high bits if the lengths differ
+ switch {
+ case n == m:
+ a2 = B.abs[n-1]<<h | B.abs[n-2]>>(_W-h)
+ case n == m+1:
+ a2 = B.abs[n-2] >> (_W - h)
+ default:
+ a2 = 0
+ }
+
+ // Since we are calculating with full words to avoid overflow,
+ // we use 'even' to track the sign of the cosequences.
+ // For even iterations: u0, v1 >= 0 && u1, v0 <= 0
+ // For odd iterations: u0, v1 <= 0 && u1, v0 >= 0
+ // The first iteration starts with k=1 (odd).
+ even = false
+ // variables to track the cosequences
+ u0, u1, u2 = 0, 1, 0
+ v0, v1, v2 = 0, 0, 1
+
+ // Calculate the quotient and cosequences using Collins' stopping condition.
+ // Note that overflow of a Word is not possible when computing the remainder
+ // sequence and cosequences since the cosequence size is bounded by the input size.
+ // See section 4.2 of Jebelean for details.
+ for a2 >= v2 && a1-a2 >= v1+v2 {
+ q, r := a1/a2, a1%a2
+ a1, a2 = a2, r
+ u0, u1, u2 = u1, u2, u1+q*u2
+ v0, v1, v2 = v1, v2, v1+q*v2
+ even = !even
+ }
+ return
+}
+
+// lehmerUpdate updates the inputs A and B such that:
+// A = u0*A + v0*B
+// B = u1*A + v1*B
+// where the signs of u0, u1, v0, v1 are given by even
+// For even == true: u0, v1 >= 0 && u1, v0 <= 0
+// For even == false: u0, v1 <= 0 && u1, v0 >= 0
+// q, r, s, t are temporary variables to avoid allocations in the multiplication
+func lehmerUpdate(A, B, q, r, s, t *Int, u0, u1, v0, v1 Word, even bool) {
+
+ t.abs = t.abs.setWord(u0)
+ s.abs = s.abs.setWord(v0)
+ t.neg = !even
+ s.neg = even
+
+ t.Mul(A, t)
+ s.Mul(B, s)
+
+ r.abs = r.abs.setWord(u1)
+ q.abs = q.abs.setWord(v1)
+ r.neg = even
+ q.neg = !even
+
+ r.Mul(A, r)
+ q.Mul(B, q)
+
+ A.Add(t, s)
+ B.Add(r, q)
+}
+
+// euclidUpdate performs a single step of the Euclidean GCD algorithm
+// if extended is true, it also updates the cosequence Ua, Ub
+func euclidUpdate(A, B, Ua, Ub, q, r, s, t *Int, extended bool) {
+ q, r = q.QuoRem(A, B, r)
+
+ *A, *B, *r = *B, *r, *A
+
+ if extended {
+ // Ua, Ub = Ub, Ua - q*Ub
+ t.Set(Ub)
+ s.Mul(Ub, q)
+ Ub.Sub(Ua, s)
+ Ua.Set(t)
+ }
+}
+
+// lehmerGCD sets z to the greatest common divisor of a and b,
+// which both must be != 0, and returns z.
+// If x or y are not nil, their values are set such that z = a*x + b*y.
+// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm L.
+// This implementation uses the improved condition by Collins requiring only one
+// quotient and avoiding the possibility of single Word overflow.
+// See Jebelean, "Improving the multiprecision Euclidean algorithm",
+// Design and Implementation of Symbolic Computation Systems, pp 45-58.
+// The cosequences are updated according to Algorithm 10.45 from
+// Cohen et al. "Handbook of Elliptic and Hyperelliptic Curve Cryptography" pp 192.
+func (z *Int) lehmerGCD(x, y, a, b *Int) *Int {
+ var A, B, Ua, Ub *Int
+
+ A = new(Int).Abs(a)
+ B = new(Int).Abs(b)
+
+ extended := x != nil || y != nil
+
+ if extended {
+ // Ua (Ub) tracks how many times input a has been accumulated into A (B).
+ Ua = new(Int).SetInt64(1)
+ Ub = new(Int)
+ }
+
+ // temp variables for multiprecision update
+ q := new(Int)
+ r := new(Int)
+ s := new(Int)
+ t := new(Int)
+
+ // ensure A >= B
+ if A.abs.cmp(B.abs) < 0 {
+ A, B = B, A
+ Ub, Ua = Ua, Ub
+ }
+
+ // loop invariant A >= B
+ for len(B.abs) > 1 {
+ // Attempt to calculate in single-precision using leading words of A and B.
+ u0, u1, v0, v1, even := lehmerSimulate(A, B)
+
+ // multiprecision Step
+ if v0 != 0 {
+ // Simulate the effect of the single-precision steps using the cosequences.
+ // A = u0*A + v0*B
+ // B = u1*A + v1*B
+ lehmerUpdate(A, B, q, r, s, t, u0, u1, v0, v1, even)
+
+ if extended {
+ // Ua = u0*Ua + v0*Ub
+ // Ub = u1*Ua + v1*Ub
+ lehmerUpdate(Ua, Ub, q, r, s, t, u0, u1, v0, v1, even)
+ }
+
+ } else {
+ // Single-digit calculations failed to simulate any quotients.
+ // Do a standard Euclidean step.
+ euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
+ }
+ }
+
+ if len(B.abs) > 0 {
+ // extended Euclidean algorithm base case if B is a single Word
+ if len(A.abs) > 1 {
+ // A is longer than a single Word, so one update is needed.
+ euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
+ }
+ if len(B.abs) > 0 {
+ // A and B are both a single Word.
+ aWord, bWord := A.abs[0], B.abs[0]
+ if extended {
+ var ua, ub, va, vb Word
+ ua, ub = 1, 0
+ va, vb = 0, 1
+ even := true
+ for bWord != 0 {
+ q, r := aWord/bWord, aWord%bWord
+ aWord, bWord = bWord, r
+ ua, ub = ub, ua+q*ub
+ va, vb = vb, va+q*vb
+ even = !even
+ }
+
+ t.abs = t.abs.setWord(ua)
+ s.abs = s.abs.setWord(va)
+ t.neg = !even
+ s.neg = even
+
+ t.Mul(Ua, t)
+ s.Mul(Ub, s)
+
+ Ua.Add(t, s)
+ } else {
+ for bWord != 0 {
+ aWord, bWord = bWord, aWord%bWord
+ }
+ }
+ A.abs[0] = aWord
+ }
+ }
+ negA := a.neg
+ if y != nil {
+ // avoid aliasing b needed in the division below
+ if y == b {
+ B.Set(b)
+ } else {
+ B = b
+ }
+ // y = (z - a*x)/b
+ y.Mul(a, Ua) // y can safely alias a
+ if negA {
+ y.neg = !y.neg
+ }
+ y.Sub(A, y)
+ y.Div(y, B)
+ }
+
+ if x != nil {
+ *x = *Ua
+ if negA {
+ x.neg = !x.neg
+ }
+ }
+
+ *z = *A
+
+ return z
+}
+
+// Rand sets z to a pseudo-random number in [0, n) and returns z.
+//
+// As this uses the math/rand package, it must not be used for
+// security-sensitive work. Use crypto/rand.Int instead.
+func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
+ z.neg = false
+ if n.neg || len(n.abs) == 0 {
+ z.abs = nil
+ return z
+ }
+ z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
+ return z
+}
+
+// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
+// and returns z. If g and n are not relatively prime, g has no multiplicative
+// inverse in the ring ℤ/nℤ. In this case, z is unchanged and the return value
+// is nil.
+func (z *Int) ModInverse(g, n *Int) *Int {
+ // GCD expects parameters a and b to be > 0.
+ if n.neg {
+ var n2 Int
+ n = n2.Neg(n)
+ }
+ if g.neg {
+ var g2 Int
+ g = g2.Mod(g, n)
+ }
+ var d, x Int
+ d.GCD(&x, nil, g, n)
+
+ // if and only if d==1, g and n are relatively prime
+ if d.Cmp(intOne) != 0 {
+ return nil
+ }
+
+ // x and y are such that g*x + n*y = 1, therefore x is the inverse element,
+ // but it may be negative, so convert to the range 0 <= z < |n|
+ if x.neg {
+ z.Add(&x, n)
+ } else {
+ z.Set(&x)
+ }
+ return z
+}
+
+// Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
+// The y argument must be an odd integer.
+func Jacobi(x, y *Int) int {
+ if len(y.abs) == 0 || y.abs[0]&1 == 0 {
+ panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y))
+ }
+
+ // We use the formulation described in chapter 2, section 2.4,
+ // "The Yacas Book of Algorithms":
+ // http://yacas.sourceforge.net/Algo.book.pdf
+
+ var a, b, c Int
+ a.Set(x)
+ b.Set(y)
+ j := 1
+
+ if b.neg {
+ if a.neg {
+ j = -1
+ }
+ b.neg = false
+ }
+
+ for {
+ if b.Cmp(intOne) == 0 {
+ return j
+ }
+ if len(a.abs) == 0 {
+ return 0
+ }
+ a.Mod(&a, &b)
+ if len(a.abs) == 0 {
+ return 0
+ }
+ // a > 0
+
+ // handle factors of 2 in 'a'
+ s := a.abs.trailingZeroBits()
+ if s&1 != 0 {
+ bmod8 := b.abs[0] & 7
+ if bmod8 == 3 || bmod8 == 5 {
+ j = -j
+ }
+ }
+ c.Rsh(&a, s) // a = 2^s*c
+
+ // swap numerator and denominator
+ if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
+ j = -j
+ }
+ a.Set(&b)
+ b.Set(&c)
+ }
+}
+
+// modSqrt3Mod4 uses the identity
+// (a^((p+1)/4))^2 mod p
+// == u^(p+1) mod p
+// == u^2 mod p
+// to calculate the square root of any quadratic residue mod p quickly for 3
+// mod 4 primes.
+func (z *Int) modSqrt3Mod4Prime(x, p *Int) *Int {
+ e := new(Int).Add(p, intOne) // e = p + 1
+ e.Rsh(e, 2) // e = (p + 1) / 4
+ z.Exp(x, e, p) // z = x^e mod p
+ return z
+}
+
+// modSqrt5Mod8 uses Atkin's observation that 2 is not a square mod p
+// alpha == (2*a)^((p-5)/8) mod p
+// beta == 2*a*alpha^2 mod p is a square root of -1
+// b == a*alpha*(beta-1) mod p is a square root of a
+// to calculate the square root of any quadratic residue mod p quickly for 5
+// mod 8 primes.
+func (z *Int) modSqrt5Mod8Prime(x, p *Int) *Int {
+ // p == 5 mod 8 implies p = e*8 + 5
+ // e is the quotient and 5 the remainder on division by 8
+ e := new(Int).Rsh(p, 3) // e = (p - 5) / 8
+ tx := new(Int).Lsh(x, 1) // tx = 2*x
+ alpha := new(Int).Exp(tx, e, p)
+ beta := new(Int).Mul(alpha, alpha)
+ beta.Mod(beta, p)
+ beta.Mul(beta, tx)
+ beta.Mod(beta, p)
+ beta.Sub(beta, intOne)
+ beta.Mul(beta, x)
+ beta.Mod(beta, p)
+ beta.Mul(beta, alpha)
+ z.Mod(beta, p)
+ return z
+}
+
+// modSqrtTonelliShanks uses the Tonelli-Shanks algorithm to find the square
+// root of a quadratic residue modulo any prime.
+func (z *Int) modSqrtTonelliShanks(x, p *Int) *Int {
+ // Break p-1 into s*2^e such that s is odd.
+ var s Int
+ s.Sub(p, intOne)
+ e := s.abs.trailingZeroBits()
+ s.Rsh(&s, e)
+
+ // find some non-square n
+ var n Int
+ n.SetInt64(2)
+ for Jacobi(&n, p) != -1 {
+ n.Add(&n, intOne)
+ }
+
+ // Core of the Tonelli-Shanks algorithm. Follows the description in
+ // section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
+ // Brown:
+ // https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020786.02p0470a.pdf
+ var y, b, g, t Int
+ y.Add(&s, intOne)
+ y.Rsh(&y, 1)
+ y.Exp(x, &y, p) // y = x^((s+1)/2)
+ b.Exp(x, &s, p) // b = x^s
+ g.Exp(&n, &s, p) // g = n^s
+ r := e
+ for {
+ // find the least m such that ord_p(b) = 2^m
+ var m uint
+ t.Set(&b)
+ for t.Cmp(intOne) != 0 {
+ t.Mul(&t, &t).Mod(&t, p)
+ m++
+ }
+
+ if m == 0 {
+ return z.Set(&y)
+ }
+
+ t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
+ // t = g^(2^(r-m-1)) mod p
+ g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
+ y.Mul(&y, &t).Mod(&y, p)
+ b.Mul(&b, &g).Mod(&b, p)
+ r = m
+ }
+}
+
+// ModSqrt sets z to a square root of x mod p if such a square root exists, and
+// returns z. The modulus p must be an odd prime. If x is not a square mod p,
+// ModSqrt leaves z unchanged and returns nil. This function panics if p is
+// not an odd integer.
+func (z *Int) ModSqrt(x, p *Int) *Int {
+ switch Jacobi(x, p) {
+ case -1:
+ return nil // x is not a square mod p
+ case 0:
+ return z.SetInt64(0) // sqrt(0) mod p = 0
+ case 1:
+ break
+ }
+ if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
+ x = new(Int).Mod(x, p)
+ }
+
+ switch {
+ case p.abs[0]%4 == 3:
+ // Check whether p is 3 mod 4, and if so, use the faster algorithm.
+ return z.modSqrt3Mod4Prime(x, p)
+ case p.abs[0]%8 == 5:
+ // Check whether p is 5 mod 8, use Atkin's algorithm.
+ return z.modSqrt5Mod8Prime(x, p)
+ default:
+ // Otherwise, use Tonelli-Shanks.
+ return z.modSqrtTonelliShanks(x, p)
+ }
+}
+
+// Lsh sets z = x << n and returns z.
+func (z *Int) Lsh(x *Int, n uint) *Int {
+ z.abs = z.abs.shl(x.abs, n)
+ z.neg = x.neg
+ return z
+}
+
+// Rsh sets z = x >> n and returns z.
+func (z *Int) Rsh(x *Int, n uint) *Int {
+ if x.neg {
+ // (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
+ t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
+ t = t.shr(t, n)
+ z.abs = t.add(t, natOne)
+ z.neg = true // z cannot be zero if x is negative
+ return z
+ }
+
+ z.abs = z.abs.shr(x.abs, n)
+ z.neg = false
+ return z
+}
+
+// Bit returns the value of the i'th bit of x. That is, it
+// returns (x>>i)&1. The bit index i must be >= 0.
+func (x *Int) Bit(i int) uint {
+ if i == 0 {
+ // optimization for common case: odd/even test of x
+ if len(x.abs) > 0 {
+ return uint(x.abs[0] & 1) // bit 0 is same for -x
+ }
+ return 0
+ }
+ if i < 0 {
+ panic("negative bit index")
+ }
+ if x.neg {
+ t := nat(nil).sub(x.abs, natOne)
+ return t.bit(uint(i)) ^ 1
+ }
+
+ return x.abs.bit(uint(i))
+}
+
+// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
+// That is, if b is 1 SetBit sets z = x | (1 << i);
+// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
+// SetBit will panic.
+func (z *Int) SetBit(x *Int, i int, b uint) *Int {
+ if i < 0 {
+ panic("negative bit index")
+ }
+ if x.neg {
+ t := z.abs.sub(x.abs, natOne)
+ t = t.setBit(t, uint(i), b^1)
+ z.abs = t.add(t, natOne)
+ z.neg = len(z.abs) > 0
+ return z
+ }
+ z.abs = z.abs.setBit(x.abs, uint(i), b)
+ z.neg = false
+ return z
+}
+
+// And sets z = x & y and returns z.
+func (z *Int) And(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
+ z.neg = true // z cannot be zero if x and y are negative
+ return z
+ }
+
+ // x & y == x & y
+ z.abs = z.abs.and(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ // x.neg != y.neg
+ if x.neg {
+ x, y = y, x // & is symmetric
+ }
+
+ // x & (-y) == x & ^(y-1) == x &^ (y-1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.andNot(x.abs, y1)
+ z.neg = false
+ return z
+}
+
+// AndNot sets z = x &^ y and returns z.
+func (z *Int) AndNot(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.andNot(y1, x1)
+ z.neg = false
+ return z
+ }
+
+ // x &^ y == x &^ y
+ z.abs = z.abs.andNot(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ if x.neg {
+ // (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
+ z.neg = true // z cannot be zero if x is negative and y is positive
+ return z
+ }
+
+ // x &^ (-y) == x &^ ^(y-1) == x & (y-1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.and(x.abs, y1)
+ z.neg = false
+ return z
+}
+
+// Or sets z = x | y and returns z.
+func (z *Int) Or(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
+ z.neg = true // z cannot be zero if x and y are negative
+ return z
+ }
+
+ // x | y == x | y
+ z.abs = z.abs.or(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ // x.neg != y.neg
+ if x.neg {
+ x, y = y, x // | is symmetric
+ }
+
+ // x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
+ z.neg = true // z cannot be zero if one of x or y is negative
+ return z
+}
+
+// Xor sets z = x ^ y and returns z.
+func (z *Int) Xor(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.xor(x1, y1)
+ z.neg = false
+ return z
+ }
+
+ // x ^ y == x ^ y
+ z.abs = z.abs.xor(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ // x.neg != y.neg
+ if x.neg {
+ x, y = y, x // ^ is symmetric
+ }
+
+ // x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
+ z.neg = true // z cannot be zero if only one of x or y is negative
+ return z
+}
+
+// Not sets z = ^x and returns z.
+func (z *Int) Not(x *Int) *Int {
+ if x.neg {
+ // ^(-x) == ^(^(x-1)) == x-1
+ z.abs = z.abs.sub(x.abs, natOne)
+ z.neg = false
+ return z
+ }
+
+ // ^x == -x-1 == -(x+1)
+ z.abs = z.abs.add(x.abs, natOne)
+ z.neg = true // z cannot be zero if x is positive
+ return z
+}
+
+// Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z.
+// It panics if x is negative.
+func (z *Int) Sqrt(x *Int) *Int {
+ if x.neg {
+ panic("square root of negative number")
+ }
+ z.neg = false
+ z.abs = z.abs.sqrt(x.abs)
+ return z
+}
diff --git a/src/math/big/int_test.go b/src/math/big/int_test.go
new file mode 100644
index 0000000..3c85573
--- /dev/null
+++ b/src/math/big/int_test.go
@@ -0,0 +1,1896 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "encoding/hex"
+ "fmt"
+ "math/rand"
+ "strconv"
+ "strings"
+ "testing"
+ "testing/quick"
+)
+
+func isNormalized(x *Int) bool {
+ if len(x.abs) == 0 {
+ return !x.neg
+ }
+ // len(x.abs) > 0
+ return x.abs[len(x.abs)-1] != 0
+}
+
+type funZZ func(z, x, y *Int) *Int
+type argZZ struct {
+ z, x, y *Int
+}
+
+var sumZZ = []argZZ{
+ {NewInt(0), NewInt(0), NewInt(0)},
+ {NewInt(1), NewInt(1), NewInt(0)},
+ {NewInt(1111111110), NewInt(123456789), NewInt(987654321)},
+ {NewInt(-1), NewInt(-1), NewInt(0)},
+ {NewInt(864197532), NewInt(-123456789), NewInt(987654321)},
+ {NewInt(-1111111110), NewInt(-123456789), NewInt(-987654321)},
+}
+
+var prodZZ = []argZZ{
+ {NewInt(0), NewInt(0), NewInt(0)},
+ {NewInt(0), NewInt(1), NewInt(0)},
+ {NewInt(1), NewInt(1), NewInt(1)},
+ {NewInt(-991 * 991), NewInt(991), NewInt(-991)},
+ // TODO(gri) add larger products
+}
+
+func TestSignZ(t *testing.T) {
+ var zero Int
+ for _, a := range sumZZ {
+ s := a.z.Sign()
+ e := a.z.Cmp(&zero)
+ if s != e {
+ t.Errorf("got %d; want %d for z = %v", s, e, a.z)
+ }
+ }
+}
+
+func TestSetZ(t *testing.T) {
+ for _, a := range sumZZ {
+ var z Int
+ z.Set(a.z)
+ if !isNormalized(&z) {
+ t.Errorf("%v is not normalized", z)
+ }
+ if (&z).Cmp(a.z) != 0 {
+ t.Errorf("got z = %v; want %v", z, a.z)
+ }
+ }
+}
+
+func TestAbsZ(t *testing.T) {
+ var zero Int
+ for _, a := range sumZZ {
+ var z Int
+ z.Abs(a.z)
+ var e Int
+ e.Set(a.z)
+ if e.Cmp(&zero) < 0 {
+ e.Sub(&zero, &e)
+ }
+ if z.Cmp(&e) != 0 {
+ t.Errorf("got z = %v; want %v", z, e)
+ }
+ }
+}
+
+func testFunZZ(t *testing.T, msg string, f funZZ, a argZZ) {
+ var z Int
+ f(&z, a.x, a.y)
+ if !isNormalized(&z) {
+ t.Errorf("%s%v is not normalized", msg, z)
+ }
+ if (&z).Cmp(a.z) != 0 {
+ t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, &z, a.z)
+ }
+}
+
+func TestSumZZ(t *testing.T) {
+ AddZZ := func(z, x, y *Int) *Int { return z.Add(x, y) }
+ SubZZ := func(z, x, y *Int) *Int { return z.Sub(x, y) }
+ for _, a := range sumZZ {
+ arg := a
+ testFunZZ(t, "AddZZ", AddZZ, arg)
+
+ arg = argZZ{a.z, a.y, a.x}
+ testFunZZ(t, "AddZZ symmetric", AddZZ, arg)
+
+ arg = argZZ{a.x, a.z, a.y}
+ testFunZZ(t, "SubZZ", SubZZ, arg)
+
+ arg = argZZ{a.y, a.z, a.x}
+ testFunZZ(t, "SubZZ symmetric", SubZZ, arg)
+ }
+}
+
+func TestProdZZ(t *testing.T) {
+ MulZZ := func(z, x, y *Int) *Int { return z.Mul(x, y) }
+ for _, a := range prodZZ {
+ arg := a
+ testFunZZ(t, "MulZZ", MulZZ, arg)
+
+ arg = argZZ{a.z, a.y, a.x}
+ testFunZZ(t, "MulZZ symmetric", MulZZ, arg)
+ }
+}
+
+// mulBytes returns x*y via grade school multiplication. Both inputs
+// and the result are assumed to be in big-endian representation (to
+// match the semantics of Int.Bytes and Int.SetBytes).
+func mulBytes(x, y []byte) []byte {
+ z := make([]byte, len(x)+len(y))
+
+ // multiply
+ k0 := len(z) - 1
+ for j := len(y) - 1; j >= 0; j-- {
+ d := int(y[j])
+ if d != 0 {
+ k := k0
+ carry := 0
+ for i := len(x) - 1; i >= 0; i-- {
+ t := int(z[k]) + int(x[i])*d + carry
+ z[k], carry = byte(t), t>>8
+ k--
+ }
+ z[k] = byte(carry)
+ }
+ k0--
+ }
+
+ // normalize (remove leading 0's)
+ i := 0
+ for i < len(z) && z[i] == 0 {
+ i++
+ }
+
+ return z[i:]
+}
+
+func checkMul(a, b []byte) bool {
+ var x, y, z1 Int
+ x.SetBytes(a)
+ y.SetBytes(b)
+ z1.Mul(&x, &y)
+
+ var z2 Int
+ z2.SetBytes(mulBytes(a, b))
+
+ return z1.Cmp(&z2) == 0
+}
+
+func TestMul(t *testing.T) {
+ if err := quick.Check(checkMul, nil); err != nil {
+ t.Error(err)
+ }
+}
+
+var mulRangesZ = []struct {
+ a, b int64
+ prod string
+}{
+ // entirely positive ranges are covered by mulRangesN
+ {-1, 1, "0"},
+ {-2, -1, "2"},
+ {-3, -2, "6"},
+ {-3, -1, "-6"},
+ {1, 3, "6"},
+ {-10, -10, "-10"},
+ {0, -1, "1"}, // empty range
+ {-1, -100, "1"}, // empty range
+ {-1, 1, "0"}, // range includes 0
+ {-1e9, 0, "0"}, // range includes 0
+ {-1e9, 1e9, "0"}, // range includes 0
+ {-10, -1, "3628800"}, // 10!
+ {-20, -2, "-2432902008176640000"}, // -20!
+ {-99, -1,
+ "-933262154439441526816992388562667004907159682643816214685929" +
+ "638952175999932299156089414639761565182862536979208272237582" +
+ "511852109168640000000000000000000000", // -99!
+ },
+}
+
+func TestMulRangeZ(t *testing.T) {
+ var tmp Int
+ // test entirely positive ranges
+ for i, r := range mulRangesN {
+ prod := tmp.MulRange(int64(r.a), int64(r.b)).String()
+ if prod != r.prod {
+ t.Errorf("#%da: got %s; want %s", i, prod, r.prod)
+ }
+ }
+ // test other ranges
+ for i, r := range mulRangesZ {
+ prod := tmp.MulRange(r.a, r.b).String()
+ if prod != r.prod {
+ t.Errorf("#%db: got %s; want %s", i, prod, r.prod)
+ }
+ }
+}
+
+func TestBinomial(t *testing.T) {
+ var z Int
+ for _, test := range []struct {
+ n, k int64
+ want string
+ }{
+ {0, 0, "1"},
+ {0, 1, "0"},
+ {1, 0, "1"},
+ {1, 1, "1"},
+ {1, 10, "0"},
+ {4, 0, "1"},
+ {4, 1, "4"},
+ {4, 2, "6"},
+ {4, 3, "4"},
+ {4, 4, "1"},
+ {10, 1, "10"},
+ {10, 9, "10"},
+ {10, 5, "252"},
+ {11, 5, "462"},
+ {11, 6, "462"},
+ {100, 10, "17310309456440"},
+ {100, 90, "17310309456440"},
+ {1000, 10, "263409560461970212832400"},
+ {1000, 990, "263409560461970212832400"},
+ } {
+ if got := z.Binomial(test.n, test.k).String(); got != test.want {
+ t.Errorf("Binomial(%d, %d) = %s; want %s", test.n, test.k, got, test.want)
+ }
+ }
+}
+
+func BenchmarkBinomial(b *testing.B) {
+ var z Int
+ for i := b.N - 1; i >= 0; i-- {
+ z.Binomial(1000, 990)
+ }
+}
+
+// Examples from the Go Language Spec, section "Arithmetic operators"
+var divisionSignsTests = []struct {
+ x, y int64
+ q, r int64 // T-division
+ d, m int64 // Euclidean division
+}{
+ {5, 3, 1, 2, 1, 2},
+ {-5, 3, -1, -2, -2, 1},
+ {5, -3, -1, 2, -1, 2},
+ {-5, -3, 1, -2, 2, 1},
+ {1, 2, 0, 1, 0, 1},
+ {8, 4, 2, 0, 2, 0},
+}
+
+func TestDivisionSigns(t *testing.T) {
+ for i, test := range divisionSignsTests {
+ x := NewInt(test.x)
+ y := NewInt(test.y)
+ q := NewInt(test.q)
+ r := NewInt(test.r)
+ d := NewInt(test.d)
+ m := NewInt(test.m)
+
+ q1 := new(Int).Quo(x, y)
+ r1 := new(Int).Rem(x, y)
+ if !isNormalized(q1) {
+ t.Errorf("#%d Quo: %v is not normalized", i, *q1)
+ }
+ if !isNormalized(r1) {
+ t.Errorf("#%d Rem: %v is not normalized", i, *r1)
+ }
+ if q1.Cmp(q) != 0 || r1.Cmp(r) != 0 {
+ t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q1, r1, q, r)
+ }
+
+ q2, r2 := new(Int).QuoRem(x, y, new(Int))
+ if !isNormalized(q2) {
+ t.Errorf("#%d Quo: %v is not normalized", i, *q2)
+ }
+ if !isNormalized(r2) {
+ t.Errorf("#%d Rem: %v is not normalized", i, *r2)
+ }
+ if q2.Cmp(q) != 0 || r2.Cmp(r) != 0 {
+ t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q2, r2, q, r)
+ }
+
+ d1 := new(Int).Div(x, y)
+ m1 := new(Int).Mod(x, y)
+ if !isNormalized(d1) {
+ t.Errorf("#%d Div: %v is not normalized", i, *d1)
+ }
+ if !isNormalized(m1) {
+ t.Errorf("#%d Mod: %v is not normalized", i, *m1)
+ }
+ if d1.Cmp(d) != 0 || m1.Cmp(m) != 0 {
+ t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d1, m1, d, m)
+ }
+
+ d2, m2 := new(Int).DivMod(x, y, new(Int))
+ if !isNormalized(d2) {
+ t.Errorf("#%d Div: %v is not normalized", i, *d2)
+ }
+ if !isNormalized(m2) {
+ t.Errorf("#%d Mod: %v is not normalized", i, *m2)
+ }
+ if d2.Cmp(d) != 0 || m2.Cmp(m) != 0 {
+ t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d2, m2, d, m)
+ }
+ }
+}
+
+func norm(x nat) nat {
+ i := len(x)
+ for i > 0 && x[i-1] == 0 {
+ i--
+ }
+ return x[:i]
+}
+
+func TestBits(t *testing.T) {
+ for _, test := range []nat{
+ nil,
+ {0},
+ {1},
+ {0, 1, 2, 3, 4},
+ {4, 3, 2, 1, 0},
+ {4, 3, 2, 1, 0, 0, 0, 0},
+ } {
+ var z Int
+ z.neg = true
+ got := z.SetBits(test)
+ want := norm(test)
+ if got.abs.cmp(want) != 0 {
+ t.Errorf("SetBits(%v) = %v; want %v", test, got.abs, want)
+ }
+
+ if got.neg {
+ t.Errorf("SetBits(%v): got negative result", test)
+ }
+
+ bits := nat(z.Bits())
+ if bits.cmp(want) != 0 {
+ t.Errorf("%v.Bits() = %v; want %v", z.abs, bits, want)
+ }
+ }
+}
+
+func checkSetBytes(b []byte) bool {
+ hex1 := hex.EncodeToString(new(Int).SetBytes(b).Bytes())
+ hex2 := hex.EncodeToString(b)
+
+ for len(hex1) < len(hex2) {
+ hex1 = "0" + hex1
+ }
+
+ for len(hex1) > len(hex2) {
+ hex2 = "0" + hex2
+ }
+
+ return hex1 == hex2
+}
+
+func TestSetBytes(t *testing.T) {
+ if err := quick.Check(checkSetBytes, nil); err != nil {
+ t.Error(err)
+ }
+}
+
+func checkBytes(b []byte) bool {
+ // trim leading zero bytes since Bytes() won't return them
+ // (was issue 12231)
+ for len(b) > 0 && b[0] == 0 {
+ b = b[1:]
+ }
+ b2 := new(Int).SetBytes(b).Bytes()
+ return bytes.Equal(b, b2)
+}
+
+func TestBytes(t *testing.T) {
+ if err := quick.Check(checkBytes, nil); err != nil {
+ t.Error(err)
+ }
+}
+
+func checkQuo(x, y []byte) bool {
+ u := new(Int).SetBytes(x)
+ v := new(Int).SetBytes(y)
+
+ if len(v.abs) == 0 {
+ return true
+ }
+
+ r := new(Int)
+ q, r := new(Int).QuoRem(u, v, r)
+
+ if r.Cmp(v) >= 0 {
+ return false
+ }
+
+ uprime := new(Int).Set(q)
+ uprime.Mul(uprime, v)
+ uprime.Add(uprime, r)
+
+ return uprime.Cmp(u) == 0
+}
+
+var quoTests = []struct {
+ x, y string
+ q, r string
+}{
+ {
+ "476217953993950760840509444250624797097991362735329973741718102894495832294430498335824897858659711275234906400899559094370964723884706254265559534144986498357",
+ "9353930466774385905609975137998169297361893554149986716853295022578535724979483772383667534691121982974895531435241089241440253066816724367338287092081996",
+ "50911",
+ "1",
+ },
+ {
+ "11510768301994997771168",
+ "1328165573307167369775",
+ "8",
+ "885443715537658812968",
+ },
+}
+
+func TestQuo(t *testing.T) {
+ if err := quick.Check(checkQuo, nil); err != nil {
+ t.Error(err)
+ }
+
+ for i, test := range quoTests {
+ x, _ := new(Int).SetString(test.x, 10)
+ y, _ := new(Int).SetString(test.y, 10)
+ expectedQ, _ := new(Int).SetString(test.q, 10)
+ expectedR, _ := new(Int).SetString(test.r, 10)
+
+ r := new(Int)
+ q, r := new(Int).QuoRem(x, y, r)
+
+ if q.Cmp(expectedQ) != 0 || r.Cmp(expectedR) != 0 {
+ t.Errorf("#%d got (%s, %s) want (%s, %s)", i, q, r, expectedQ, expectedR)
+ }
+ }
+}
+
+func TestQuoStepD6(t *testing.T) {
+ // See Knuth, Volume 2, section 4.3.1, exercise 21. This code exercises
+ // a code path which only triggers 1 in 10^{-19} cases.
+
+ u := &Int{false, nat{0, 0, 1 + 1<<(_W-1), _M ^ (1 << (_W - 1))}}
+ v := &Int{false, nat{5, 2 + 1<<(_W-1), 1 << (_W - 1)}}
+
+ r := new(Int)
+ q, r := new(Int).QuoRem(u, v, r)
+ const expectedQ64 = "18446744073709551613"
+ const expectedR64 = "3138550867693340382088035895064302439801311770021610913807"
+ const expectedQ32 = "4294967293"
+ const expectedR32 = "39614081266355540837921718287"
+ if q.String() != expectedQ64 && q.String() != expectedQ32 ||
+ r.String() != expectedR64 && r.String() != expectedR32 {
+ t.Errorf("got (%s, %s) want (%s, %s) or (%s, %s)", q, r, expectedQ64, expectedR64, expectedQ32, expectedR32)
+ }
+}
+
+func BenchmarkQuoRem(b *testing.B) {
+ x, _ := new(Int).SetString("153980389784927331788354528594524332344709972855165340650588877572729725338415474372475094155672066328274535240275856844648695200875763869073572078279316458648124537905600131008790701752441155668003033945258023841165089852359980273279085783159654751552359397986180318708491098942831252291841441726305535546071", 0)
+ y, _ := new(Int).SetString("7746362281539803897849273317883545285945243323447099728551653406505888775727297253384154743724750941556720663282745352402758568446486952008757638690735720782793164586481245379056001310087907017524411556680030339452580238411650898523599802732790857831596547515523593979861803187084910989428312522918414417263055355460715745539358014631136245887418412633787074173796862711588221766398229333338511838891484974940633857861775630560092874987828057333663969469797013996401149696897591265769095952887917296740109742927689053276850469671231961384715398038978492733178835452859452433234470997285516534065058887757272972533841547437247509415567206632827453524027585684464869520087576386907357207827931645864812453790560013100879070175244115566800303394525802384116508985235998027327908578315965475155235939798618031870849109894283125229184144172630553554607112725169432413343763989564437170644270643461665184965150423819594083121075825", 0)
+ q := new(Int)
+ r := new(Int)
+
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ q.QuoRem(y, x, r)
+ }
+}
+
+var bitLenTests = []struct {
+ in string
+ out int
+}{
+ {"-1", 1},
+ {"0", 0},
+ {"1", 1},
+ {"2", 2},
+ {"4", 3},
+ {"0xabc", 12},
+ {"0x8000", 16},
+ {"0x80000000", 32},
+ {"0x800000000000", 48},
+ {"0x8000000000000000", 64},
+ {"0x80000000000000000000", 80},
+ {"-0x4000000000000000000000", 87},
+}
+
+func TestBitLen(t *testing.T) {
+ for i, test := range bitLenTests {
+ x, ok := new(Int).SetString(test.in, 0)
+ if !ok {
+ t.Errorf("#%d test input invalid: %s", i, test.in)
+ continue
+ }
+
+ if n := x.BitLen(); n != test.out {
+ t.Errorf("#%d got %d want %d", i, n, test.out)
+ }
+ }
+}
+
+var expTests = []struct {
+ x, y, m string
+ out string
+}{
+ // y <= 0
+ {"0", "0", "", "1"},
+ {"1", "0", "", "1"},
+ {"-10", "0", "", "1"},
+ {"1234", "-1", "", "1"},
+ {"1234", "-1", "0", "1"},
+ {"17", "-100", "1234", "865"},
+ {"2", "-100", "1234", ""},
+
+ // m == 1
+ {"0", "0", "1", "0"},
+ {"1", "0", "1", "0"},
+ {"-10", "0", "1", "0"},
+ {"1234", "-1", "1", "0"},
+
+ // misc
+ {"5", "1", "3", "2"},
+ {"5", "-7", "", "1"},
+ {"-5", "-7", "", "1"},
+ {"5", "0", "", "1"},
+ {"-5", "0", "", "1"},
+ {"5", "1", "", "5"},
+ {"-5", "1", "", "-5"},
+ {"-5", "1", "7", "2"},
+ {"-2", "3", "2", "0"},
+ {"5", "2", "", "25"},
+ {"1", "65537", "2", "1"},
+ {"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
+ {"0x8000000000000000", "2", "6719", "4944"},
+ {"0x8000000000000000", "3", "6719", "5447"},
+ {"0x8000000000000000", "1000", "6719", "1603"},
+ {"0x8000000000000000", "1000000", "6719", "3199"},
+ {"0x8000000000000000", "-1000000", "6719", "3663"}, // 3663 = ModInverse(3199, 6719) Issue #25865
+
+ {"0xffffffffffffffffffffffffffffffff", "0x12345678123456781234567812345678123456789", "0x01112222333344445555666677778889", "0x36168FA1DB3AAE6C8CE647E137F97A"},
+
+ {
+ "2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
+ "298472983472983471903246121093472394872319615612417471234712061",
+ "29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
+ "23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
+ },
+ // test case for issue 8822
+ {
+ "11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865",
+ "0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+ "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
+ "21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
+ },
+ {
+ "-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
+ "0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+ "0xAC6BDB41324A9A9BF166DE5E1389582FAF72B6651987EE07FC3192943DB56050A37329CBB4A099ED8193E0757767A13DD52312AB4B03310DCD7F48A9DA04FD50E8083969EDB767B0CF6095179A163AB3661A05FBD5FAAAE82918A9962F0B93B855F97993EC975EEAA80D740ADBF4FF747359D041D5C33EA71D281E446B14773BCA97B43A23FB801676BD207A436C6481F1D2B9078717461A5B9D32E688F87748544523B524B0D57D5EA77A2775D2ECFA032CFBDBF52FB3786160279004E57AE6AF874E7303CE53299CCC041C7BC308D82A5698F3A8D0C38271AE35F8E9DBFBB694B5C803D89F7AE435DE236D525F54759B65E372FCD68EF20FA7111F9E4AFF73",
+ "21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
+ },
+
+ // test cases for issue 13907
+ {"0xffffffff00000001", "0xffffffff00000001", "0xffffffff00000001", "0"},
+ {"0xffffffffffffffff00000001", "0xffffffffffffffff00000001", "0xffffffffffffffff00000001", "0"},
+ {"0xffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffff00000001", "0"},
+ {"0xffffffffffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffffffffffff00000001", "0"},
+
+ {
+ "2",
+ "0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+ "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odd
+ "0x6AADD3E3E424D5B713FCAA8D8945B1E055166132038C57BBD2D51C833F0C5EA2007A2324CE514F8E8C2F008A2F36F44005A4039CB55830986F734C93DAF0EB4BAB54A6A8C7081864F44346E9BC6F0A3EB9F2C0146A00C6A05187D0C101E1F2D038CDB70CB5E9E05A2D188AB6CBB46286624D4415E7D4DBFAD3BCC6009D915C406EED38F468B940F41E6BEDC0430DD78E6F19A7DA3A27498A4181E24D738B0072D8F6ADB8C9809A5B033A09785814FD9919F6EF9F83EEA519BEC593855C4C10CBEEC582D4AE0792158823B0275E6AEC35242740468FAF3D5C60FD1E376362B6322F78B7ED0CA1C5BBCD2B49734A56C0967A1D01A100932C837B91D592CE08ABFF",
+ },
+ {
+ "2",
+ "0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+ "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even
+ "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
+ },
+}
+
+func TestExp(t *testing.T) {
+ for i, test := range expTests {
+ x, ok1 := new(Int).SetString(test.x, 0)
+ y, ok2 := new(Int).SetString(test.y, 0)
+
+ var ok3, ok4 bool
+ var out, m *Int
+
+ if len(test.out) == 0 {
+ out, ok3 = nil, true
+ } else {
+ out, ok3 = new(Int).SetString(test.out, 0)
+ }
+
+ if len(test.m) == 0 {
+ m, ok4 = nil, true
+ } else {
+ m, ok4 = new(Int).SetString(test.m, 0)
+ }
+
+ if !ok1 || !ok2 || !ok3 || !ok4 {
+ t.Errorf("#%d: error in input", i)
+ continue
+ }
+
+ z1 := new(Int).Exp(x, y, m)
+ if z1 != nil && !isNormalized(z1) {
+ t.Errorf("#%d: %v is not normalized", i, *z1)
+ }
+ if !(z1 == nil && out == nil || z1.Cmp(out) == 0) {
+ t.Errorf("#%d: got %x want %x", i, z1, out)
+ }
+
+ if m == nil {
+ // The result should be the same as for m == 0;
+ // specifically, there should be no div-zero panic.
+ m = &Int{abs: nat{}} // m != nil && len(m.abs) == 0
+ z2 := new(Int).Exp(x, y, m)
+ if z2.Cmp(z1) != 0 {
+ t.Errorf("#%d: got %x want %x", i, z2, z1)
+ }
+ }
+ }
+}
+
+func BenchmarkExp(b *testing.B) {
+ x, _ := new(Int).SetString("11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865", 0)
+ y, _ := new(Int).SetString("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
+ n, _ := new(Int).SetString("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
+ out := new(Int)
+ for i := 0; i < b.N; i++ {
+ out.Exp(x, y, n)
+ }
+}
+
+func BenchmarkExp2(b *testing.B) {
+ x, _ := new(Int).SetString("2", 0)
+ y, _ := new(Int).SetString("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
+ n, _ := new(Int).SetString("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
+ out := new(Int)
+ for i := 0; i < b.N; i++ {
+ out.Exp(x, y, n)
+ }
+}
+
+func checkGcd(aBytes, bBytes []byte) bool {
+ x := new(Int)
+ y := new(Int)
+ a := new(Int).SetBytes(aBytes)
+ b := new(Int).SetBytes(bBytes)
+
+ d := new(Int).GCD(x, y, a, b)
+ x.Mul(x, a)
+ y.Mul(y, b)
+ x.Add(x, y)
+
+ return x.Cmp(d) == 0
+}
+
+// euclidExtGCD is a reference implementation of Euclid's
+// extended GCD algorithm for testing against optimized algorithms.
+// Requirements: a, b > 0
+func euclidExtGCD(a, b *Int) (g, x, y *Int) {
+ A := new(Int).Set(a)
+ B := new(Int).Set(b)
+
+ // A = Ua*a + Va*b
+ // B = Ub*a + Vb*b
+ Ua := new(Int).SetInt64(1)
+ Va := new(Int)
+
+ Ub := new(Int)
+ Vb := new(Int).SetInt64(1)
+
+ q := new(Int)
+ temp := new(Int)
+
+ r := new(Int)
+ for len(B.abs) > 0 {
+ q, r = q.QuoRem(A, B, r)
+
+ A, B, r = B, r, A
+
+ // Ua, Ub = Ub, Ua-q*Ub
+ temp.Set(Ub)
+ Ub.Mul(Ub, q)
+ Ub.Sub(Ua, Ub)
+ Ua.Set(temp)
+
+ // Va, Vb = Vb, Va-q*Vb
+ temp.Set(Vb)
+ Vb.Mul(Vb, q)
+ Vb.Sub(Va, Vb)
+ Va.Set(temp)
+ }
+ return A, Ua, Va
+}
+
+func checkLehmerGcd(aBytes, bBytes []byte) bool {
+ a := new(Int).SetBytes(aBytes)
+ b := new(Int).SetBytes(bBytes)
+
+ if a.Sign() <= 0 || b.Sign() <= 0 {
+ return true // can only test positive arguments
+ }
+
+ d := new(Int).lehmerGCD(nil, nil, a, b)
+ d0, _, _ := euclidExtGCD(a, b)
+
+ return d.Cmp(d0) == 0
+}
+
+func checkLehmerExtGcd(aBytes, bBytes []byte) bool {
+ a := new(Int).SetBytes(aBytes)
+ b := new(Int).SetBytes(bBytes)
+ x := new(Int)
+ y := new(Int)
+
+ if a.Sign() <= 0 || b.Sign() <= 0 {
+ return true // can only test positive arguments
+ }
+
+ d := new(Int).lehmerGCD(x, y, a, b)
+ d0, x0, y0 := euclidExtGCD(a, b)
+
+ return d.Cmp(d0) == 0 && x.Cmp(x0) == 0 && y.Cmp(y0) == 0
+}
+
+var gcdTests = []struct {
+ d, x, y, a, b string
+}{
+ // a <= 0 || b <= 0
+ {"0", "0", "0", "0", "0"},
+ {"7", "0", "1", "0", "7"},
+ {"7", "0", "-1", "0", "-7"},
+ {"11", "1", "0", "11", "0"},
+ {"7", "-1", "-2", "-77", "35"},
+ {"935", "-3", "8", "64515", "24310"},
+ {"935", "-3", "-8", "64515", "-24310"},
+ {"935", "3", "-8", "-64515", "-24310"},
+
+ {"1", "-9", "47", "120", "23"},
+ {"7", "1", "-2", "77", "35"},
+ {"935", "-3", "8", "64515", "24310"},
+ {"935000000000000000", "-3", "8", "64515000000000000000", "24310000000000000000"},
+ {"1", "-221", "22059940471369027483332068679400581064239780177629666810348940098015901108344", "98920366548084643601728869055592650835572950932266967461790948584315647051443", "991"},
+}
+
+func testGcd(t *testing.T, d, x, y, a, b *Int) {
+ var X *Int
+ if x != nil {
+ X = new(Int)
+ }
+ var Y *Int
+ if y != nil {
+ Y = new(Int)
+ }
+
+ D := new(Int).GCD(X, Y, a, b)
+ if D.Cmp(d) != 0 {
+ t.Errorf("GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, D, d)
+ }
+ if x != nil && X.Cmp(x) != 0 {
+ t.Errorf("GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, X, x)
+ }
+ if y != nil && Y.Cmp(y) != 0 {
+ t.Errorf("GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, Y, y)
+ }
+
+ // check results in presence of aliasing (issue #11284)
+ a2 := new(Int).Set(a)
+ b2 := new(Int).Set(b)
+ a2.GCD(X, Y, a2, b2) // result is same as 1st argument
+ if a2.Cmp(d) != 0 {
+ t.Errorf("aliased z = a GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, a2, d)
+ }
+ if x != nil && X.Cmp(x) != 0 {
+ t.Errorf("aliased z = a GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, X, x)
+ }
+ if y != nil && Y.Cmp(y) != 0 {
+ t.Errorf("aliased z = a GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, Y, y)
+ }
+
+ a2 = new(Int).Set(a)
+ b2 = new(Int).Set(b)
+ b2.GCD(X, Y, a2, b2) // result is same as 2nd argument
+ if b2.Cmp(d) != 0 {
+ t.Errorf("aliased z = b GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, b2, d)
+ }
+ if x != nil && X.Cmp(x) != 0 {
+ t.Errorf("aliased z = b GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, X, x)
+ }
+ if y != nil && Y.Cmp(y) != 0 {
+ t.Errorf("aliased z = b GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, Y, y)
+ }
+
+ a2 = new(Int).Set(a)
+ b2 = new(Int).Set(b)
+ D = new(Int).GCD(a2, b2, a2, b2) // x = a, y = b
+ if D.Cmp(d) != 0 {
+ t.Errorf("aliased x = a, y = b GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, D, d)
+ }
+ if x != nil && a2.Cmp(x) != 0 {
+ t.Errorf("aliased x = a, y = b GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, a2, x)
+ }
+ if y != nil && b2.Cmp(y) != 0 {
+ t.Errorf("aliased x = a, y = b GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, b2, y)
+ }
+
+ a2 = new(Int).Set(a)
+ b2 = new(Int).Set(b)
+ D = new(Int).GCD(b2, a2, a2, b2) // x = b, y = a
+ if D.Cmp(d) != 0 {
+ t.Errorf("aliased x = b, y = a GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, D, d)
+ }
+ if x != nil && b2.Cmp(x) != 0 {
+ t.Errorf("aliased x = b, y = a GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, b2, x)
+ }
+ if y != nil && a2.Cmp(y) != 0 {
+ t.Errorf("aliased x = b, y = a GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, a2, y)
+ }
+}
+
+func TestGcd(t *testing.T) {
+ for _, test := range gcdTests {
+ d, _ := new(Int).SetString(test.d, 0)
+ x, _ := new(Int).SetString(test.x, 0)
+ y, _ := new(Int).SetString(test.y, 0)
+ a, _ := new(Int).SetString(test.a, 0)
+ b, _ := new(Int).SetString(test.b, 0)
+
+ testGcd(t, d, nil, nil, a, b)
+ testGcd(t, d, x, nil, a, b)
+ testGcd(t, d, nil, y, a, b)
+ testGcd(t, d, x, y, a, b)
+ }
+
+ if err := quick.Check(checkGcd, nil); err != nil {
+ t.Error(err)
+ }
+
+ if err := quick.Check(checkLehmerGcd, nil); err != nil {
+ t.Error(err)
+ }
+
+ if err := quick.Check(checkLehmerExtGcd, nil); err != nil {
+ t.Error(err)
+ }
+}
+
+type intShiftTest struct {
+ in string
+ shift uint
+ out string
+}
+
+var rshTests = []intShiftTest{
+ {"0", 0, "0"},
+ {"-0", 0, "0"},
+ {"0", 1, "0"},
+ {"0", 2, "0"},
+ {"1", 0, "1"},
+ {"1", 1, "0"},
+ {"1", 2, "0"},
+ {"2", 0, "2"},
+ {"2", 1, "1"},
+ {"-1", 0, "-1"},
+ {"-1", 1, "-1"},
+ {"-1", 10, "-1"},
+ {"-100", 2, "-25"},
+ {"-100", 3, "-13"},
+ {"-100", 100, "-1"},
+ {"4294967296", 0, "4294967296"},
+ {"4294967296", 1, "2147483648"},
+ {"4294967296", 2, "1073741824"},
+ {"18446744073709551616", 0, "18446744073709551616"},
+ {"18446744073709551616", 1, "9223372036854775808"},
+ {"18446744073709551616", 2, "4611686018427387904"},
+ {"18446744073709551616", 64, "1"},
+ {"340282366920938463463374607431768211456", 64, "18446744073709551616"},
+ {"340282366920938463463374607431768211456", 128, "1"},
+}
+
+func TestRsh(t *testing.T) {
+ for i, test := range rshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ out := new(Int).Rsh(in, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if out.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, expected)
+ }
+ }
+}
+
+func TestRshSelf(t *testing.T) {
+ for i, test := range rshTests {
+ z, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ z.Rsh(z, test.shift)
+
+ if !isNormalized(z) {
+ t.Errorf("#%d: %v is not normalized", i, *z)
+ }
+ if z.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, z, expected)
+ }
+ }
+}
+
+var lshTests = []intShiftTest{
+ {"0", 0, "0"},
+ {"0", 1, "0"},
+ {"0", 2, "0"},
+ {"1", 0, "1"},
+ {"1", 1, "2"},
+ {"1", 2, "4"},
+ {"2", 0, "2"},
+ {"2", 1, "4"},
+ {"2", 2, "8"},
+ {"-87", 1, "-174"},
+ {"4294967296", 0, "4294967296"},
+ {"4294967296", 1, "8589934592"},
+ {"4294967296", 2, "17179869184"},
+ {"18446744073709551616", 0, "18446744073709551616"},
+ {"9223372036854775808", 1, "18446744073709551616"},
+ {"4611686018427387904", 2, "18446744073709551616"},
+ {"1", 64, "18446744073709551616"},
+ {"18446744073709551616", 64, "340282366920938463463374607431768211456"},
+ {"1", 128, "340282366920938463463374607431768211456"},
+}
+
+func TestLsh(t *testing.T) {
+ for i, test := range lshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ out := new(Int).Lsh(in, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if out.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, expected)
+ }
+ }
+}
+
+func TestLshSelf(t *testing.T) {
+ for i, test := range lshTests {
+ z, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ z.Lsh(z, test.shift)
+
+ if !isNormalized(z) {
+ t.Errorf("#%d: %v is not normalized", i, *z)
+ }
+ if z.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, z, expected)
+ }
+ }
+}
+
+func TestLshRsh(t *testing.T) {
+ for i, test := range rshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ out := new(Int).Lsh(in, test.shift)
+ out = out.Rsh(out, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if in.Cmp(out) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, in)
+ }
+ }
+ for i, test := range lshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ out := new(Int).Lsh(in, test.shift)
+ out.Rsh(out, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if in.Cmp(out) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, in)
+ }
+ }
+}
+
+// Entries must be sorted by value in ascending order.
+var cmpAbsTests = []string{
+ "0",
+ "1",
+ "2",
+ "10",
+ "10000000",
+ "2783678367462374683678456387645876387564783686583485",
+ "2783678367462374683678456387645876387564783686583486",
+ "32957394867987420967976567076075976570670947609750670956097509670576075067076027578341538",
+}
+
+func TestCmpAbs(t *testing.T) {
+ values := make([]*Int, len(cmpAbsTests))
+ var prev *Int
+ for i, s := range cmpAbsTests {
+ x, ok := new(Int).SetString(s, 0)
+ if !ok {
+ t.Fatalf("SetString(%s, 0) failed", s)
+ }
+ if prev != nil && prev.Cmp(x) >= 0 {
+ t.Fatal("cmpAbsTests entries not sorted in ascending order")
+ }
+ values[i] = x
+ prev = x
+ }
+
+ for i, x := range values {
+ for j, y := range values {
+ // try all combinations of signs for x, y
+ for k := 0; k < 4; k++ {
+ var a, b Int
+ a.Set(x)
+ b.Set(y)
+ if k&1 != 0 {
+ a.Neg(&a)
+ }
+ if k&2 != 0 {
+ b.Neg(&b)
+ }
+
+ got := a.CmpAbs(&b)
+ want := 0
+ switch {
+ case i > j:
+ want = 1
+ case i < j:
+ want = -1
+ }
+ if got != want {
+ t.Errorf("absCmp |%s|, |%s|: got %d; want %d", &a, &b, got, want)
+ }
+ }
+ }
+ }
+}
+
+func TestIntCmpSelf(t *testing.T) {
+ for _, s := range cmpAbsTests {
+ x, ok := new(Int).SetString(s, 0)
+ if !ok {
+ t.Fatalf("SetString(%s, 0) failed", s)
+ }
+ got := x.Cmp(x)
+ want := 0
+ if got != want {
+ t.Errorf("x = %s: x.Cmp(x): got %d; want %d", x, got, want)
+ }
+ }
+}
+
+var int64Tests = []string{
+ // int64
+ "0",
+ "1",
+ "-1",
+ "4294967295",
+ "-4294967295",
+ "4294967296",
+ "-4294967296",
+ "9223372036854775807",
+ "-9223372036854775807",
+ "-9223372036854775808",
+
+ // not int64
+ "0x8000000000000000",
+ "-0x8000000000000001",
+ "38579843757496759476987459679745",
+ "-38579843757496759476987459679745",
+}
+
+func TestInt64(t *testing.T) {
+ for _, s := range int64Tests {
+ var x Int
+ _, ok := x.SetString(s, 0)
+ if !ok {
+ t.Errorf("SetString(%s, 0) failed", s)
+ continue
+ }
+
+ want, err := strconv.ParseInt(s, 0, 64)
+ if err != nil {
+ if err.(*strconv.NumError).Err == strconv.ErrRange {
+ if x.IsInt64() {
+ t.Errorf("IsInt64(%s) succeeded unexpectedly", s)
+ }
+ } else {
+ t.Errorf("ParseInt(%s) failed", s)
+ }
+ continue
+ }
+
+ if !x.IsInt64() {
+ t.Errorf("IsInt64(%s) failed unexpectedly", s)
+ }
+
+ got := x.Int64()
+ if got != want {
+ t.Errorf("Int64(%s) = %d; want %d", s, got, want)
+ }
+ }
+}
+
+var uint64Tests = []string{
+ // uint64
+ "0",
+ "1",
+ "4294967295",
+ "4294967296",
+ "8589934591",
+ "8589934592",
+ "9223372036854775807",
+ "9223372036854775808",
+ "0x08000000000000000",
+
+ // not uint64
+ "0x10000000000000000",
+ "-0x08000000000000000",
+ "-1",
+}
+
+func TestUint64(t *testing.T) {
+ for _, s := range uint64Tests {
+ var x Int
+ _, ok := x.SetString(s, 0)
+ if !ok {
+ t.Errorf("SetString(%s, 0) failed", s)
+ continue
+ }
+
+ want, err := strconv.ParseUint(s, 0, 64)
+ if err != nil {
+ // check for sign explicitly (ErrRange doesn't cover signed input)
+ if s[0] == '-' || err.(*strconv.NumError).Err == strconv.ErrRange {
+ if x.IsUint64() {
+ t.Errorf("IsUint64(%s) succeeded unexpectedly", s)
+ }
+ } else {
+ t.Errorf("ParseUint(%s) failed", s)
+ }
+ continue
+ }
+
+ if !x.IsUint64() {
+ t.Errorf("IsUint64(%s) failed unexpectedly", s)
+ }
+
+ got := x.Uint64()
+ if got != want {
+ t.Errorf("Uint64(%s) = %d; want %d", s, got, want)
+ }
+ }
+}
+
+var bitwiseTests = []struct {
+ x, y string
+ and, or, xor, andNot string
+}{
+ {"0x00", "0x00", "0x00", "0x00", "0x00", "0x00"},
+ {"0x00", "0x01", "0x00", "0x01", "0x01", "0x00"},
+ {"0x01", "0x00", "0x00", "0x01", "0x01", "0x01"},
+ {"-0x01", "0x00", "0x00", "-0x01", "-0x01", "-0x01"},
+ {"-0xaf", "-0x50", "-0xf0", "-0x0f", "0xe1", "0x41"},
+ {"0x00", "-0x01", "0x00", "-0x01", "-0x01", "0x00"},
+ {"0x01", "0x01", "0x01", "0x01", "0x00", "0x00"},
+ {"-0x01", "-0x01", "-0x01", "-0x01", "0x00", "0x00"},
+ {"0x07", "0x08", "0x00", "0x0f", "0x0f", "0x07"},
+ {"0x05", "0x0f", "0x05", "0x0f", "0x0a", "0x00"},
+ {"0xff", "-0x0a", "0xf6", "-0x01", "-0xf7", "0x09"},
+ {"0x013ff6", "0x9a4e", "0x1a46", "0x01bffe", "0x01a5b8", "0x0125b0"},
+ {"-0x013ff6", "0x9a4e", "0x800a", "-0x0125b2", "-0x01a5bc", "-0x01c000"},
+ {"-0x013ff6", "-0x9a4e", "-0x01bffe", "-0x1a46", "0x01a5b8", "0x8008"},
+ {
+ "0x1000009dc6e3d9822cba04129bcbe3401",
+ "0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+ "0x1000001186210100001000009048c2001",
+ "0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
+ "0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
+ "0x8c40c2d8822caa04120b8321400",
+ },
+ {
+ "0x1000009dc6e3d9822cba04129bcbe3401",
+ "-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+ "0x8c40c2d8822caa04120b8321401",
+ "-0xb9bd7d543685789d57ca918e82229142459020483cd2014001fd",
+ "-0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fe",
+ "0x1000001186210100001000009048c2000",
+ },
+ {
+ "-0x1000009dc6e3d9822cba04129bcbe3401",
+ "-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+ "-0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
+ "-0x1000001186210100001000009048c2001",
+ "0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
+ "0xb9bd7d543685789d57ca918e82229142459020483cd2014001fc",
+ },
+}
+
+type bitFun func(z, x, y *Int) *Int
+
+func testBitFun(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
+ expected := new(Int)
+ expected.SetString(exp, 0)
+
+ out := f(new(Int), x, y)
+ if out.Cmp(expected) != 0 {
+ t.Errorf("%s: got %s want %s", msg, out, expected)
+ }
+}
+
+func testBitFunSelf(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
+ self := new(Int)
+ self.Set(x)
+ expected := new(Int)
+ expected.SetString(exp, 0)
+
+ self = f(self, self, y)
+ if self.Cmp(expected) != 0 {
+ t.Errorf("%s: got %s want %s", msg, self, expected)
+ }
+}
+
+func altBit(x *Int, i int) uint {
+ z := new(Int).Rsh(x, uint(i))
+ z = z.And(z, NewInt(1))
+ if z.Cmp(new(Int)) != 0 {
+ return 1
+ }
+ return 0
+}
+
+func altSetBit(z *Int, x *Int, i int, b uint) *Int {
+ one := NewInt(1)
+ m := one.Lsh(one, uint(i))
+ switch b {
+ case 1:
+ return z.Or(x, m)
+ case 0:
+ return z.AndNot(x, m)
+ }
+ panic("set bit is not 0 or 1")
+}
+
+func testBitset(t *testing.T, x *Int) {
+ n := x.BitLen()
+ z := new(Int).Set(x)
+ z1 := new(Int).Set(x)
+ for i := 0; i < n+10; i++ {
+ old := z.Bit(i)
+ old1 := altBit(z1, i)
+ if old != old1 {
+ t.Errorf("bitset: inconsistent value for Bit(%s, %d), got %v want %v", z1, i, old, old1)
+ }
+ z := new(Int).SetBit(z, i, 1)
+ z1 := altSetBit(new(Int), z1, i, 1)
+ if z.Bit(i) == 0 {
+ t.Errorf("bitset: bit %d of %s got 0 want 1", i, x)
+ }
+ if z.Cmp(z1) != 0 {
+ t.Errorf("bitset: inconsistent value after SetBit 1, got %s want %s", z, z1)
+ }
+ z.SetBit(z, i, 0)
+ altSetBit(z1, z1, i, 0)
+ if z.Bit(i) != 0 {
+ t.Errorf("bitset: bit %d of %s got 1 want 0", i, x)
+ }
+ if z.Cmp(z1) != 0 {
+ t.Errorf("bitset: inconsistent value after SetBit 0, got %s want %s", z, z1)
+ }
+ altSetBit(z1, z1, i, old)
+ z.SetBit(z, i, old)
+ if z.Cmp(z1) != 0 {
+ t.Errorf("bitset: inconsistent value after SetBit old, got %s want %s", z, z1)
+ }
+ }
+ if z.Cmp(x) != 0 {
+ t.Errorf("bitset: got %s want %s", z, x)
+ }
+}
+
+var bitsetTests = []struct {
+ x string
+ i int
+ b uint
+}{
+ {"0", 0, 0},
+ {"0", 200, 0},
+ {"1", 0, 1},
+ {"1", 1, 0},
+ {"-1", 0, 1},
+ {"-1", 200, 1},
+ {"0x2000000000000000000000000000", 108, 0},
+ {"0x2000000000000000000000000000", 109, 1},
+ {"0x2000000000000000000000000000", 110, 0},
+ {"-0x2000000000000000000000000001", 108, 1},
+ {"-0x2000000000000000000000000001", 109, 0},
+ {"-0x2000000000000000000000000001", 110, 1},
+}
+
+func TestBitSet(t *testing.T) {
+ for _, test := range bitwiseTests {
+ x := new(Int)
+ x.SetString(test.x, 0)
+ testBitset(t, x)
+ x = new(Int)
+ x.SetString(test.y, 0)
+ testBitset(t, x)
+ }
+ for i, test := range bitsetTests {
+ x := new(Int)
+ x.SetString(test.x, 0)
+ b := x.Bit(test.i)
+ if b != test.b {
+ t.Errorf("#%d got %v want %v", i, b, test.b)
+ }
+ }
+ z := NewInt(1)
+ z.SetBit(NewInt(0), 2, 1)
+ if z.Cmp(NewInt(4)) != 0 {
+ t.Errorf("destination leaked into result; got %s want 4", z)
+ }
+}
+
+var tzbTests = []struct {
+ in string
+ out uint
+}{
+ {"0", 0},
+ {"1", 0},
+ {"-1", 0},
+ {"4", 2},
+ {"-8", 3},
+ {"0x4000000000000000000", 74},
+ {"-0x8000000000000000000", 75},
+}
+
+func TestTrailingZeroBits(t *testing.T) {
+ for i, test := range tzbTests {
+ in, _ := new(Int).SetString(test.in, 0)
+ want := test.out
+ got := in.TrailingZeroBits()
+
+ if got != want {
+ t.Errorf("#%d: got %v want %v", i, got, want)
+ }
+ }
+}
+
+func BenchmarkBitset(b *testing.B) {
+ z := new(Int)
+ z.SetBit(z, 512, 1)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ z.SetBit(z, i&512, 1)
+ }
+}
+
+func BenchmarkBitsetNeg(b *testing.B) {
+ z := NewInt(-1)
+ z.SetBit(z, 512, 0)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ z.SetBit(z, i&512, 0)
+ }
+}
+
+func BenchmarkBitsetOrig(b *testing.B) {
+ z := new(Int)
+ altSetBit(z, z, 512, 1)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ altSetBit(z, z, i&512, 1)
+ }
+}
+
+func BenchmarkBitsetNegOrig(b *testing.B) {
+ z := NewInt(-1)
+ altSetBit(z, z, 512, 0)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ altSetBit(z, z, i&512, 0)
+ }
+}
+
+// tri generates the trinomial 2**(n*2) - 2**n - 1, which is always 3 mod 4 and
+// 7 mod 8, so that 2 is always a quadratic residue.
+func tri(n uint) *Int {
+ x := NewInt(1)
+ x.Lsh(x, n)
+ x2 := new(Int).Lsh(x, n)
+ x2.Sub(x2, x)
+ x2.Sub(x2, intOne)
+ return x2
+}
+
+func BenchmarkModSqrt225_Tonelli(b *testing.B) {
+ p := tri(225)
+ x := NewInt(2)
+ for i := 0; i < b.N; i++ {
+ x.SetUint64(2)
+ x.modSqrtTonelliShanks(x, p)
+ }
+}
+
+func BenchmarkModSqrt225_3Mod4(b *testing.B) {
+ p := tri(225)
+ x := new(Int).SetUint64(2)
+ for i := 0; i < b.N; i++ {
+ x.SetUint64(2)
+ x.modSqrt3Mod4Prime(x, p)
+ }
+}
+
+func BenchmarkModSqrt231_Tonelli(b *testing.B) {
+ p := tri(231)
+ p.Sub(p, intOne)
+ p.Sub(p, intOne) // tri(231) - 2 is a prime == 5 mod 8
+ x := new(Int).SetUint64(7)
+ for i := 0; i < b.N; i++ {
+ x.SetUint64(7)
+ x.modSqrtTonelliShanks(x, p)
+ }
+}
+
+func BenchmarkModSqrt231_5Mod8(b *testing.B) {
+ p := tri(231)
+ p.Sub(p, intOne)
+ p.Sub(p, intOne) // tri(231) - 2 is a prime == 5 mod 8
+ x := new(Int).SetUint64(7)
+ for i := 0; i < b.N; i++ {
+ x.SetUint64(7)
+ x.modSqrt5Mod8Prime(x, p)
+ }
+}
+
+func TestBitwise(t *testing.T) {
+ x := new(Int)
+ y := new(Int)
+ for _, test := range bitwiseTests {
+ x.SetString(test.x, 0)
+ y.SetString(test.y, 0)
+
+ testBitFun(t, "and", (*Int).And, x, y, test.and)
+ testBitFunSelf(t, "and", (*Int).And, x, y, test.and)
+ testBitFun(t, "andNot", (*Int).AndNot, x, y, test.andNot)
+ testBitFunSelf(t, "andNot", (*Int).AndNot, x, y, test.andNot)
+ testBitFun(t, "or", (*Int).Or, x, y, test.or)
+ testBitFunSelf(t, "or", (*Int).Or, x, y, test.or)
+ testBitFun(t, "xor", (*Int).Xor, x, y, test.xor)
+ testBitFunSelf(t, "xor", (*Int).Xor, x, y, test.xor)
+ }
+}
+
+var notTests = []struct {
+ in string
+ out string
+}{
+ {"0", "-1"},
+ {"1", "-2"},
+ {"7", "-8"},
+ {"0", "-1"},
+ {"-81910", "81909"},
+ {
+ "298472983472983471903246121093472394872319615612417471234712061",
+ "-298472983472983471903246121093472394872319615612417471234712062",
+ },
+}
+
+func TestNot(t *testing.T) {
+ in := new(Int)
+ out := new(Int)
+ expected := new(Int)
+ for i, test := range notTests {
+ in.SetString(test.in, 10)
+ expected.SetString(test.out, 10)
+ out = out.Not(in)
+ if out.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, expected)
+ }
+ out = out.Not(out)
+ if out.Cmp(in) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, in)
+ }
+ }
+}
+
+var modInverseTests = []struct {
+ element string
+ modulus string
+}{
+ {"1234567", "458948883992"},
+ {"239487239847", "2410312426921032588552076022197566074856950548502459942654116941958108831682612228890093858261341614673227141477904012196503648957050582631942730706805009223062734745341073406696246014589361659774041027169249453200378729434170325843778659198143763193776859869524088940195577346119843545301547043747207749969763750084308926339295559968882457872412993810129130294592999947926365264059284647209730384947211681434464714438488520940127459844288859336526896320919633919"},
+ {"-10", "13"}, // issue #16984
+ {"10", "-13"},
+ {"-17", "-13"},
+}
+
+func TestModInverse(t *testing.T) {
+ var element, modulus, gcd, inverse Int
+ one := NewInt(1)
+ for _, test := range modInverseTests {
+ (&element).SetString(test.element, 10)
+ (&modulus).SetString(test.modulus, 10)
+ (&inverse).ModInverse(&element, &modulus)
+ (&inverse).Mul(&inverse, &element)
+ (&inverse).Mod(&inverse, &modulus)
+ if (&inverse).Cmp(one) != 0 {
+ t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
+ }
+ }
+ // exhaustive test for small values
+ for n := 2; n < 100; n++ {
+ (&modulus).SetInt64(int64(n))
+ for x := 1; x < n; x++ {
+ (&element).SetInt64(int64(x))
+ (&gcd).GCD(nil, nil, &element, &modulus)
+ if (&gcd).Cmp(one) != 0 {
+ continue
+ }
+ (&inverse).ModInverse(&element, &modulus)
+ (&inverse).Mul(&inverse, &element)
+ (&inverse).Mod(&inverse, &modulus)
+ if (&inverse).Cmp(one) != 0 {
+ t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
+ }
+ }
+ }
+}
+
+func BenchmarkModInverse(b *testing.B) {
+ p := new(Int).SetInt64(1) // Mersenne prime 2**1279 -1
+ p.abs = p.abs.shl(p.abs, 1279)
+ p.Sub(p, intOne)
+ x := new(Int).Sub(p, intOne)
+ z := new(Int)
+ for i := 0; i < b.N; i++ {
+ z.ModInverse(x, p)
+ }
+}
+
+// testModSqrt is a helper for TestModSqrt,
+// which checks that ModSqrt can compute a square-root of elt^2.
+func testModSqrt(t *testing.T, elt, mod, sq, sqrt *Int) bool {
+ var sqChk, sqrtChk, sqrtsq Int
+ sq.Mul(elt, elt)
+ sq.Mod(sq, mod)
+ z := sqrt.ModSqrt(sq, mod)
+ if z != sqrt {
+ t.Errorf("ModSqrt returned wrong value %s", z)
+ }
+
+ // test ModSqrt arguments outside the range [0,mod)
+ sqChk.Add(sq, mod)
+ z = sqrtChk.ModSqrt(&sqChk, mod)
+ if z != &sqrtChk || z.Cmp(sqrt) != 0 {
+ t.Errorf("ModSqrt returned inconsistent value %s", z)
+ }
+ sqChk.Sub(sq, mod)
+ z = sqrtChk.ModSqrt(&sqChk, mod)
+ if z != &sqrtChk || z.Cmp(sqrt) != 0 {
+ t.Errorf("ModSqrt returned inconsistent value %s", z)
+ }
+
+ // test x aliasing z
+ z = sqrtChk.ModSqrt(sqrtChk.Set(sq), mod)
+ if z != &sqrtChk || z.Cmp(sqrt) != 0 {
+ t.Errorf("ModSqrt returned inconsistent value %s", z)
+ }
+
+ // make sure we actually got a square root
+ if sqrt.Cmp(elt) == 0 {
+ return true // we found the "desired" square root
+ }
+ sqrtsq.Mul(sqrt, sqrt) // make sure we found the "other" one
+ sqrtsq.Mod(&sqrtsq, mod)
+ return sq.Cmp(&sqrtsq) == 0
+}
+
+func TestModSqrt(t *testing.T) {
+ var elt, mod, modx4, sq, sqrt Int
+ r := rand.New(rand.NewSource(9))
+ for i, s := range primes[1:] { // skip 2, use only odd primes
+ mod.SetString(s, 10)
+ modx4.Lsh(&mod, 2)
+
+ // test a few random elements per prime
+ for x := 1; x < 5; x++ {
+ elt.Rand(r, &modx4)
+ elt.Sub(&elt, &mod) // test range [-mod, 3*mod)
+ if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
+ t.Errorf("#%d: failed (sqrt(e) = %s)", i, &sqrt)
+ }
+ }
+
+ if testing.Short() && i > 2 {
+ break
+ }
+ }
+
+ if testing.Short() {
+ return
+ }
+
+ // exhaustive test for small values
+ for n := 3; n < 100; n++ {
+ mod.SetInt64(int64(n))
+ if !mod.ProbablyPrime(10) {
+ continue
+ }
+ isSquare := make([]bool, n)
+
+ // test all the squares
+ for x := 1; x < n; x++ {
+ elt.SetInt64(int64(x))
+ if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
+ t.Errorf("#%d: failed (sqrt(%d,%d) = %s)", x, &elt, &mod, &sqrt)
+ }
+ isSquare[sq.Uint64()] = true
+ }
+
+ // test all non-squares
+ for x := 1; x < n; x++ {
+ sq.SetInt64(int64(x))
+ z := sqrt.ModSqrt(&sq, &mod)
+ if !isSquare[x] && z != nil {
+ t.Errorf("#%d: failed (sqrt(%d,%d) = nil)", x, &sqrt, &mod)
+ }
+ }
+ }
+}
+
+func TestJacobi(t *testing.T) {
+ testCases := []struct {
+ x, y int64
+ result int
+ }{
+ {0, 1, 1},
+ {0, -1, 1},
+ {1, 1, 1},
+ {1, -1, 1},
+ {0, 5, 0},
+ {1, 5, 1},
+ {2, 5, -1},
+ {-2, 5, -1},
+ {2, -5, -1},
+ {-2, -5, 1},
+ {3, 5, -1},
+ {5, 5, 0},
+ {-5, 5, 0},
+ {6, 5, 1},
+ {6, -5, 1},
+ {-6, 5, 1},
+ {-6, -5, -1},
+ }
+
+ var x, y Int
+
+ for i, test := range testCases {
+ x.SetInt64(test.x)
+ y.SetInt64(test.y)
+ expected := test.result
+ actual := Jacobi(&x, &y)
+ if actual != expected {
+ t.Errorf("#%d: Jacobi(%d, %d) = %d, but expected %d", i, test.x, test.y, actual, expected)
+ }
+ }
+}
+
+func TestJacobiPanic(t *testing.T) {
+ const failureMsg = "test failure"
+ defer func() {
+ msg := recover()
+ if msg == nil || msg == failureMsg {
+ panic(msg)
+ }
+ t.Log(msg)
+ }()
+ x := NewInt(1)
+ y := NewInt(2)
+ // Jacobi should panic when the second argument is even.
+ Jacobi(x, y)
+ panic(failureMsg)
+}
+
+func TestIssue2607(t *testing.T) {
+ // This code sequence used to hang.
+ n := NewInt(10)
+ n.Rand(rand.New(rand.NewSource(9)), n)
+}
+
+func TestSqrt(t *testing.T) {
+ root := 0
+ r := new(Int)
+ for i := 0; i < 10000; i++ {
+ if (root+1)*(root+1) <= i {
+ root++
+ }
+ n := NewInt(int64(i))
+ r.SetInt64(-2)
+ r.Sqrt(n)
+ if r.Cmp(NewInt(int64(root))) != 0 {
+ t.Errorf("Sqrt(%v) = %v, want %v", n, r, root)
+ }
+ }
+
+ for i := 0; i < 1000; i += 10 {
+ n, _ := new(Int).SetString("1"+strings.Repeat("0", i), 10)
+ r := new(Int).Sqrt(n)
+ root, _ := new(Int).SetString("1"+strings.Repeat("0", i/2), 10)
+ if r.Cmp(root) != 0 {
+ t.Errorf("Sqrt(1e%d) = %v, want 1e%d", i, r, i/2)
+ }
+ }
+
+ // Test aliasing.
+ r.SetInt64(100)
+ r.Sqrt(r)
+ if r.Int64() != 10 {
+ t.Errorf("Sqrt(100) = %v, want 10 (aliased output)", r.Int64())
+ }
+}
+
+// We can't test this together with the other Exp tests above because
+// it requires a different receiver setup.
+func TestIssue22830(t *testing.T) {
+ one := new(Int).SetInt64(1)
+ base, _ := new(Int).SetString("84555555300000000000", 10)
+ mod, _ := new(Int).SetString("66666670001111111111", 10)
+ want, _ := new(Int).SetString("17888885298888888889", 10)
+
+ var tests = []int64{
+ 0, 1, -1,
+ }
+
+ for _, n := range tests {
+ m := NewInt(n)
+ if got := m.Exp(base, one, mod); got.Cmp(want) != 0 {
+ t.Errorf("(%v).Exp(%s, 1, %s) = %s, want %s", n, base, mod, got, want)
+ }
+ }
+}
+
+func BenchmarkSqrt(b *testing.B) {
+ n, _ := new(Int).SetString("1"+strings.Repeat("0", 1001), 10)
+ b.ResetTimer()
+ t := new(Int)
+ for i := 0; i < b.N; i++ {
+ t.Sqrt(n)
+ }
+}
+
+func benchmarkIntSqr(b *testing.B, nwords int) {
+ x := new(Int)
+ x.abs = rndNat(nwords)
+ t := new(Int)
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ t.Mul(x, x)
+ }
+}
+
+func BenchmarkIntSqr(b *testing.B) {
+ for _, n := range sqrBenchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
+ benchmarkIntSqr(b, n)
+ })
+ }
+}
+
+func benchmarkDiv(b *testing.B, aSize, bSize int) {
+ var r = rand.New(rand.NewSource(1234))
+ aa := randInt(r, uint(aSize))
+ bb := randInt(r, uint(bSize))
+ if aa.Cmp(bb) < 0 {
+ aa, bb = bb, aa
+ }
+ x := new(Int)
+ y := new(Int)
+
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ x.DivMod(aa, bb, y)
+ }
+}
+
+func BenchmarkDiv(b *testing.B) {
+ sizes := []int{
+ 10, 20, 50, 100, 200, 500, 1000,
+ 1e4, 1e5, 1e6, 1e7,
+ }
+ for _, i := range sizes {
+ j := 2 * i
+ b.Run(fmt.Sprintf("%d/%d", j, i), func(b *testing.B) {
+ benchmarkDiv(b, j, i)
+ })
+ }
+}
+
+func TestFillBytes(t *testing.T) {
+ checkResult := func(t *testing.T, buf []byte, want *Int) {
+ t.Helper()
+ got := new(Int).SetBytes(buf)
+ if got.CmpAbs(want) != 0 {
+ t.Errorf("got 0x%x, want 0x%x: %x", got, want, buf)
+ }
+ }
+ panics := func(f func()) (panic bool) {
+ defer func() { panic = recover() != nil }()
+ f()
+ return
+ }
+
+ for _, n := range []string{
+ "0",
+ "1000",
+ "0xffffffff",
+ "-0xffffffff",
+ "0xffffffffffffffff",
+ "0x10000000000000000",
+ "0xabababababababababababababababababababababababababa",
+ "0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+ } {
+ t.Run(n, func(t *testing.T) {
+ t.Logf(n)
+ x, ok := new(Int).SetString(n, 0)
+ if !ok {
+ panic("invalid test entry")
+ }
+
+ // Perfectly sized buffer.
+ byteLen := (x.BitLen() + 7) / 8
+ buf := make([]byte, byteLen)
+ checkResult(t, x.FillBytes(buf), x)
+
+ // Way larger, checking all bytes get zeroed.
+ buf = make([]byte, 100)
+ for i := range buf {
+ buf[i] = 0xff
+ }
+ checkResult(t, x.FillBytes(buf), x)
+
+ // Too small.
+ if byteLen > 0 {
+ buf = make([]byte, byteLen-1)
+ if !panics(func() { x.FillBytes(buf) }) {
+ t.Errorf("expected panic for small buffer and value %x", x)
+ }
+ }
+ })
+ }
+}
diff --git a/src/math/big/intconv.go b/src/math/big/intconv.go
new file mode 100644
index 0000000..0567284
--- /dev/null
+++ b/src/math/big/intconv.go
@@ -0,0 +1,257 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements int-to-string conversion functions.
+
+package big
+
+import (
+ "errors"
+ "fmt"
+ "io"
+)
+
+// Text returns the string representation of x in the given base.
+// Base must be between 2 and 62, inclusive. The result uses the
+// lower-case letters 'a' to 'z' for digit values 10 to 35, and
+// the upper-case letters 'A' to 'Z' for digit values 36 to 61.
+// No prefix (such as "0x") is added to the string. If x is a nil
+// pointer it returns "<nil>".
+func (x *Int) Text(base int) string {
+ if x == nil {
+ return "<nil>"
+ }
+ return string(x.abs.itoa(x.neg, base))
+}
+
+// Append appends the string representation of x, as generated by
+// x.Text(base), to buf and returns the extended buffer.
+func (x *Int) Append(buf []byte, base int) []byte {
+ if x == nil {
+ return append(buf, "<nil>"...)
+ }
+ return append(buf, x.abs.itoa(x.neg, base)...)
+}
+
+// String returns the decimal representation of x as generated by
+// x.Text(10).
+func (x *Int) String() string {
+ return x.Text(10)
+}
+
+// write count copies of text to s
+func writeMultiple(s fmt.State, text string, count int) {
+ if len(text) > 0 {
+ b := []byte(text)
+ for ; count > 0; count-- {
+ s.Write(b)
+ }
+ }
+}
+
+var _ fmt.Formatter = intOne // *Int must implement fmt.Formatter
+
+// Format implements fmt.Formatter. It accepts the formats
+// 'b' (binary), 'o' (octal with 0 prefix), 'O' (octal with 0o prefix),
+// 'd' (decimal), 'x' (lowercase hexadecimal), and
+// 'X' (uppercase hexadecimal).
+// Also supported are the full suite of package fmt's format
+// flags for integral types, including '+' and ' ' for sign
+// control, '#' for leading zero in octal and for hexadecimal,
+// a leading "0x" or "0X" for "%#x" and "%#X" respectively,
+// specification of minimum digits precision, output field
+// width, space or zero padding, and '-' for left or right
+// justification.
+//
+func (x *Int) Format(s fmt.State, ch rune) {
+ // determine base
+ var base int
+ switch ch {
+ case 'b':
+ base = 2
+ case 'o', 'O':
+ base = 8
+ case 'd', 's', 'v':
+ base = 10
+ case 'x', 'X':
+ base = 16
+ default:
+ // unknown format
+ fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())
+ return
+ }
+
+ if x == nil {
+ fmt.Fprint(s, "<nil>")
+ return
+ }
+
+ // determine sign character
+ sign := ""
+ switch {
+ case x.neg:
+ sign = "-"
+ case s.Flag('+'): // supersedes ' ' when both specified
+ sign = "+"
+ case s.Flag(' '):
+ sign = " "
+ }
+
+ // determine prefix characters for indicating output base
+ prefix := ""
+ if s.Flag('#') {
+ switch ch {
+ case 'b': // binary
+ prefix = "0b"
+ case 'o': // octal
+ prefix = "0"
+ case 'x': // hexadecimal
+ prefix = "0x"
+ case 'X':
+ prefix = "0X"
+ }
+ }
+ if ch == 'O' {
+ prefix = "0o"
+ }
+
+ digits := x.abs.utoa(base)
+ if ch == 'X' {
+ // faster than bytes.ToUpper
+ for i, d := range digits {
+ if 'a' <= d && d <= 'z' {
+ digits[i] = 'A' + (d - 'a')
+ }
+ }
+ }
+
+ // number of characters for the three classes of number padding
+ var left int // space characters to left of digits for right justification ("%8d")
+ var zeros int // zero characters (actually cs[0]) as left-most digits ("%.8d")
+ var right int // space characters to right of digits for left justification ("%-8d")
+
+ // determine number padding from precision: the least number of digits to output
+ precision, precisionSet := s.Precision()
+ if precisionSet {
+ switch {
+ case len(digits) < precision:
+ zeros = precision - len(digits) // count of zero padding
+ case len(digits) == 1 && digits[0] == '0' && precision == 0:
+ return // print nothing if zero value (x == 0) and zero precision ("." or ".0")
+ }
+ }
+
+ // determine field pad from width: the least number of characters to output
+ length := len(sign) + len(prefix) + zeros + len(digits)
+ if width, widthSet := s.Width(); widthSet && length < width { // pad as specified
+ switch d := width - length; {
+ case s.Flag('-'):
+ // pad on the right with spaces; supersedes '0' when both specified
+ right = d
+ case s.Flag('0') && !precisionSet:
+ // pad with zeros unless precision also specified
+ zeros = d
+ default:
+ // pad on the left with spaces
+ left = d
+ }
+ }
+
+ // print number as [left pad][sign][prefix][zero pad][digits][right pad]
+ writeMultiple(s, " ", left)
+ writeMultiple(s, sign, 1)
+ writeMultiple(s, prefix, 1)
+ writeMultiple(s, "0", zeros)
+ s.Write(digits)
+ writeMultiple(s, " ", right)
+}
+
+// scan sets z to the integer value corresponding to the longest possible prefix
+// read from r representing a signed integer number in a given conversion base.
+// It returns z, the actual conversion base used, and an error, if any. In the
+// error case, the value of z is undefined but the returned value is nil. The
+// syntax follows the syntax of integer literals in Go.
+//
+// The base argument must be 0 or a value from 2 through MaxBase. If the base
+// is 0, the string prefix determines the actual conversion base. A prefix of
+// ``0b'' or ``0B'' selects base 2; a ``0'', ``0o'', or ``0O'' prefix selects
+// base 8, and a ``0x'' or ``0X'' prefix selects base 16. Otherwise the selected
+// base is 10.
+//
+func (z *Int) scan(r io.ByteScanner, base int) (*Int, int, error) {
+ // determine sign
+ neg, err := scanSign(r)
+ if err != nil {
+ return nil, 0, err
+ }
+
+ // determine mantissa
+ z.abs, base, _, err = z.abs.scan(r, base, false)
+ if err != nil {
+ return nil, base, err
+ }
+ z.neg = len(z.abs) > 0 && neg // 0 has no sign
+
+ return z, base, nil
+}
+
+func scanSign(r io.ByteScanner) (neg bool, err error) {
+ var ch byte
+ if ch, err = r.ReadByte(); err != nil {
+ return false, err
+ }
+ switch ch {
+ case '-':
+ neg = true
+ case '+':
+ // nothing to do
+ default:
+ r.UnreadByte()
+ }
+ return
+}
+
+// byteReader is a local wrapper around fmt.ScanState;
+// it implements the ByteReader interface.
+type byteReader struct {
+ fmt.ScanState
+}
+
+func (r byteReader) ReadByte() (byte, error) {
+ ch, size, err := r.ReadRune()
+ if size != 1 && err == nil {
+ err = fmt.Errorf("invalid rune %#U", ch)
+ }
+ return byte(ch), err
+}
+
+func (r byteReader) UnreadByte() error {
+ return r.UnreadRune()
+}
+
+var _ fmt.Scanner = intOne // *Int must implement fmt.Scanner
+
+// Scan is a support routine for fmt.Scanner; it sets z to the value of
+// the scanned number. It accepts the formats 'b' (binary), 'o' (octal),
+// 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
+func (z *Int) Scan(s fmt.ScanState, ch rune) error {
+ s.SkipSpace() // skip leading space characters
+ base := 0
+ switch ch {
+ case 'b':
+ base = 2
+ case 'o':
+ base = 8
+ case 'd':
+ base = 10
+ case 'x', 'X':
+ base = 16
+ case 's', 'v':
+ // let scan determine the base
+ default:
+ return errors.New("Int.Scan: invalid verb")
+ }
+ _, _, err := z.scan(byteReader{s}, base)
+ return err
+}
diff --git a/src/math/big/intconv_test.go b/src/math/big/intconv_test.go
new file mode 100644
index 0000000..5ba2926
--- /dev/null
+++ b/src/math/big/intconv_test.go
@@ -0,0 +1,431 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "fmt"
+ "testing"
+)
+
+var stringTests = []struct {
+ in string
+ out string
+ base int
+ val int64
+ ok bool
+}{
+ // invalid inputs
+ {in: ""},
+ {in: "a"},
+ {in: "z"},
+ {in: "+"},
+ {in: "-"},
+ {in: "0b"},
+ {in: "0o"},
+ {in: "0x"},
+ {in: "0y"},
+ {in: "2", base: 2},
+ {in: "0b2", base: 0},
+ {in: "08"},
+ {in: "8", base: 8},
+ {in: "0xg", base: 0},
+ {in: "g", base: 16},
+
+ // invalid inputs with separators
+ // (smoke tests only - a comprehensive set of tests is in natconv_test.go)
+ {in: "_"},
+ {in: "0_"},
+ {in: "_0"},
+ {in: "-1__0"},
+ {in: "0x10_"},
+ {in: "1_000", base: 10}, // separators are not permitted for bases != 0
+ {in: "d_e_a_d", base: 16},
+
+ // valid inputs
+ {"0", "0", 0, 0, true},
+ {"0", "0", 10, 0, true},
+ {"0", "0", 16, 0, true},
+ {"+0", "0", 0, 0, true},
+ {"-0", "0", 0, 0, true},
+ {"10", "10", 0, 10, true},
+ {"10", "10", 10, 10, true},
+ {"10", "10", 16, 16, true},
+ {"-10", "-10", 16, -16, true},
+ {"+10", "10", 16, 16, true},
+ {"0b10", "2", 0, 2, true},
+ {"0o10", "8", 0, 8, true},
+ {"0x10", "16", 0, 16, true},
+ {in: "0x10", base: 16},
+ {"-0x10", "-16", 0, -16, true},
+ {"+0x10", "16", 0, 16, true},
+ {"00", "0", 0, 0, true},
+ {"0", "0", 8, 0, true},
+ {"07", "7", 0, 7, true},
+ {"7", "7", 8, 7, true},
+ {"023", "19", 0, 19, true},
+ {"23", "23", 8, 19, true},
+ {"cafebabe", "cafebabe", 16, 0xcafebabe, true},
+ {"0b0", "0", 0, 0, true},
+ {"-111", "-111", 2, -7, true},
+ {"-0b111", "-7", 0, -7, true},
+ {"0b1001010111", "599", 0, 0x257, true},
+ {"1001010111", "1001010111", 2, 0x257, true},
+ {"A", "a", 36, 10, true},
+ {"A", "A", 37, 36, true},
+ {"ABCXYZ", "abcxyz", 36, 623741435, true},
+ {"ABCXYZ", "ABCXYZ", 62, 33536793425, true},
+
+ // valid input with separators
+ // (smoke tests only - a comprehensive set of tests is in natconv_test.go)
+ {"1_000", "1000", 0, 1000, true},
+ {"0b_1010", "10", 0, 10, true},
+ {"+0o_660", "432", 0, 0660, true},
+ {"-0xF00D_1E", "-15731998", 0, -0xf00d1e, true},
+}
+
+func TestIntText(t *testing.T) {
+ z := new(Int)
+ for _, test := range stringTests {
+ if !test.ok {
+ continue
+ }
+
+ _, ok := z.SetString(test.in, test.base)
+ if !ok {
+ t.Errorf("%v: failed to parse", test)
+ continue
+ }
+
+ base := test.base
+ if base == 0 {
+ base = 10
+ }
+
+ if got := z.Text(base); got != test.out {
+ t.Errorf("%v: got %s; want %s", test, got, test.out)
+ }
+ }
+}
+
+func TestAppendText(t *testing.T) {
+ z := new(Int)
+ var buf []byte
+ for _, test := range stringTests {
+ if !test.ok {
+ continue
+ }
+
+ _, ok := z.SetString(test.in, test.base)
+ if !ok {
+ t.Errorf("%v: failed to parse", test)
+ continue
+ }
+
+ base := test.base
+ if base == 0 {
+ base = 10
+ }
+
+ i := len(buf)
+ buf = z.Append(buf, base)
+ if got := string(buf[i:]); got != test.out {
+ t.Errorf("%v: got %s; want %s", test, got, test.out)
+ }
+ }
+}
+
+func format(base int) string {
+ switch base {
+ case 2:
+ return "%b"
+ case 8:
+ return "%o"
+ case 16:
+ return "%x"
+ }
+ return "%d"
+}
+
+func TestGetString(t *testing.T) {
+ z := new(Int)
+ for i, test := range stringTests {
+ if !test.ok {
+ continue
+ }
+ z.SetInt64(test.val)
+
+ if test.base == 10 {
+ if got := z.String(); got != test.out {
+ t.Errorf("#%da got %s; want %s", i, got, test.out)
+ }
+ }
+
+ f := format(test.base)
+ got := fmt.Sprintf(f, z)
+ if f == "%d" {
+ if got != fmt.Sprintf("%d", test.val) {
+ t.Errorf("#%db got %s; want %d", i, got, test.val)
+ }
+ } else {
+ if got != test.out {
+ t.Errorf("#%dc got %s; want %s", i, got, test.out)
+ }
+ }
+ }
+}
+
+func TestSetString(t *testing.T) {
+ tmp := new(Int)
+ for i, test := range stringTests {
+ // initialize to a non-zero value so that issues with parsing
+ // 0 are detected
+ tmp.SetInt64(1234567890)
+ n1, ok1 := new(Int).SetString(test.in, test.base)
+ n2, ok2 := tmp.SetString(test.in, test.base)
+ expected := NewInt(test.val)
+ if ok1 != test.ok || ok2 != test.ok {
+ t.Errorf("#%d (input '%s') ok incorrect (should be %t)", i, test.in, test.ok)
+ continue
+ }
+ if !ok1 {
+ if n1 != nil {
+ t.Errorf("#%d (input '%s') n1 != nil", i, test.in)
+ }
+ continue
+ }
+ if !ok2 {
+ if n2 != nil {
+ t.Errorf("#%d (input '%s') n2 != nil", i, test.in)
+ }
+ continue
+ }
+
+ if ok1 && !isNormalized(n1) {
+ t.Errorf("#%d (input '%s'): %v is not normalized", i, test.in, *n1)
+ }
+ if ok2 && !isNormalized(n2) {
+ t.Errorf("#%d (input '%s'): %v is not normalized", i, test.in, *n2)
+ }
+
+ if n1.Cmp(expected) != 0 {
+ t.Errorf("#%d (input '%s') got: %s want: %d", i, test.in, n1, test.val)
+ }
+ if n2.Cmp(expected) != 0 {
+ t.Errorf("#%d (input '%s') got: %s want: %d", i, test.in, n2, test.val)
+ }
+ }
+}
+
+var formatTests = []struct {
+ input string
+ format string
+ output string
+}{
+ {"<nil>", "%x", "<nil>"},
+ {"<nil>", "%#x", "<nil>"},
+ {"<nil>", "%#y", "%!y(big.Int=<nil>)"},
+
+ {"10", "%b", "1010"},
+ {"10", "%o", "12"},
+ {"10", "%d", "10"},
+ {"10", "%v", "10"},
+ {"10", "%x", "a"},
+ {"10", "%X", "A"},
+ {"-10", "%X", "-A"},
+ {"10", "%y", "%!y(big.Int=10)"},
+ {"-10", "%y", "%!y(big.Int=-10)"},
+
+ {"10", "%#b", "0b1010"},
+ {"10", "%#o", "012"},
+ {"10", "%O", "0o12"},
+ {"-10", "%#b", "-0b1010"},
+ {"-10", "%#o", "-012"},
+ {"-10", "%O", "-0o12"},
+ {"10", "%#d", "10"},
+ {"10", "%#v", "10"},
+ {"10", "%#x", "0xa"},
+ {"10", "%#X", "0XA"},
+ {"-10", "%#X", "-0XA"},
+ {"10", "%#y", "%!y(big.Int=10)"},
+ {"-10", "%#y", "%!y(big.Int=-10)"},
+
+ {"1234", "%d", "1234"},
+ {"1234", "%3d", "1234"},
+ {"1234", "%4d", "1234"},
+ {"-1234", "%d", "-1234"},
+ {"1234", "% 5d", " 1234"},
+ {"1234", "%+5d", "+1234"},
+ {"1234", "%-5d", "1234 "},
+ {"1234", "%x", "4d2"},
+ {"1234", "%X", "4D2"},
+ {"-1234", "%3x", "-4d2"},
+ {"-1234", "%4x", "-4d2"},
+ {"-1234", "%5x", " -4d2"},
+ {"-1234", "%-5x", "-4d2 "},
+ {"1234", "%03d", "1234"},
+ {"1234", "%04d", "1234"},
+ {"1234", "%05d", "01234"},
+ {"1234", "%06d", "001234"},
+ {"-1234", "%06d", "-01234"},
+ {"1234", "%+06d", "+01234"},
+ {"1234", "% 06d", " 01234"},
+ {"1234", "%-6d", "1234 "},
+ {"1234", "%-06d", "1234 "},
+ {"-1234", "%-06d", "-1234 "},
+
+ {"1234", "%.3d", "1234"},
+ {"1234", "%.4d", "1234"},
+ {"1234", "%.5d", "01234"},
+ {"1234", "%.6d", "001234"},
+ {"-1234", "%.3d", "-1234"},
+ {"-1234", "%.4d", "-1234"},
+ {"-1234", "%.5d", "-01234"},
+ {"-1234", "%.6d", "-001234"},
+
+ {"1234", "%8.3d", " 1234"},
+ {"1234", "%8.4d", " 1234"},
+ {"1234", "%8.5d", " 01234"},
+ {"1234", "%8.6d", " 001234"},
+ {"-1234", "%8.3d", " -1234"},
+ {"-1234", "%8.4d", " -1234"},
+ {"-1234", "%8.5d", " -01234"},
+ {"-1234", "%8.6d", " -001234"},
+
+ {"1234", "%+8.3d", " +1234"},
+ {"1234", "%+8.4d", " +1234"},
+ {"1234", "%+8.5d", " +01234"},
+ {"1234", "%+8.6d", " +001234"},
+ {"-1234", "%+8.3d", " -1234"},
+ {"-1234", "%+8.4d", " -1234"},
+ {"-1234", "%+8.5d", " -01234"},
+ {"-1234", "%+8.6d", " -001234"},
+
+ {"1234", "% 8.3d", " 1234"},
+ {"1234", "% 8.4d", " 1234"},
+ {"1234", "% 8.5d", " 01234"},
+ {"1234", "% 8.6d", " 001234"},
+ {"-1234", "% 8.3d", " -1234"},
+ {"-1234", "% 8.4d", " -1234"},
+ {"-1234", "% 8.5d", " -01234"},
+ {"-1234", "% 8.6d", " -001234"},
+
+ {"1234", "%.3x", "4d2"},
+ {"1234", "%.4x", "04d2"},
+ {"1234", "%.5x", "004d2"},
+ {"1234", "%.6x", "0004d2"},
+ {"-1234", "%.3x", "-4d2"},
+ {"-1234", "%.4x", "-04d2"},
+ {"-1234", "%.5x", "-004d2"},
+ {"-1234", "%.6x", "-0004d2"},
+
+ {"1234", "%8.3x", " 4d2"},
+ {"1234", "%8.4x", " 04d2"},
+ {"1234", "%8.5x", " 004d2"},
+ {"1234", "%8.6x", " 0004d2"},
+ {"-1234", "%8.3x", " -4d2"},
+ {"-1234", "%8.4x", " -04d2"},
+ {"-1234", "%8.5x", " -004d2"},
+ {"-1234", "%8.6x", " -0004d2"},
+
+ {"1234", "%+8.3x", " +4d2"},
+ {"1234", "%+8.4x", " +04d2"},
+ {"1234", "%+8.5x", " +004d2"},
+ {"1234", "%+8.6x", " +0004d2"},
+ {"-1234", "%+8.3x", " -4d2"},
+ {"-1234", "%+8.4x", " -04d2"},
+ {"-1234", "%+8.5x", " -004d2"},
+ {"-1234", "%+8.6x", " -0004d2"},
+
+ {"1234", "% 8.3x", " 4d2"},
+ {"1234", "% 8.4x", " 04d2"},
+ {"1234", "% 8.5x", " 004d2"},
+ {"1234", "% 8.6x", " 0004d2"},
+ {"1234", "% 8.7x", " 00004d2"},
+ {"1234", "% 8.8x", " 000004d2"},
+ {"-1234", "% 8.3x", " -4d2"},
+ {"-1234", "% 8.4x", " -04d2"},
+ {"-1234", "% 8.5x", " -004d2"},
+ {"-1234", "% 8.6x", " -0004d2"},
+ {"-1234", "% 8.7x", "-00004d2"},
+ {"-1234", "% 8.8x", "-000004d2"},
+
+ {"1234", "%-8.3d", "1234 "},
+ {"1234", "%-8.4d", "1234 "},
+ {"1234", "%-8.5d", "01234 "},
+ {"1234", "%-8.6d", "001234 "},
+ {"1234", "%-8.7d", "0001234 "},
+ {"1234", "%-8.8d", "00001234"},
+ {"-1234", "%-8.3d", "-1234 "},
+ {"-1234", "%-8.4d", "-1234 "},
+ {"-1234", "%-8.5d", "-01234 "},
+ {"-1234", "%-8.6d", "-001234 "},
+ {"-1234", "%-8.7d", "-0001234"},
+ {"-1234", "%-8.8d", "-00001234"},
+
+ {"16777215", "%b", "111111111111111111111111"}, // 2**24 - 1
+
+ {"0", "%.d", ""},
+ {"0", "%.0d", ""},
+ {"0", "%3.d", ""},
+}
+
+func TestFormat(t *testing.T) {
+ for i, test := range formatTests {
+ var x *Int
+ if test.input != "<nil>" {
+ var ok bool
+ x, ok = new(Int).SetString(test.input, 0)
+ if !ok {
+ t.Errorf("#%d failed reading input %s", i, test.input)
+ }
+ }
+ output := fmt.Sprintf(test.format, x)
+ if output != test.output {
+ t.Errorf("#%d got %q; want %q, {%q, %q, %q}", i, output, test.output, test.input, test.format, test.output)
+ }
+ }
+}
+
+var scanTests = []struct {
+ input string
+ format string
+ output string
+ remaining int
+}{
+ {"1010", "%b", "10", 0},
+ {"0b1010", "%v", "10", 0},
+ {"12", "%o", "10", 0},
+ {"012", "%v", "10", 0},
+ {"10", "%d", "10", 0},
+ {"10", "%v", "10", 0},
+ {"a", "%x", "10", 0},
+ {"0xa", "%v", "10", 0},
+ {"A", "%X", "10", 0},
+ {"-A", "%X", "-10", 0},
+ {"+0b1011001", "%v", "89", 0},
+ {"0xA", "%v", "10", 0},
+ {"0 ", "%v", "0", 1},
+ {"2+3", "%v", "2", 2},
+ {"0XABC 12", "%v", "2748", 3},
+}
+
+func TestScan(t *testing.T) {
+ var buf bytes.Buffer
+ for i, test := range scanTests {
+ x := new(Int)
+ buf.Reset()
+ buf.WriteString(test.input)
+ if _, err := fmt.Fscanf(&buf, test.format, x); err != nil {
+ t.Errorf("#%d error: %s", i, err)
+ }
+ if x.String() != test.output {
+ t.Errorf("#%d got %s; want %s", i, x.String(), test.output)
+ }
+ if buf.Len() != test.remaining {
+ t.Errorf("#%d got %d bytes remaining; want %d", i, buf.Len(), test.remaining)
+ }
+ }
+}
diff --git a/src/math/big/intmarsh.go b/src/math/big/intmarsh.go
new file mode 100644
index 0000000..c1422e2
--- /dev/null
+++ b/src/math/big/intmarsh.go
@@ -0,0 +1,80 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements encoding/decoding of Ints.
+
+package big
+
+import (
+ "bytes"
+ "fmt"
+)
+
+// Gob codec version. Permits backward-compatible changes to the encoding.
+const intGobVersion byte = 1
+
+// GobEncode implements the gob.GobEncoder interface.
+func (x *Int) GobEncode() ([]byte, error) {
+ if x == nil {
+ return nil, nil
+ }
+ buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
+ i := x.abs.bytes(buf) - 1 // i >= 0
+ b := intGobVersion << 1 // make space for sign bit
+ if x.neg {
+ b |= 1
+ }
+ buf[i] = b
+ return buf[i:], nil
+}
+
+// GobDecode implements the gob.GobDecoder interface.
+func (z *Int) GobDecode(buf []byte) error {
+ if len(buf) == 0 {
+ // Other side sent a nil or default value.
+ *z = Int{}
+ return nil
+ }
+ b := buf[0]
+ if b>>1 != intGobVersion {
+ return fmt.Errorf("Int.GobDecode: encoding version %d not supported", b>>1)
+ }
+ z.neg = b&1 != 0
+ z.abs = z.abs.setBytes(buf[1:])
+ return nil
+}
+
+// MarshalText implements the encoding.TextMarshaler interface.
+func (x *Int) MarshalText() (text []byte, err error) {
+ if x == nil {
+ return []byte("<nil>"), nil
+ }
+ return x.abs.itoa(x.neg, 10), nil
+}
+
+// UnmarshalText implements the encoding.TextUnmarshaler interface.
+func (z *Int) UnmarshalText(text []byte) error {
+ if _, ok := z.setFromScanner(bytes.NewReader(text), 0); !ok {
+ return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
+ }
+ return nil
+}
+
+// The JSON marshalers are only here for API backward compatibility
+// (programs that explicitly look for these two methods). JSON works
+// fine with the TextMarshaler only.
+
+// MarshalJSON implements the json.Marshaler interface.
+func (x *Int) MarshalJSON() ([]byte, error) {
+ return x.MarshalText()
+}
+
+// UnmarshalJSON implements the json.Unmarshaler interface.
+func (z *Int) UnmarshalJSON(text []byte) error {
+ // Ignore null, like in the main JSON package.
+ if string(text) == "null" {
+ return nil
+ }
+ return z.UnmarshalText(text)
+}
diff --git a/src/math/big/intmarsh_test.go b/src/math/big/intmarsh_test.go
new file mode 100644
index 0000000..f82956c
--- /dev/null
+++ b/src/math/big/intmarsh_test.go
@@ -0,0 +1,121 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "encoding/gob"
+ "encoding/json"
+ "encoding/xml"
+ "testing"
+)
+
+var encodingTests = []string{
+ "0",
+ "1",
+ "2",
+ "10",
+ "1000",
+ "1234567890",
+ "298472983472983471903246121093472394872319615612417471234712061",
+}
+
+func TestIntGobEncoding(t *testing.T) {
+ var medium bytes.Buffer
+ enc := gob.NewEncoder(&medium)
+ dec := gob.NewDecoder(&medium)
+ for _, test := range encodingTests {
+ for _, sign := range []string{"", "+", "-"} {
+ x := sign + test
+ medium.Reset() // empty buffer for each test case (in case of failures)
+ var tx Int
+ tx.SetString(x, 10)
+ if err := enc.Encode(&tx); err != nil {
+ t.Errorf("encoding of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Int
+ if err := dec.Decode(&rx); err != nil {
+ t.Errorf("decoding of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("transmission of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+ }
+}
+
+// Sending a nil Int pointer (inside a slice) on a round trip through gob should yield a zero.
+// TODO: top-level nils.
+func TestGobEncodingNilIntInSlice(t *testing.T) {
+ buf := new(bytes.Buffer)
+ enc := gob.NewEncoder(buf)
+ dec := gob.NewDecoder(buf)
+
+ var in = make([]*Int, 1)
+ err := enc.Encode(&in)
+ if err != nil {
+ t.Errorf("gob encode failed: %q", err)
+ }
+ var out []*Int
+ err = dec.Decode(&out)
+ if err != nil {
+ t.Fatalf("gob decode failed: %q", err)
+ }
+ if len(out) != 1 {
+ t.Fatalf("wrong len; want 1 got %d", len(out))
+ }
+ var zero Int
+ if out[0].Cmp(&zero) != 0 {
+ t.Fatalf("transmission of (*Int)(nil) failed: got %s want 0", out)
+ }
+}
+
+func TestIntJSONEncoding(t *testing.T) {
+ for _, test := range encodingTests {
+ for _, sign := range []string{"", "+", "-"} {
+ x := sign + test
+ var tx Int
+ tx.SetString(x, 10)
+ b, err := json.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Int
+ if err := json.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+ }
+}
+
+func TestIntXMLEncoding(t *testing.T) {
+ for _, test := range encodingTests {
+ for _, sign := range []string{"", "+", "-"} {
+ x := sign + test
+ var tx Int
+ tx.SetString(x, 0)
+ b, err := xml.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Int
+ if err := xml.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("XML encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+ }
+}
diff --git a/src/math/big/link_test.go b/src/math/big/link_test.go
new file mode 100644
index 0000000..42f9cef
--- /dev/null
+++ b/src/math/big/link_test.go
@@ -0,0 +1,63 @@
+// Copyright 2020 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "internal/testenv"
+ "os"
+ "os/exec"
+ "path/filepath"
+ "testing"
+)
+
+// Tests that the linker is able to remove references to Float, Rat,
+// and Int if unused (notably, not used by init).
+func TestLinkerGC(t *testing.T) {
+ if testing.Short() {
+ t.Skip("skipping in short mode")
+ }
+ t.Parallel()
+ tmp := t.TempDir()
+ goBin := testenv.GoToolPath(t)
+ goFile := filepath.Join(tmp, "x.go")
+ file := []byte(`package main
+import _ "math/big"
+func main() {}
+`)
+ if err := os.WriteFile(goFile, file, 0644); err != nil {
+ t.Fatal(err)
+ }
+ cmd := exec.Command(goBin, "build", "-o", "x.exe", "x.go")
+ cmd.Dir = tmp
+ if out, err := cmd.CombinedOutput(); err != nil {
+ t.Fatalf("compile: %v, %s", err, out)
+ }
+
+ cmd = exec.Command(goBin, "tool", "nm", "x.exe")
+ cmd.Dir = tmp
+ nm, err := cmd.CombinedOutput()
+ if err != nil {
+ t.Fatalf("nm: %v, %s", err, nm)
+ }
+ const want = "runtime.(*Frames).Next"
+ if !bytes.Contains(nm, []byte(want)) {
+ // Test the test.
+ t.Errorf("expected symbol %q not found", want)
+ }
+ bad := []string{
+ "math/big.(*Float)",
+ "math/big.(*Rat)",
+ "math/big.(*Int)",
+ }
+ for _, sym := range bad {
+ if bytes.Contains(nm, []byte(sym)) {
+ t.Errorf("unexpected symbol %q found", sym)
+ }
+ }
+ if t.Failed() {
+ t.Logf("Got: %s", nm)
+ }
+}
diff --git a/src/math/big/nat.go b/src/math/big/nat.go
new file mode 100644
index 0000000..bbd6c88
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+++ b/src/math/big/nat.go
@@ -0,0 +1,1569 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements unsigned multi-precision integers (natural
+// numbers). They are the building blocks for the implementation
+// of signed integers, rationals, and floating-point numbers.
+//
+// Caution: This implementation relies on the function "alias"
+// which assumes that (nat) slice capacities are never
+// changed (no 3-operand slice expressions). If that
+// changes, alias needs to be updated for correctness.
+
+package big
+
+import (
+ "encoding/binary"
+ "math/bits"
+ "math/rand"
+ "sync"
+)
+
+// An unsigned integer x of the form
+//
+// x = x[n-1]*_B^(n-1) + x[n-2]*_B^(n-2) + ... + x[1]*_B + x[0]
+//
+// with 0 <= x[i] < _B and 0 <= i < n is stored in a slice of length n,
+// with the digits x[i] as the slice elements.
+//
+// A number is normalized if the slice contains no leading 0 digits.
+// During arithmetic operations, denormalized values may occur but are
+// always normalized before returning the final result. The normalized
+// representation of 0 is the empty or nil slice (length = 0).
+//
+type nat []Word
+
+var (
+ natOne = nat{1}
+ natTwo = nat{2}
+ natFive = nat{5}
+ natTen = nat{10}
+)
+
+func (z nat) clear() {
+ for i := range z {
+ z[i] = 0
+ }
+}
+
+func (z nat) norm() nat {
+ i := len(z)
+ for i > 0 && z[i-1] == 0 {
+ i--
+ }
+ return z[0:i]
+}
+
+func (z nat) make(n int) nat {
+ if n <= cap(z) {
+ return z[:n] // reuse z
+ }
+ if n == 1 {
+ // Most nats start small and stay that way; don't over-allocate.
+ return make(nat, 1)
+ }
+ // Choosing a good value for e has significant performance impact
+ // because it increases the chance that a value can be reused.
+ const e = 4 // extra capacity
+ return make(nat, n, n+e)
+}
+
+func (z nat) setWord(x Word) nat {
+ if x == 0 {
+ return z[:0]
+ }
+ z = z.make(1)
+ z[0] = x
+ return z
+}
+
+func (z nat) setUint64(x uint64) nat {
+ // single-word value
+ if w := Word(x); uint64(w) == x {
+ return z.setWord(w)
+ }
+ // 2-word value
+ z = z.make(2)
+ z[1] = Word(x >> 32)
+ z[0] = Word(x)
+ return z
+}
+
+func (z nat) set(x nat) nat {
+ z = z.make(len(x))
+ copy(z, x)
+ return z
+}
+
+func (z nat) add(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+
+ switch {
+ case m < n:
+ return z.add(y, x)
+ case m == 0:
+ // n == 0 because m >= n; result is 0
+ return z[:0]
+ case n == 0:
+ // result is x
+ return z.set(x)
+ }
+ // m > 0
+
+ z = z.make(m + 1)
+ c := addVV(z[0:n], x, y)
+ if m > n {
+ c = addVW(z[n:m], x[n:], c)
+ }
+ z[m] = c
+
+ return z.norm()
+}
+
+func (z nat) sub(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+
+ switch {
+ case m < n:
+ panic("underflow")
+ case m == 0:
+ // n == 0 because m >= n; result is 0
+ return z[:0]
+ case n == 0:
+ // result is x
+ return z.set(x)
+ }
+ // m > 0
+
+ z = z.make(m)
+ c := subVV(z[0:n], x, y)
+ if m > n {
+ c = subVW(z[n:], x[n:], c)
+ }
+ if c != 0 {
+ panic("underflow")
+ }
+
+ return z.norm()
+}
+
+func (x nat) cmp(y nat) (r int) {
+ m := len(x)
+ n := len(y)
+ if m != n || m == 0 {
+ switch {
+ case m < n:
+ r = -1
+ case m > n:
+ r = 1
+ }
+ return
+ }
+
+ i := m - 1
+ for i > 0 && x[i] == y[i] {
+ i--
+ }
+
+ switch {
+ case x[i] < y[i]:
+ r = -1
+ case x[i] > y[i]:
+ r = 1
+ }
+ return
+}
+
+func (z nat) mulAddWW(x nat, y, r Word) nat {
+ m := len(x)
+ if m == 0 || y == 0 {
+ return z.setWord(r) // result is r
+ }
+ // m > 0
+
+ z = z.make(m + 1)
+ z[m] = mulAddVWW(z[0:m], x, y, r)
+
+ return z.norm()
+}
+
+// basicMul multiplies x and y and leaves the result in z.
+// The (non-normalized) result is placed in z[0 : len(x) + len(y)].
+func basicMul(z, x, y nat) {
+ z[0 : len(x)+len(y)].clear() // initialize z
+ for i, d := range y {
+ if d != 0 {
+ z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
+ }
+ }
+}
+
+// montgomery computes z mod m = x*y*2**(-n*_W) mod m,
+// assuming k = -1/m mod 2**_W.
+// z is used for storing the result which is returned;
+// z must not alias x, y or m.
+// See Gueron, "Efficient Software Implementations of Modular Exponentiation".
+// https://eprint.iacr.org/2011/239.pdf
+// In the terminology of that paper, this is an "Almost Montgomery Multiplication":
+// x and y are required to satisfy 0 <= z < 2**(n*_W) and then the result
+// z is guaranteed to satisfy 0 <= z < 2**(n*_W), but it may not be < m.
+func (z nat) montgomery(x, y, m nat, k Word, n int) nat {
+ // This code assumes x, y, m are all the same length, n.
+ // (required by addMulVVW and the for loop).
+ // It also assumes that x, y are already reduced mod m,
+ // or else the result will not be properly reduced.
+ if len(x) != n || len(y) != n || len(m) != n {
+ panic("math/big: mismatched montgomery number lengths")
+ }
+ z = z.make(n * 2)
+ z.clear()
+ var c Word
+ for i := 0; i < n; i++ {
+ d := y[i]
+ c2 := addMulVVW(z[i:n+i], x, d)
+ t := z[i] * k
+ c3 := addMulVVW(z[i:n+i], m, t)
+ cx := c + c2
+ cy := cx + c3
+ z[n+i] = cy
+ if cx < c2 || cy < c3 {
+ c = 1
+ } else {
+ c = 0
+ }
+ }
+ if c != 0 {
+ subVV(z[:n], z[n:], m)
+ } else {
+ copy(z[:n], z[n:])
+ }
+ return z[:n]
+}
+
+// Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
+// Factored out for readability - do not use outside karatsuba.
+func karatsubaAdd(z, x nat, n int) {
+ if c := addVV(z[0:n], z, x); c != 0 {
+ addVW(z[n:n+n>>1], z[n:], c)
+ }
+}
+
+// Like karatsubaAdd, but does subtract.
+func karatsubaSub(z, x nat, n int) {
+ if c := subVV(z[0:n], z, x); c != 0 {
+ subVW(z[n:n+n>>1], z[n:], c)
+ }
+}
+
+// Operands that are shorter than karatsubaThreshold are multiplied using
+// "grade school" multiplication; for longer operands the Karatsuba algorithm
+// is used.
+var karatsubaThreshold = 40 // computed by calibrate_test.go
+
+// karatsuba multiplies x and y and leaves the result in z.
+// Both x and y must have the same length n and n must be a
+// power of 2. The result vector z must have len(z) >= 6*n.
+// The (non-normalized) result is placed in z[0 : 2*n].
+func karatsuba(z, x, y nat) {
+ n := len(y)
+
+ // Switch to basic multiplication if numbers are odd or small.
+ // (n is always even if karatsubaThreshold is even, but be
+ // conservative)
+ if n&1 != 0 || n < karatsubaThreshold || n < 2 {
+ basicMul(z, x, y)
+ return
+ }
+ // n&1 == 0 && n >= karatsubaThreshold && n >= 2
+
+ // Karatsuba multiplication is based on the observation that
+ // for two numbers x and y with:
+ //
+ // x = x1*b + x0
+ // y = y1*b + y0
+ //
+ // the product x*y can be obtained with 3 products z2, z1, z0
+ // instead of 4:
+ //
+ // x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0
+ // = z2*b*b + z1*b + z0
+ //
+ // with:
+ //
+ // xd = x1 - x0
+ // yd = y0 - y1
+ //
+ // z1 = xd*yd + z2 + z0
+ // = (x1-x0)*(y0 - y1) + z2 + z0
+ // = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z2 + z0
+ // = x1*y0 - z2 - z0 + x0*y1 + z2 + z0
+ // = x1*y0 + x0*y1
+
+ // split x, y into "digits"
+ n2 := n >> 1 // n2 >= 1
+ x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0
+ y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0
+
+ // z is used for the result and temporary storage:
+ //
+ // 6*n 5*n 4*n 3*n 2*n 1*n 0*n
+ // z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ]
+ //
+ // For each recursive call of karatsuba, an unused slice of
+ // z is passed in that has (at least) half the length of the
+ // caller's z.
+
+ // compute z0 and z2 with the result "in place" in z
+ karatsuba(z, x0, y0) // z0 = x0*y0
+ karatsuba(z[n:], x1, y1) // z2 = x1*y1
+
+ // compute xd (or the negative value if underflow occurs)
+ s := 1 // sign of product xd*yd
+ xd := z[2*n : 2*n+n2]
+ if subVV(xd, x1, x0) != 0 { // x1-x0
+ s = -s
+ subVV(xd, x0, x1) // x0-x1
+ }
+
+ // compute yd (or the negative value if underflow occurs)
+ yd := z[2*n+n2 : 3*n]
+ if subVV(yd, y0, y1) != 0 { // y0-y1
+ s = -s
+ subVV(yd, y1, y0) // y1-y0
+ }
+
+ // p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0
+ // p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0
+ p := z[n*3:]
+ karatsuba(p, xd, yd)
+
+ // save original z2:z0
+ // (ok to use upper half of z since we're done recursing)
+ r := z[n*4:]
+ copy(r, z[:n*2])
+
+ // add up all partial products
+ //
+ // 2*n n 0
+ // z = [ z2 | z0 ]
+ // + [ z0 ]
+ // + [ z2 ]
+ // + [ p ]
+ //
+ karatsubaAdd(z[n2:], r, n)
+ karatsubaAdd(z[n2:], r[n:], n)
+ if s > 0 {
+ karatsubaAdd(z[n2:], p, n)
+ } else {
+ karatsubaSub(z[n2:], p, n)
+ }
+}
+
+// alias reports whether x and y share the same base array.
+// Note: alias assumes that the capacity of underlying arrays
+// is never changed for nat values; i.e. that there are
+// no 3-operand slice expressions in this code (or worse,
+// reflect-based operations to the same effect).
+func alias(x, y nat) bool {
+ return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
+}
+
+// addAt implements z += x<<(_W*i); z must be long enough.
+// (we don't use nat.add because we need z to stay the same
+// slice, and we don't need to normalize z after each addition)
+func addAt(z, x nat, i int) {
+ if n := len(x); n > 0 {
+ if c := addVV(z[i:i+n], z[i:], x); c != 0 {
+ j := i + n
+ if j < len(z) {
+ addVW(z[j:], z[j:], c)
+ }
+ }
+ }
+}
+
+func max(x, y int) int {
+ if x > y {
+ return x
+ }
+ return y
+}
+
+// karatsubaLen computes an approximation to the maximum k <= n such that
+// k = p<<i for a number p <= threshold and an i >= 0. Thus, the
+// result is the largest number that can be divided repeatedly by 2 before
+// becoming about the value of threshold.
+func karatsubaLen(n, threshold int) int {
+ i := uint(0)
+ for n > threshold {
+ n >>= 1
+ i++
+ }
+ return n << i
+}
+
+func (z nat) mul(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+
+ switch {
+ case m < n:
+ return z.mul(y, x)
+ case m == 0 || n == 0:
+ return z[:0]
+ case n == 1:
+ return z.mulAddWW(x, y[0], 0)
+ }
+ // m >= n > 1
+
+ // determine if z can be reused
+ if alias(z, x) || alias(z, y) {
+ z = nil // z is an alias for x or y - cannot reuse
+ }
+
+ // use basic multiplication if the numbers are small
+ if n < karatsubaThreshold {
+ z = z.make(m + n)
+ basicMul(z, x, y)
+ return z.norm()
+ }
+ // m >= n && n >= karatsubaThreshold && n >= 2
+
+ // determine Karatsuba length k such that
+ //
+ // x = xh*b + x0 (0 <= x0 < b)
+ // y = yh*b + y0 (0 <= y0 < b)
+ // b = 1<<(_W*k) ("base" of digits xi, yi)
+ //
+ k := karatsubaLen(n, karatsubaThreshold)
+ // k <= n
+
+ // multiply x0 and y0 via Karatsuba
+ x0 := x[0:k] // x0 is not normalized
+ y0 := y[0:k] // y0 is not normalized
+ z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y
+ karatsuba(z, x0, y0)
+ z = z[0 : m+n] // z has final length but may be incomplete
+ z[2*k:].clear() // upper portion of z is garbage (and 2*k <= m+n since k <= n <= m)
+
+ // If xh != 0 or yh != 0, add the missing terms to z. For
+ //
+ // xh = xi*b^i + ... + x2*b^2 + x1*b (0 <= xi < b)
+ // yh = y1*b (0 <= y1 < b)
+ //
+ // the missing terms are
+ //
+ // x0*y1*b and xi*y0*b^i, xi*y1*b^(i+1) for i > 0
+ //
+ // since all the yi for i > 1 are 0 by choice of k: If any of them
+ // were > 0, then yh >= b^2 and thus y >= b^2. Then k' = k*2 would
+ // be a larger valid threshold contradicting the assumption about k.
+ //
+ if k < n || m != n {
+ tp := getNat(3 * k)
+ t := *tp
+
+ // add x0*y1*b
+ x0 := x0.norm()
+ y1 := y[k:] // y1 is normalized because y is
+ t = t.mul(x0, y1) // update t so we don't lose t's underlying array
+ addAt(z, t, k)
+
+ // add xi*y0<<i, xi*y1*b<<(i+k)
+ y0 := y0.norm()
+ for i := k; i < len(x); i += k {
+ xi := x[i:]
+ if len(xi) > k {
+ xi = xi[:k]
+ }
+ xi = xi.norm()
+ t = t.mul(xi, y0)
+ addAt(z, t, i)
+ t = t.mul(xi, y1)
+ addAt(z, t, i+k)
+ }
+
+ putNat(tp)
+ }
+
+ return z.norm()
+}
+
+// basicSqr sets z = x*x and is asymptotically faster than basicMul
+// by about a factor of 2, but slower for small arguments due to overhead.
+// Requirements: len(x) > 0, len(z) == 2*len(x)
+// The (non-normalized) result is placed in z.
+func basicSqr(z, x nat) {
+ n := len(x)
+ tp := getNat(2 * n)
+ t := *tp // temporary variable to hold the products
+ t.clear()
+ z[1], z[0] = mulWW(x[0], x[0]) // the initial square
+ for i := 1; i < n; i++ {
+ d := x[i]
+ // z collects the squares x[i] * x[i]
+ z[2*i+1], z[2*i] = mulWW(d, d)
+ // t collects the products x[i] * x[j] where j < i
+ t[2*i] = addMulVVW(t[i:2*i], x[0:i], d)
+ }
+ t[2*n-1] = shlVU(t[1:2*n-1], t[1:2*n-1], 1) // double the j < i products
+ addVV(z, z, t) // combine the result
+ putNat(tp)
+}
+
+// karatsubaSqr squares x and leaves the result in z.
+// len(x) must be a power of 2 and len(z) >= 6*len(x).
+// The (non-normalized) result is placed in z[0 : 2*len(x)].
+//
+// The algorithm and the layout of z are the same as for karatsuba.
+func karatsubaSqr(z, x nat) {
+ n := len(x)
+
+ if n&1 != 0 || n < karatsubaSqrThreshold || n < 2 {
+ basicSqr(z[:2*n], x)
+ return
+ }
+
+ n2 := n >> 1
+ x1, x0 := x[n2:], x[0:n2]
+
+ karatsubaSqr(z, x0)
+ karatsubaSqr(z[n:], x1)
+
+ // s = sign(xd*yd) == -1 for xd != 0; s == 1 for xd == 0
+ xd := z[2*n : 2*n+n2]
+ if subVV(xd, x1, x0) != 0 {
+ subVV(xd, x0, x1)
+ }
+
+ p := z[n*3:]
+ karatsubaSqr(p, xd)
+
+ r := z[n*4:]
+ copy(r, z[:n*2])
+
+ karatsubaAdd(z[n2:], r, n)
+ karatsubaAdd(z[n2:], r[n:], n)
+ karatsubaSub(z[n2:], p, n) // s == -1 for p != 0; s == 1 for p == 0
+}
+
+// Operands that are shorter than basicSqrThreshold are squared using
+// "grade school" multiplication; for operands longer than karatsubaSqrThreshold
+// we use the Karatsuba algorithm optimized for x == y.
+var basicSqrThreshold = 20 // computed by calibrate_test.go
+var karatsubaSqrThreshold = 260 // computed by calibrate_test.go
+
+// z = x*x
+func (z nat) sqr(x nat) nat {
+ n := len(x)
+ switch {
+ case n == 0:
+ return z[:0]
+ case n == 1:
+ d := x[0]
+ z = z.make(2)
+ z[1], z[0] = mulWW(d, d)
+ return z.norm()
+ }
+
+ if alias(z, x) {
+ z = nil // z is an alias for x - cannot reuse
+ }
+
+ if n < basicSqrThreshold {
+ z = z.make(2 * n)
+ basicMul(z, x, x)
+ return z.norm()
+ }
+ if n < karatsubaSqrThreshold {
+ z = z.make(2 * n)
+ basicSqr(z, x)
+ return z.norm()
+ }
+
+ // Use Karatsuba multiplication optimized for x == y.
+ // The algorithm and layout of z are the same as for mul.
+
+ // z = (x1*b + x0)^2 = x1^2*b^2 + 2*x1*x0*b + x0^2
+
+ k := karatsubaLen(n, karatsubaSqrThreshold)
+
+ x0 := x[0:k]
+ z = z.make(max(6*k, 2*n))
+ karatsubaSqr(z, x0) // z = x0^2
+ z = z[0 : 2*n]
+ z[2*k:].clear()
+
+ if k < n {
+ tp := getNat(2 * k)
+ t := *tp
+ x0 := x0.norm()
+ x1 := x[k:]
+ t = t.mul(x0, x1)
+ addAt(z, t, k)
+ addAt(z, t, k) // z = 2*x1*x0*b + x0^2
+ t = t.sqr(x1)
+ addAt(z, t, 2*k) // z = x1^2*b^2 + 2*x1*x0*b + x0^2
+ putNat(tp)
+ }
+
+ return z.norm()
+}
+
+// mulRange computes the product of all the unsigned integers in the
+// range [a, b] inclusively. If a > b (empty range), the result is 1.
+func (z nat) mulRange(a, b uint64) nat {
+ switch {
+ case a == 0:
+ // cut long ranges short (optimization)
+ return z.setUint64(0)
+ case a > b:
+ return z.setUint64(1)
+ case a == b:
+ return z.setUint64(a)
+ case a+1 == b:
+ return z.mul(nat(nil).setUint64(a), nat(nil).setUint64(b))
+ }
+ m := (a + b) / 2
+ return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
+}
+
+// q = (x-r)/y, with 0 <= r < y
+func (z nat) divW(x nat, y Word) (q nat, r Word) {
+ m := len(x)
+ switch {
+ case y == 0:
+ panic("division by zero")
+ case y == 1:
+ q = z.set(x) // result is x
+ return
+ case m == 0:
+ q = z[:0] // result is 0
+ return
+ }
+ // m > 0
+ z = z.make(m)
+ r = divWVW(z, 0, x, y)
+ q = z.norm()
+ return
+}
+
+func (z nat) div(z2, u, v nat) (q, r nat) {
+ if len(v) == 0 {
+ panic("division by zero")
+ }
+
+ if u.cmp(v) < 0 {
+ q = z[:0]
+ r = z2.set(u)
+ return
+ }
+
+ if len(v) == 1 {
+ var r2 Word
+ q, r2 = z.divW(u, v[0])
+ r = z2.setWord(r2)
+ return
+ }
+
+ q, r = z.divLarge(z2, u, v)
+ return
+}
+
+// getNat returns a *nat of len n. The contents may not be zero.
+// The pool holds *nat to avoid allocation when converting to interface{}.
+func getNat(n int) *nat {
+ var z *nat
+ if v := natPool.Get(); v != nil {
+ z = v.(*nat)
+ }
+ if z == nil {
+ z = new(nat)
+ }
+ *z = z.make(n)
+ return z
+}
+
+func putNat(x *nat) {
+ natPool.Put(x)
+}
+
+var natPool sync.Pool
+
+// q = (uIn-r)/vIn, with 0 <= r < vIn
+// Uses z as storage for q, and u as storage for r if possible.
+// See Knuth, Volume 2, section 4.3.1, Algorithm D.
+// Preconditions:
+// len(vIn) >= 2
+// len(uIn) >= len(vIn)
+// u must not alias z
+func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
+ n := len(vIn)
+ m := len(uIn) - n
+
+ // D1.
+ shift := nlz(vIn[n-1])
+ // do not modify vIn, it may be used by another goroutine simultaneously
+ vp := getNat(n)
+ v := *vp
+ shlVU(v, vIn, shift)
+
+ // u may safely alias uIn or vIn, the value of uIn is used to set u and vIn was already used
+ u = u.make(len(uIn) + 1)
+ u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
+
+ // z may safely alias uIn or vIn, both values were used already
+ if alias(z, u) {
+ z = nil // z is an alias for u - cannot reuse
+ }
+ q = z.make(m + 1)
+
+ if n < divRecursiveThreshold {
+ q.divBasic(u, v)
+ } else {
+ q.divRecursive(u, v)
+ }
+ putNat(vp)
+
+ q = q.norm()
+ shrVU(u, u, shift)
+ r = u.norm()
+
+ return q, r
+}
+
+// divBasic performs word-by-word division of u by v.
+// The quotient is written in pre-allocated q.
+// The remainder overwrites input u.
+//
+// Precondition:
+// - q is large enough to hold the quotient u / v
+// which has a maximum length of len(u)-len(v)+1.
+func (q nat) divBasic(u, v nat) {
+ n := len(v)
+ m := len(u) - n
+
+ qhatvp := getNat(n + 1)
+ qhatv := *qhatvp
+
+ // D2.
+ vn1 := v[n-1]
+ rec := reciprocalWord(vn1)
+ for j := m; j >= 0; j-- {
+ // D3.
+ qhat := Word(_M)
+ var ujn Word
+ if j+n < len(u) {
+ ujn = u[j+n]
+ }
+ if ujn != vn1 {
+ var rhat Word
+ qhat, rhat = divWW(ujn, u[j+n-1], vn1, rec)
+
+ // x1 | x2 = q̂v_{n-2}
+ vn2 := v[n-2]
+ x1, x2 := mulWW(qhat, vn2)
+ // test if q̂v_{n-2} > br̂ + u_{j+n-2}
+ ujn2 := u[j+n-2]
+ for greaterThan(x1, x2, rhat, ujn2) {
+ qhat--
+ prevRhat := rhat
+ rhat += vn1
+ // v[n-1] >= 0, so this tests for overflow.
+ if rhat < prevRhat {
+ break
+ }
+ x1, x2 = mulWW(qhat, vn2)
+ }
+ }
+
+ // D4.
+ // Compute the remainder u - (q̂*v) << (_W*j).
+ // The subtraction may overflow if q̂ estimate was off by one.
+ qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
+ qhl := len(qhatv)
+ if j+qhl > len(u) && qhatv[n] == 0 {
+ qhl--
+ }
+ c := subVV(u[j:j+qhl], u[j:], qhatv)
+ if c != 0 {
+ c := addVV(u[j:j+n], u[j:], v)
+ // If n == qhl, the carry from subVV and the carry from addVV
+ // cancel out and don't affect u[j+n].
+ if n < qhl {
+ u[j+n] += c
+ }
+ qhat--
+ }
+
+ if j == m && m == len(q) && qhat == 0 {
+ continue
+ }
+ q[j] = qhat
+ }
+
+ putNat(qhatvp)
+}
+
+const divRecursiveThreshold = 100
+
+// divRecursive performs word-by-word division of u by v.
+// The quotient is written in pre-allocated z.
+// The remainder overwrites input u.
+//
+// Precondition:
+// - len(z) >= len(u)-len(v)
+//
+// See Burnikel, Ziegler, "Fast Recursive Division", Algorithm 1 and 2.
+func (z nat) divRecursive(u, v nat) {
+ // Recursion depth is less than 2 log2(len(v))
+ // Allocate a slice of temporaries to be reused across recursion.
+ recDepth := 2 * bits.Len(uint(len(v)))
+ // large enough to perform Karatsuba on operands as large as v
+ tmp := getNat(3 * len(v))
+ temps := make([]*nat, recDepth)
+ z.clear()
+ z.divRecursiveStep(u, v, 0, tmp, temps)
+ for _, n := range temps {
+ if n != nil {
+ putNat(n)
+ }
+ }
+ putNat(tmp)
+}
+
+// divRecursiveStep computes the division of u by v.
+// - z must be large enough to hold the quotient
+// - the quotient will overwrite z
+// - the remainder will overwrite u
+func (z nat) divRecursiveStep(u, v nat, depth int, tmp *nat, temps []*nat) {
+ u = u.norm()
+ v = v.norm()
+
+ if len(u) == 0 {
+ z.clear()
+ return
+ }
+ n := len(v)
+ if n < divRecursiveThreshold {
+ z.divBasic(u, v)
+ return
+ }
+ m := len(u) - n
+ if m < 0 {
+ return
+ }
+
+ // Produce the quotient by blocks of B words.
+ // Division by v (length n) is done using a length n/2 division
+ // and a length n/2 multiplication for each block. The final
+ // complexity is driven by multiplication complexity.
+ B := n / 2
+
+ // Allocate a nat for qhat below.
+ if temps[depth] == nil {
+ temps[depth] = getNat(n)
+ } else {
+ *temps[depth] = temps[depth].make(B + 1)
+ }
+
+ j := m
+ for j > B {
+ // Divide u[j-B:j+n] by vIn. Keep remainder in u
+ // for next block.
+ //
+ // The following property will be used (Lemma 2):
+ // if u = u1 << s + u0
+ // v = v1 << s + v0
+ // then floor(u1/v1) >= floor(u/v)
+ //
+ // Moreover, the difference is at most 2 if len(v1) >= len(u/v)
+ // We choose s = B-1 since len(v)-s >= B+1 >= len(u/v)
+ s := (B - 1)
+ // Except for the first step, the top bits are always
+ // a division remainder, so the quotient length is <= n.
+ uu := u[j-B:]
+
+ qhat := *temps[depth]
+ qhat.clear()
+ qhat.divRecursiveStep(uu[s:B+n], v[s:], depth+1, tmp, temps)
+ qhat = qhat.norm()
+ // Adjust the quotient:
+ // u = u_h << s + u_l
+ // v = v_h << s + v_l
+ // u_h = q̂ v_h + rh
+ // u = q̂ (v - v_l) + rh << s + u_l
+ // After the above step, u contains a remainder:
+ // u = rh << s + u_l
+ // and we need to subtract q̂ v_l
+ //
+ // But it may be a bit too large, in which case q̂ needs to be smaller.
+ qhatv := tmp.make(3 * n)
+ qhatv.clear()
+ qhatv = qhatv.mul(qhat, v[:s])
+ for i := 0; i < 2; i++ {
+ e := qhatv.cmp(uu.norm())
+ if e <= 0 {
+ break
+ }
+ subVW(qhat, qhat, 1)
+ c := subVV(qhatv[:s], qhatv[:s], v[:s])
+ if len(qhatv) > s {
+ subVW(qhatv[s:], qhatv[s:], c)
+ }
+ addAt(uu[s:], v[s:], 0)
+ }
+ if qhatv.cmp(uu.norm()) > 0 {
+ panic("impossible")
+ }
+ c := subVV(uu[:len(qhatv)], uu[:len(qhatv)], qhatv)
+ if c > 0 {
+ subVW(uu[len(qhatv):], uu[len(qhatv):], c)
+ }
+ addAt(z, qhat, j-B)
+ j -= B
+ }
+
+ // Now u < (v<<B), compute lower bits in the same way.
+ // Choose shift = B-1 again.
+ s := B - 1
+ qhat := *temps[depth]
+ qhat.clear()
+ qhat.divRecursiveStep(u[s:].norm(), v[s:], depth+1, tmp, temps)
+ qhat = qhat.norm()
+ qhatv := tmp.make(3 * n)
+ qhatv.clear()
+ qhatv = qhatv.mul(qhat, v[:s])
+ // Set the correct remainder as before.
+ for i := 0; i < 2; i++ {
+ if e := qhatv.cmp(u.norm()); e > 0 {
+ subVW(qhat, qhat, 1)
+ c := subVV(qhatv[:s], qhatv[:s], v[:s])
+ if len(qhatv) > s {
+ subVW(qhatv[s:], qhatv[s:], c)
+ }
+ addAt(u[s:], v[s:], 0)
+ }
+ }
+ if qhatv.cmp(u.norm()) > 0 {
+ panic("impossible")
+ }
+ c := subVV(u[0:len(qhatv)], u[0:len(qhatv)], qhatv)
+ if c > 0 {
+ c = subVW(u[len(qhatv):], u[len(qhatv):], c)
+ }
+ if c > 0 {
+ panic("impossible")
+ }
+
+ // Done!
+ addAt(z, qhat.norm(), 0)
+}
+
+// Length of x in bits. x must be normalized.
+func (x nat) bitLen() int {
+ if i := len(x) - 1; i >= 0 {
+ return i*_W + bits.Len(uint(x[i]))
+ }
+ return 0
+}
+
+// trailingZeroBits returns the number of consecutive least significant zero
+// bits of x.
+func (x nat) trailingZeroBits() uint {
+ if len(x) == 0 {
+ return 0
+ }
+ var i uint
+ for x[i] == 0 {
+ i++
+ }
+ // x[i] != 0
+ return i*_W + uint(bits.TrailingZeros(uint(x[i])))
+}
+
+func same(x, y nat) bool {
+ return len(x) == len(y) && len(x) > 0 && &x[0] == &y[0]
+}
+
+// z = x << s
+func (z nat) shl(x nat, s uint) nat {
+ if s == 0 {
+ if same(z, x) {
+ return z
+ }
+ if !alias(z, x) {
+ return z.set(x)
+ }
+ }
+
+ m := len(x)
+ if m == 0 {
+ return z[:0]
+ }
+ // m > 0
+
+ n := m + int(s/_W)
+ z = z.make(n + 1)
+ z[n] = shlVU(z[n-m:n], x, s%_W)
+ z[0 : n-m].clear()
+
+ return z.norm()
+}
+
+// z = x >> s
+func (z nat) shr(x nat, s uint) nat {
+ if s == 0 {
+ if same(z, x) {
+ return z
+ }
+ if !alias(z, x) {
+ return z.set(x)
+ }
+ }
+
+ m := len(x)
+ n := m - int(s/_W)
+ if n <= 0 {
+ return z[:0]
+ }
+ // n > 0
+
+ z = z.make(n)
+ shrVU(z, x[m-n:], s%_W)
+
+ return z.norm()
+}
+
+func (z nat) setBit(x nat, i uint, b uint) nat {
+ j := int(i / _W)
+ m := Word(1) << (i % _W)
+ n := len(x)
+ switch b {
+ case 0:
+ z = z.make(n)
+ copy(z, x)
+ if j >= n {
+ // no need to grow
+ return z
+ }
+ z[j] &^= m
+ return z.norm()
+ case 1:
+ if j >= n {
+ z = z.make(j + 1)
+ z[n:].clear()
+ } else {
+ z = z.make(n)
+ }
+ copy(z, x)
+ z[j] |= m
+ // no need to normalize
+ return z
+ }
+ panic("set bit is not 0 or 1")
+}
+
+// bit returns the value of the i'th bit, with lsb == bit 0.
+func (x nat) bit(i uint) uint {
+ j := i / _W
+ if j >= uint(len(x)) {
+ return 0
+ }
+ // 0 <= j < len(x)
+ return uint(x[j] >> (i % _W) & 1)
+}
+
+// sticky returns 1 if there's a 1 bit within the
+// i least significant bits, otherwise it returns 0.
+func (x nat) sticky(i uint) uint {
+ j := i / _W
+ if j >= uint(len(x)) {
+ if len(x) == 0 {
+ return 0
+ }
+ return 1
+ }
+ // 0 <= j < len(x)
+ for _, x := range x[:j] {
+ if x != 0 {
+ return 1
+ }
+ }
+ if x[j]<<(_W-i%_W) != 0 {
+ return 1
+ }
+ return 0
+}
+
+func (z nat) and(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ if m > n {
+ m = n
+ }
+ // m <= n
+
+ z = z.make(m)
+ for i := 0; i < m; i++ {
+ z[i] = x[i] & y[i]
+ }
+
+ return z.norm()
+}
+
+func (z nat) andNot(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ if n > m {
+ n = m
+ }
+ // m >= n
+
+ z = z.make(m)
+ for i := 0; i < n; i++ {
+ z[i] = x[i] &^ y[i]
+ }
+ copy(z[n:m], x[n:m])
+
+ return z.norm()
+}
+
+func (z nat) or(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ s := x
+ if m < n {
+ n, m = m, n
+ s = y
+ }
+ // m >= n
+
+ z = z.make(m)
+ for i := 0; i < n; i++ {
+ z[i] = x[i] | y[i]
+ }
+ copy(z[n:m], s[n:m])
+
+ return z.norm()
+}
+
+func (z nat) xor(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ s := x
+ if m < n {
+ n, m = m, n
+ s = y
+ }
+ // m >= n
+
+ z = z.make(m)
+ for i := 0; i < n; i++ {
+ z[i] = x[i] ^ y[i]
+ }
+ copy(z[n:m], s[n:m])
+
+ return z.norm()
+}
+
+// greaterThan reports whether (x1<<_W + x2) > (y1<<_W + y2)
+func greaterThan(x1, x2, y1, y2 Word) bool {
+ return x1 > y1 || x1 == y1 && x2 > y2
+}
+
+// modW returns x % d.
+func (x nat) modW(d Word) (r Word) {
+ // TODO(agl): we don't actually need to store the q value.
+ var q nat
+ q = q.make(len(x))
+ return divWVW(q, 0, x, d)
+}
+
+// random creates a random integer in [0..limit), using the space in z if
+// possible. n is the bit length of limit.
+func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
+ if alias(z, limit) {
+ z = nil // z is an alias for limit - cannot reuse
+ }
+ z = z.make(len(limit))
+
+ bitLengthOfMSW := uint(n % _W)
+ if bitLengthOfMSW == 0 {
+ bitLengthOfMSW = _W
+ }
+ mask := Word((1 << bitLengthOfMSW) - 1)
+
+ for {
+ switch _W {
+ case 32:
+ for i := range z {
+ z[i] = Word(rand.Uint32())
+ }
+ case 64:
+ for i := range z {
+ z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
+ }
+ default:
+ panic("unknown word size")
+ }
+ z[len(limit)-1] &= mask
+ if z.cmp(limit) < 0 {
+ break
+ }
+ }
+
+ return z.norm()
+}
+
+// If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
+// otherwise it sets z to x**y. The result is the value of z.
+func (z nat) expNN(x, y, m nat) nat {
+ if alias(z, x) || alias(z, y) {
+ // We cannot allow in-place modification of x or y.
+ z = nil
+ }
+
+ // x**y mod 1 == 0
+ if len(m) == 1 && m[0] == 1 {
+ return z.setWord(0)
+ }
+ // m == 0 || m > 1
+
+ // x**0 == 1
+ if len(y) == 0 {
+ return z.setWord(1)
+ }
+ // y > 0
+
+ // x**1 mod m == x mod m
+ if len(y) == 1 && y[0] == 1 && len(m) != 0 {
+ _, z = nat(nil).div(z, x, m)
+ return z
+ }
+ // y > 1
+
+ if len(m) != 0 {
+ // We likely end up being as long as the modulus.
+ z = z.make(len(m))
+ }
+ z = z.set(x)
+
+ // If the base is non-trivial and the exponent is large, we use
+ // 4-bit, windowed exponentiation. This involves precomputing 14 values
+ // (x^2...x^15) but then reduces the number of multiply-reduces by a
+ // third. Even for a 32-bit exponent, this reduces the number of
+ // operations. Uses Montgomery method for odd moduli.
+ if x.cmp(natOne) > 0 && len(y) > 1 && len(m) > 0 {
+ if m[0]&1 == 1 {
+ return z.expNNMontgomery(x, y, m)
+ }
+ return z.expNNWindowed(x, y, m)
+ }
+
+ v := y[len(y)-1] // v > 0 because y is normalized and y > 0
+ shift := nlz(v) + 1
+ v <<= shift
+ var q nat
+
+ const mask = 1 << (_W - 1)
+
+ // We walk through the bits of the exponent one by one. Each time we
+ // see a bit, we square, thus doubling the power. If the bit is a one,
+ // we also multiply by x, thus adding one to the power.
+
+ w := _W - int(shift)
+ // zz and r are used to avoid allocating in mul and div as
+ // otherwise the arguments would alias.
+ var zz, r nat
+ for j := 0; j < w; j++ {
+ zz = zz.sqr(z)
+ zz, z = z, zz
+
+ if v&mask != 0 {
+ zz = zz.mul(z, x)
+ zz, z = z, zz
+ }
+
+ if len(m) != 0 {
+ zz, r = zz.div(r, z, m)
+ zz, r, q, z = q, z, zz, r
+ }
+
+ v <<= 1
+ }
+
+ for i := len(y) - 2; i >= 0; i-- {
+ v = y[i]
+
+ for j := 0; j < _W; j++ {
+ zz = zz.sqr(z)
+ zz, z = z, zz
+
+ if v&mask != 0 {
+ zz = zz.mul(z, x)
+ zz, z = z, zz
+ }
+
+ if len(m) != 0 {
+ zz, r = zz.div(r, z, m)
+ zz, r, q, z = q, z, zz, r
+ }
+
+ v <<= 1
+ }
+ }
+
+ return z.norm()
+}
+
+// expNNWindowed calculates x**y mod m using a fixed, 4-bit window.
+func (z nat) expNNWindowed(x, y, m nat) nat {
+ // zz and r are used to avoid allocating in mul and div as otherwise
+ // the arguments would alias.
+ var zz, r nat
+
+ const n = 4
+ // powers[i] contains x^i.
+ var powers [1 << n]nat
+ powers[0] = natOne
+ powers[1] = x
+ for i := 2; i < 1<<n; i += 2 {
+ p2, p, p1 := &powers[i/2], &powers[i], &powers[i+1]
+ *p = p.sqr(*p2)
+ zz, r = zz.div(r, *p, m)
+ *p, r = r, *p
+ *p1 = p1.mul(*p, x)
+ zz, r = zz.div(r, *p1, m)
+ *p1, r = r, *p1
+ }
+
+ z = z.setWord(1)
+
+ for i := len(y) - 1; i >= 0; i-- {
+ yi := y[i]
+ for j := 0; j < _W; j += n {
+ if i != len(y)-1 || j != 0 {
+ // Unrolled loop for significant performance
+ // gain. Use go test -bench=".*" in crypto/rsa
+ // to check performance before making changes.
+ zz = zz.sqr(z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ zz = zz.sqr(z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ zz = zz.sqr(z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ zz = zz.sqr(z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+ }
+
+ zz = zz.mul(z, powers[yi>>(_W-n)])
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ yi <<= n
+ }
+ }
+
+ return z.norm()
+}
+
+// expNNMontgomery calculates x**y mod m using a fixed, 4-bit window.
+// Uses Montgomery representation.
+func (z nat) expNNMontgomery(x, y, m nat) nat {
+ numWords := len(m)
+
+ // We want the lengths of x and m to be equal.
+ // It is OK if x >= m as long as len(x) == len(m).
+ if len(x) > numWords {
+ _, x = nat(nil).div(nil, x, m)
+ // Note: now len(x) <= numWords, not guaranteed ==.
+ }
+ if len(x) < numWords {
+ rr := make(nat, numWords)
+ copy(rr, x)
+ x = rr
+ }
+
+ // Ideally the precomputations would be performed outside, and reused
+ // k0 = -m**-1 mod 2**_W. Algorithm from: Dumas, J.G. "On Newton–Raphson
+ // Iteration for Multiplicative Inverses Modulo Prime Powers".
+ k0 := 2 - m[0]
+ t := m[0] - 1
+ for i := 1; i < _W; i <<= 1 {
+ t *= t
+ k0 *= (t + 1)
+ }
+ k0 = -k0
+
+ // RR = 2**(2*_W*len(m)) mod m
+ RR := nat(nil).setWord(1)
+ zz := nat(nil).shl(RR, uint(2*numWords*_W))
+ _, RR = nat(nil).div(RR, zz, m)
+ if len(RR) < numWords {
+ zz = zz.make(numWords)
+ copy(zz, RR)
+ RR = zz
+ }
+ // one = 1, with equal length to that of m
+ one := make(nat, numWords)
+ one[0] = 1
+
+ const n = 4
+ // powers[i] contains x^i
+ var powers [1 << n]nat
+ powers[0] = powers[0].montgomery(one, RR, m, k0, numWords)
+ powers[1] = powers[1].montgomery(x, RR, m, k0, numWords)
+ for i := 2; i < 1<<n; i++ {
+ powers[i] = powers[i].montgomery(powers[i-1], powers[1], m, k0, numWords)
+ }
+
+ // initialize z = 1 (Montgomery 1)
+ z = z.make(numWords)
+ copy(z, powers[0])
+
+ zz = zz.make(numWords)
+
+ // same windowed exponent, but with Montgomery multiplications
+ for i := len(y) - 1; i >= 0; i-- {
+ yi := y[i]
+ for j := 0; j < _W; j += n {
+ if i != len(y)-1 || j != 0 {
+ zz = zz.montgomery(z, z, m, k0, numWords)
+ z = z.montgomery(zz, zz, m, k0, numWords)
+ zz = zz.montgomery(z, z, m, k0, numWords)
+ z = z.montgomery(zz, zz, m, k0, numWords)
+ }
+ zz = zz.montgomery(z, powers[yi>>(_W-n)], m, k0, numWords)
+ z, zz = zz, z
+ yi <<= n
+ }
+ }
+ // convert to regular number
+ zz = zz.montgomery(z, one, m, k0, numWords)
+
+ // One last reduction, just in case.
+ // See golang.org/issue/13907.
+ if zz.cmp(m) >= 0 {
+ // Common case is m has high bit set; in that case,
+ // since zz is the same length as m, there can be just
+ // one multiple of m to remove. Just subtract.
+ // We think that the subtract should be sufficient in general,
+ // so do that unconditionally, but double-check,
+ // in case our beliefs are wrong.
+ // The div is not expected to be reached.
+ zz = zz.sub(zz, m)
+ if zz.cmp(m) >= 0 {
+ _, zz = nat(nil).div(nil, zz, m)
+ }
+ }
+
+ return zz.norm()
+}
+
+// bytes writes the value of z into buf using big-endian encoding.
+// The value of z is encoded in the slice buf[i:]. If the value of z
+// cannot be represented in buf, bytes panics. The number i of unused
+// bytes at the beginning of buf is returned as result.
+func (z nat) bytes(buf []byte) (i int) {
+ i = len(buf)
+ for _, d := range z {
+ for j := 0; j < _S; j++ {
+ i--
+ if i >= 0 {
+ buf[i] = byte(d)
+ } else if byte(d) != 0 {
+ panic("math/big: buffer too small to fit value")
+ }
+ d >>= 8
+ }
+ }
+
+ if i < 0 {
+ i = 0
+ }
+ for i < len(buf) && buf[i] == 0 {
+ i++
+ }
+
+ return
+}
+
+// bigEndianWord returns the contents of buf interpreted as a big-endian encoded Word value.
+func bigEndianWord(buf []byte) Word {
+ if _W == 64 {
+ return Word(binary.BigEndian.Uint64(buf))
+ }
+ return Word(binary.BigEndian.Uint32(buf))
+}
+
+// setBytes interprets buf as the bytes of a big-endian unsigned
+// integer, sets z to that value, and returns z.
+func (z nat) setBytes(buf []byte) nat {
+ z = z.make((len(buf) + _S - 1) / _S)
+
+ i := len(buf)
+ for k := 0; i >= _S; k++ {
+ z[k] = bigEndianWord(buf[i-_S : i])
+ i -= _S
+ }
+ if i > 0 {
+ var d Word
+ for s := uint(0); i > 0; s += 8 {
+ d |= Word(buf[i-1]) << s
+ i--
+ }
+ z[len(z)-1] = d
+ }
+
+ return z.norm()
+}
+
+// sqrt sets z = ⌊√x⌋
+func (z nat) sqrt(x nat) nat {
+ if x.cmp(natOne) <= 0 {
+ return z.set(x)
+ }
+ if alias(z, x) {
+ z = nil
+ }
+
+ // Start with value known to be too large and repeat "z = ⌊(z + ⌊x/z⌋)/2⌋" until it stops getting smaller.
+ // See Brent and Zimmermann, Modern Computer Arithmetic, Algorithm 1.13 (SqrtInt).
+ // https://members.loria.fr/PZimmermann/mca/pub226.html
+ // If x is one less than a perfect square, the sequence oscillates between the correct z and z+1;
+ // otherwise it converges to the correct z and stays there.
+ var z1, z2 nat
+ z1 = z
+ z1 = z1.setUint64(1)
+ z1 = z1.shl(z1, uint(x.bitLen()+1)/2) // must be ≥ √x
+ for n := 0; ; n++ {
+ z2, _ = z2.div(nil, x, z1)
+ z2 = z2.add(z2, z1)
+ z2 = z2.shr(z2, 1)
+ if z2.cmp(z1) >= 0 {
+ // z1 is answer.
+ // Figure out whether z1 or z2 is currently aliased to z by looking at loop count.
+ if n&1 == 0 {
+ return z1
+ }
+ return z.set(z1)
+ }
+ z1, z2 = z2, z1
+ }
+}
diff --git a/src/math/big/nat_test.go b/src/math/big/nat_test.go
new file mode 100644
index 0000000..0850818
--- /dev/null
+++ b/src/math/big/nat_test.go
@@ -0,0 +1,816 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "fmt"
+ "runtime"
+ "strings"
+ "testing"
+)
+
+var cmpTests = []struct {
+ x, y nat
+ r int
+}{
+ {nil, nil, 0},
+ {nil, nat(nil), 0},
+ {nat(nil), nil, 0},
+ {nat(nil), nat(nil), 0},
+ {nat{0}, nat{0}, 0},
+ {nat{0}, nat{1}, -1},
+ {nat{1}, nat{0}, 1},
+ {nat{1}, nat{1}, 0},
+ {nat{0, _M}, nat{1}, 1},
+ {nat{1}, nat{0, _M}, -1},
+ {nat{1, _M}, nat{0, _M}, 1},
+ {nat{0, _M}, nat{1, _M}, -1},
+ {nat{16, 571956, 8794, 68}, nat{837, 9146, 1, 754489}, -1},
+ {nat{34986, 41, 105, 1957}, nat{56, 7458, 104, 1957}, 1},
+}
+
+func TestCmp(t *testing.T) {
+ for i, a := range cmpTests {
+ r := a.x.cmp(a.y)
+ if r != a.r {
+ t.Errorf("#%d got r = %v; want %v", i, r, a.r)
+ }
+ }
+}
+
+type funNN func(z, x, y nat) nat
+type argNN struct {
+ z, x, y nat
+}
+
+var sumNN = []argNN{
+ {},
+ {nat{1}, nil, nat{1}},
+ {nat{1111111110}, nat{123456789}, nat{987654321}},
+ {nat{0, 0, 0, 1}, nil, nat{0, 0, 0, 1}},
+ {nat{0, 0, 0, 1111111110}, nat{0, 0, 0, 123456789}, nat{0, 0, 0, 987654321}},
+ {nat{0, 0, 0, 1}, nat{0, 0, _M}, nat{0, 0, 1}},
+}
+
+var prodNN = []argNN{
+ {},
+ {nil, nil, nil},
+ {nil, nat{991}, nil},
+ {nat{991}, nat{991}, nat{1}},
+ {nat{991 * 991}, nat{991}, nat{991}},
+ {nat{0, 0, 991 * 991}, nat{0, 991}, nat{0, 991}},
+ {nat{1 * 991, 2 * 991, 3 * 991, 4 * 991}, nat{1, 2, 3, 4}, nat{991}},
+ {nat{4, 11, 20, 30, 20, 11, 4}, nat{1, 2, 3, 4}, nat{4, 3, 2, 1}},
+ // 3^100 * 3^28 = 3^128
+ {
+ natFromString("11790184577738583171520872861412518665678211592275841109096961"),
+ natFromString("515377520732011331036461129765621272702107522001"),
+ natFromString("22876792454961"),
+ },
+ // z = 111....1 (70000 digits)
+ // x = 10^(99*700) + ... + 10^1400 + 10^700 + 1
+ // y = 111....1 (700 digits, larger than Karatsuba threshold on 32-bit and 64-bit)
+ {
+ natFromString(strings.Repeat("1", 70000)),
+ natFromString("1" + strings.Repeat(strings.Repeat("0", 699)+"1", 99)),
+ natFromString(strings.Repeat("1", 700)),
+ },
+ // z = 111....1 (20000 digits)
+ // x = 10^10000 + 1
+ // y = 111....1 (10000 digits)
+ {
+ natFromString(strings.Repeat("1", 20000)),
+ natFromString("1" + strings.Repeat("0", 9999) + "1"),
+ natFromString(strings.Repeat("1", 10000)),
+ },
+}
+
+func natFromString(s string) nat {
+ x, _, _, err := nat(nil).scan(strings.NewReader(s), 0, false)
+ if err != nil {
+ panic(err)
+ }
+ return x
+}
+
+func TestSet(t *testing.T) {
+ for _, a := range sumNN {
+ z := nat(nil).set(a.z)
+ if z.cmp(a.z) != 0 {
+ t.Errorf("got z = %v; want %v", z, a.z)
+ }
+ }
+}
+
+func testFunNN(t *testing.T, msg string, f funNN, a argNN) {
+ z := f(nil, a.x, a.y)
+ if z.cmp(a.z) != 0 {
+ t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, z, a.z)
+ }
+}
+
+func TestFunNN(t *testing.T) {
+ for _, a := range sumNN {
+ arg := a
+ testFunNN(t, "add", nat.add, arg)
+
+ arg = argNN{a.z, a.y, a.x}
+ testFunNN(t, "add symmetric", nat.add, arg)
+
+ arg = argNN{a.x, a.z, a.y}
+ testFunNN(t, "sub", nat.sub, arg)
+
+ arg = argNN{a.y, a.z, a.x}
+ testFunNN(t, "sub symmetric", nat.sub, arg)
+ }
+
+ for _, a := range prodNN {
+ arg := a
+ testFunNN(t, "mul", nat.mul, arg)
+
+ arg = argNN{a.z, a.y, a.x}
+ testFunNN(t, "mul symmetric", nat.mul, arg)
+ }
+}
+
+var mulRangesN = []struct {
+ a, b uint64
+ prod string
+}{
+ {0, 0, "0"},
+ {1, 1, "1"},
+ {1, 2, "2"},
+ {1, 3, "6"},
+ {10, 10, "10"},
+ {0, 100, "0"},
+ {0, 1e9, "0"},
+ {1, 0, "1"}, // empty range
+ {100, 1, "1"}, // empty range
+ {1, 10, "3628800"}, // 10!
+ {1, 20, "2432902008176640000"}, // 20!
+ {1, 100,
+ "933262154439441526816992388562667004907159682643816214685929" +
+ "638952175999932299156089414639761565182862536979208272237582" +
+ "51185210916864000000000000000000000000", // 100!
+ },
+}
+
+func TestMulRangeN(t *testing.T) {
+ for i, r := range mulRangesN {
+ prod := string(nat(nil).mulRange(r.a, r.b).utoa(10))
+ if prod != r.prod {
+ t.Errorf("#%d: got %s; want %s", i, prod, r.prod)
+ }
+ }
+}
+
+// allocBytes returns the number of bytes allocated by invoking f.
+func allocBytes(f func()) uint64 {
+ var stats runtime.MemStats
+ runtime.ReadMemStats(&stats)
+ t := stats.TotalAlloc
+ f()
+ runtime.ReadMemStats(&stats)
+ return stats.TotalAlloc - t
+}
+
+// TestMulUnbalanced tests that multiplying numbers of different lengths
+// does not cause deep recursion and in turn allocate too much memory.
+// Test case for issue 3807.
+func TestMulUnbalanced(t *testing.T) {
+ defer runtime.GOMAXPROCS(runtime.GOMAXPROCS(1))
+ x := rndNat(50000)
+ y := rndNat(40)
+ allocSize := allocBytes(func() {
+ nat(nil).mul(x, y)
+ })
+ inputSize := uint64(len(x)+len(y)) * _S
+ if ratio := allocSize / uint64(inputSize); ratio > 10 {
+ t.Errorf("multiplication uses too much memory (%d > %d times the size of inputs)", allocSize, ratio)
+ }
+}
+
+// rndNat returns a random nat value >= 0 of (usually) n words in length.
+// In extremely unlikely cases it may be smaller than n words if the top-
+// most words are 0.
+func rndNat(n int) nat {
+ return nat(rndV(n)).norm()
+}
+
+// rndNat1 is like rndNat but the result is guaranteed to be > 0.
+func rndNat1(n int) nat {
+ x := nat(rndV(n)).norm()
+ if len(x) == 0 {
+ x.setWord(1)
+ }
+ return x
+}
+
+func BenchmarkMul(b *testing.B) {
+ mulx := rndNat(1e4)
+ muly := rndNat(1e4)
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ var z nat
+ z.mul(mulx, muly)
+ }
+}
+
+func benchmarkNatMul(b *testing.B, nwords int) {
+ x := rndNat(nwords)
+ y := rndNat(nwords)
+ var z nat
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ z.mul(x, y)
+ }
+}
+
+var mulBenchSizes = []int{10, 100, 1000, 10000, 100000}
+
+func BenchmarkNatMul(b *testing.B) {
+ for _, n := range mulBenchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
+ benchmarkNatMul(b, n)
+ })
+ }
+}
+
+func TestNLZ(t *testing.T) {
+ var x Word = _B >> 1
+ for i := 0; i <= _W; i++ {
+ if int(nlz(x)) != i {
+ t.Errorf("failed at %x: got %d want %d", x, nlz(x), i)
+ }
+ x >>= 1
+ }
+}
+
+type shiftTest struct {
+ in nat
+ shift uint
+ out nat
+}
+
+var leftShiftTests = []shiftTest{
+ {nil, 0, nil},
+ {nil, 1, nil},
+ {natOne, 0, natOne},
+ {natOne, 1, natTwo},
+ {nat{1 << (_W - 1)}, 1, nat{0}},
+ {nat{1 << (_W - 1), 0}, 1, nat{0, 1}},
+}
+
+func TestShiftLeft(t *testing.T) {
+ for i, test := range leftShiftTests {
+ var z nat
+ z = z.shl(test.in, test.shift)
+ for j, d := range test.out {
+ if j >= len(z) || z[j] != d {
+ t.Errorf("#%d: got: %v want: %v", i, z, test.out)
+ break
+ }
+ }
+ }
+}
+
+var rightShiftTests = []shiftTest{
+ {nil, 0, nil},
+ {nil, 1, nil},
+ {natOne, 0, natOne},
+ {natOne, 1, nil},
+ {natTwo, 1, natOne},
+ {nat{0, 1}, 1, nat{1 << (_W - 1)}},
+ {nat{2, 1, 1}, 1, nat{1<<(_W-1) + 1, 1 << (_W - 1)}},
+}
+
+func TestShiftRight(t *testing.T) {
+ for i, test := range rightShiftTests {
+ var z nat
+ z = z.shr(test.in, test.shift)
+ for j, d := range test.out {
+ if j >= len(z) || z[j] != d {
+ t.Errorf("#%d: got: %v want: %v", i, z, test.out)
+ break
+ }
+ }
+ }
+}
+
+func BenchmarkZeroShifts(b *testing.B) {
+ x := rndNat(800)
+
+ b.Run("Shl", func(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ var z nat
+ z.shl(x, 0)
+ }
+ })
+ b.Run("ShlSame", func(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ x.shl(x, 0)
+ }
+ })
+
+ b.Run("Shr", func(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ var z nat
+ z.shr(x, 0)
+ }
+ })
+ b.Run("ShrSame", func(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ x.shr(x, 0)
+ }
+ })
+}
+
+type modWTest struct {
+ in string
+ dividend string
+ out string
+}
+
+var modWTests32 = []modWTest{
+ {"23492635982634928349238759823742", "252341", "220170"},
+}
+
+var modWTests64 = []modWTest{
+ {"6527895462947293856291561095690465243862946", "524326975699234", "375066989628668"},
+}
+
+func runModWTests(t *testing.T, tests []modWTest) {
+ for i, test := range tests {
+ in, _ := new(Int).SetString(test.in, 10)
+ d, _ := new(Int).SetString(test.dividend, 10)
+ out, _ := new(Int).SetString(test.out, 10)
+
+ r := in.abs.modW(d.abs[0])
+ if r != out.abs[0] {
+ t.Errorf("#%d failed: got %d want %s", i, r, out)
+ }
+ }
+}
+
+func TestModW(t *testing.T) {
+ if _W >= 32 {
+ runModWTests(t, modWTests32)
+ }
+ if _W >= 64 {
+ runModWTests(t, modWTests64)
+ }
+}
+
+var montgomeryTests = []struct {
+ x, y, m string
+ k0 uint64
+ out32, out64 string
+}{
+ {
+ "0xffffffffffffffffffffffffffffffffffffffffffffffffe",
+ "0xffffffffffffffffffffffffffffffffffffffffffffffffe",
+ "0xfffffffffffffffffffffffffffffffffffffffffffffffff",
+ 1,
+ "0x1000000000000000000000000000000000000000000",
+ "0x10000000000000000000000000000000000",
+ },
+ {
+ "0x000000000ffffff5",
+ "0x000000000ffffff0",
+ "0x0000000010000001",
+ 0xff0000000fffffff,
+ "0x000000000bfffff4",
+ "0x0000000003400001",
+ },
+ {
+ "0x0000000080000000",
+ "0x00000000ffffffff",
+ "0x1000000000000001",
+ 0xfffffffffffffff,
+ "0x0800000008000001",
+ "0x0800000008000001",
+ },
+ {
+ "0x0000000080000000",
+ "0x0000000080000000",
+ "0xffffffff00000001",
+ 0xfffffffeffffffff,
+ "0xbfffffff40000001",
+ "0xbfffffff40000001",
+ },
+ {
+ "0x0000000080000000",
+ "0x0000000080000000",
+ "0x00ffffff00000001",
+ 0xfffffeffffffff,
+ "0xbfffff40000001",
+ "0xbfffff40000001",
+ },
+ {
+ "0x0000000080000000",
+ "0x0000000080000000",
+ "0x0000ffff00000001",
+ 0xfffeffffffff,
+ "0xbfff40000001",
+ "0xbfff40000001",
+ },
+ {
+ "0x3321ffffffffffffffffffffffffffff00000000000022222623333333332bbbb888c0",
+ "0x3321ffffffffffffffffffffffffffff00000000000022222623333333332bbbb888c0",
+ "0x33377fffffffffffffffffffffffffffffffffffffffffffff0000000000022222eee1",
+ 0xdecc8f1249812adf,
+ "0x04eb0e11d72329dc0915f86784820fc403275bf2f6620a20e0dd344c5cd0875e50deb5",
+ "0x0d7144739a7d8e11d72329dc0915f86784820fc403275bf2f61ed96f35dd34dbb3d6a0",
+ },
+ {
+ "0x10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffff00000000000022222223333333333444444444",
+ "0x10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffff999999999999999aaabbbbbbbbcccccccccccc",
+ "0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff33377fffffffffffffffffffffffffffffffffffffffffffff0000000000022222eee1",
+ 0xdecc8f1249812adf,
+ "0x5c0d52f451aec609b15da8e5e5626c4eaa88723bdeac9d25ca9b961269400410ca208a16af9c2fb07d7a11c7772cba02c22f9711078d51a3797eb18e691295293284d988e349fa6deba46b25a4ecd9f715",
+ "0x92fcad4b5c0d52f451aec609b15da8e5e5626c4eaa88723bdeac9d25ca9b961269400410ca208a16af9c2fb07d799c32fe2f3cc5422f9711078d51a3797eb18e691295293284d8f5e69caf6decddfe1df6",
+ },
+}
+
+func TestMontgomery(t *testing.T) {
+ one := NewInt(1)
+ _B := new(Int).Lsh(one, _W)
+ for i, test := range montgomeryTests {
+ x := natFromString(test.x)
+ y := natFromString(test.y)
+ m := natFromString(test.m)
+ for len(x) < len(m) {
+ x = append(x, 0)
+ }
+ for len(y) < len(m) {
+ y = append(y, 0)
+ }
+
+ if x.cmp(m) > 0 {
+ _, r := nat(nil).div(nil, x, m)
+ t.Errorf("#%d: x > m (0x%s > 0x%s; use 0x%s)", i, x.utoa(16), m.utoa(16), r.utoa(16))
+ }
+ if y.cmp(m) > 0 {
+ _, r := nat(nil).div(nil, x, m)
+ t.Errorf("#%d: y > m (0x%s > 0x%s; use 0x%s)", i, y.utoa(16), m.utoa(16), r.utoa(16))
+ }
+
+ var out nat
+ if _W == 32 {
+ out = natFromString(test.out32)
+ } else {
+ out = natFromString(test.out64)
+ }
+
+ // t.Logf("#%d: len=%d\n", i, len(m))
+
+ // check output in table
+ xi := &Int{abs: x}
+ yi := &Int{abs: y}
+ mi := &Int{abs: m}
+ p := new(Int).Mod(new(Int).Mul(xi, new(Int).Mul(yi, new(Int).ModInverse(new(Int).Lsh(one, uint(len(m))*_W), mi))), mi)
+ if out.cmp(p.abs.norm()) != 0 {
+ t.Errorf("#%d: out in table=0x%s, computed=0x%s", i, out.utoa(16), p.abs.norm().utoa(16))
+ }
+
+ // check k0 in table
+ k := new(Int).Mod(&Int{abs: m}, _B)
+ k = new(Int).Sub(_B, k)
+ k = new(Int).Mod(k, _B)
+ k0 := Word(new(Int).ModInverse(k, _B).Uint64())
+ if k0 != Word(test.k0) {
+ t.Errorf("#%d: k0 in table=%#x, computed=%#x\n", i, test.k0, k0)
+ }
+
+ // check montgomery with correct k0 produces correct output
+ z := nat(nil).montgomery(x, y, m, k0, len(m))
+ z = z.norm()
+ if z.cmp(out) != 0 {
+ t.Errorf("#%d: got 0x%s want 0x%s", i, z.utoa(16), out.utoa(16))
+ }
+ }
+}
+
+var expNNTests = []struct {
+ x, y, m string
+ out string
+}{
+ {"0", "0", "0", "1"},
+ {"0", "0", "1", "0"},
+ {"1", "1", "1", "0"},
+ {"2", "1", "1", "0"},
+ {"2", "2", "1", "0"},
+ {"10", "100000000000", "1", "0"},
+ {"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
+ {"0x8000000000000000", "2", "6719", "4944"},
+ {"0x8000000000000000", "3", "6719", "5447"},
+ {"0x8000000000000000", "1000", "6719", "1603"},
+ {"0x8000000000000000", "1000000", "6719", "3199"},
+ {
+ "2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
+ "298472983472983471903246121093472394872319615612417471234712061",
+ "29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
+ "23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
+ },
+ {
+ "11521922904531591643048817447554701904414021819823889996244743037378330903763518501116638828335352811871131385129455853417360623007349090150042001944696604737499160174391019030572483602867266711107136838523916077674888297896995042968746762200926853379",
+ "426343618817810911523",
+ "444747819283133684179",
+ "42",
+ },
+}
+
+func TestExpNN(t *testing.T) {
+ for i, test := range expNNTests {
+ x := natFromString(test.x)
+ y := natFromString(test.y)
+ out := natFromString(test.out)
+
+ var m nat
+ if len(test.m) > 0 {
+ m = natFromString(test.m)
+ }
+
+ z := nat(nil).expNN(x, y, m)
+ if z.cmp(out) != 0 {
+ t.Errorf("#%d got %s want %s", i, z.utoa(10), out.utoa(10))
+ }
+ }
+}
+
+func BenchmarkExp3Power(b *testing.B) {
+ const x = 3
+ for _, y := range []Word{
+ 0x10, 0x40, 0x100, 0x400, 0x1000, 0x4000, 0x10000, 0x40000, 0x100000, 0x400000,
+ } {
+ b.Run(fmt.Sprintf("%#x", y), func(b *testing.B) {
+ var z nat
+ for i := 0; i < b.N; i++ {
+ z.expWW(x, y)
+ }
+ })
+ }
+}
+
+func fibo(n int) nat {
+ switch n {
+ case 0:
+ return nil
+ case 1:
+ return nat{1}
+ }
+ f0 := fibo(0)
+ f1 := fibo(1)
+ var f2 nat
+ for i := 1; i < n; i++ {
+ f2 = f2.add(f0, f1)
+ f0, f1, f2 = f1, f2, f0
+ }
+ return f1
+}
+
+var fiboNums = []string{
+ "0",
+ "55",
+ "6765",
+ "832040",
+ "102334155",
+ "12586269025",
+ "1548008755920",
+ "190392490709135",
+ "23416728348467685",
+ "2880067194370816120",
+ "354224848179261915075",
+}
+
+func TestFibo(t *testing.T) {
+ for i, want := range fiboNums {
+ n := i * 10
+ got := string(fibo(n).utoa(10))
+ if got != want {
+ t.Errorf("fibo(%d) failed: got %s want %s", n, got, want)
+ }
+ }
+}
+
+func BenchmarkFibo(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ fibo(1e0)
+ fibo(1e1)
+ fibo(1e2)
+ fibo(1e3)
+ fibo(1e4)
+ fibo(1e5)
+ }
+}
+
+var bitTests = []struct {
+ x string
+ i uint
+ want uint
+}{
+ {"0", 0, 0},
+ {"0", 1, 0},
+ {"0", 1000, 0},
+
+ {"0x1", 0, 1},
+ {"0x10", 0, 0},
+ {"0x10", 3, 0},
+ {"0x10", 4, 1},
+ {"0x10", 5, 0},
+
+ {"0x8000000000000000", 62, 0},
+ {"0x8000000000000000", 63, 1},
+ {"0x8000000000000000", 64, 0},
+
+ {"0x3" + strings.Repeat("0", 32), 127, 0},
+ {"0x3" + strings.Repeat("0", 32), 128, 1},
+ {"0x3" + strings.Repeat("0", 32), 129, 1},
+ {"0x3" + strings.Repeat("0", 32), 130, 0},
+}
+
+func TestBit(t *testing.T) {
+ for i, test := range bitTests {
+ x := natFromString(test.x)
+ if got := x.bit(test.i); got != test.want {
+ t.Errorf("#%d: %s.bit(%d) = %v; want %v", i, test.x, test.i, got, test.want)
+ }
+ }
+}
+
+var stickyTests = []struct {
+ x string
+ i uint
+ want uint
+}{
+ {"0", 0, 0},
+ {"0", 1, 0},
+ {"0", 1000, 0},
+
+ {"0x1", 0, 0},
+ {"0x1", 1, 1},
+
+ {"0x1350", 0, 0},
+ {"0x1350", 4, 0},
+ {"0x1350", 5, 1},
+
+ {"0x8000000000000000", 63, 0},
+ {"0x8000000000000000", 64, 1},
+
+ {"0x1" + strings.Repeat("0", 100), 400, 0},
+ {"0x1" + strings.Repeat("0", 100), 401, 1},
+}
+
+func TestSticky(t *testing.T) {
+ for i, test := range stickyTests {
+ x := natFromString(test.x)
+ if got := x.sticky(test.i); got != test.want {
+ t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i, got, test.want)
+ }
+ if test.want == 1 {
+ // all subsequent i's should also return 1
+ for d := uint(1); d <= 3; d++ {
+ if got := x.sticky(test.i + d); got != 1 {
+ t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i+d, got, 1)
+ }
+ }
+ }
+ }
+}
+
+func testSqr(t *testing.T, x nat) {
+ got := make(nat, 2*len(x))
+ want := make(nat, 2*len(x))
+ got = got.sqr(x)
+ want = want.mul(x, x)
+ if got.cmp(want) != 0 {
+ t.Errorf("basicSqr(%v), got %v, want %v", x, got, want)
+ }
+}
+
+func TestSqr(t *testing.T) {
+ for _, a := range prodNN {
+ if a.x != nil {
+ testSqr(t, a.x)
+ }
+ if a.y != nil {
+ testSqr(t, a.y)
+ }
+ if a.z != nil {
+ testSqr(t, a.z)
+ }
+ }
+}
+
+func benchmarkNatSqr(b *testing.B, nwords int) {
+ x := rndNat(nwords)
+ var z nat
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ z.sqr(x)
+ }
+}
+
+var sqrBenchSizes = []int{
+ 1, 2, 3, 5, 8, 10, 20, 30, 50, 80,
+ 100, 200, 300, 500, 800,
+ 1000, 10000, 100000,
+}
+
+func BenchmarkNatSqr(b *testing.B) {
+ for _, n := range sqrBenchSizes {
+ if isRaceBuilder && n > 1e3 {
+ continue
+ }
+ b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
+ benchmarkNatSqr(b, n)
+ })
+ }
+}
+
+func BenchmarkNatSetBytes(b *testing.B) {
+ const maxLength = 128
+ lengths := []int{
+ // No remainder:
+ 8, 24, maxLength,
+ // With remainder:
+ 7, 23, maxLength - 1,
+ }
+ n := make(nat, maxLength/_W) // ensure n doesn't need to grow during the test
+ buf := make([]byte, maxLength)
+ for _, l := range lengths {
+ b.Run(fmt.Sprint(l), func(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ n.setBytes(buf[:l])
+ }
+ })
+ }
+}
+
+func TestNatDiv(t *testing.T) {
+ sizes := []int{
+ 1, 2, 5, 8, 15, 25, 40, 65, 100,
+ 200, 500, 800, 1500, 2500, 4000, 6500, 10000,
+ }
+ for _, i := range sizes {
+ for _, j := range sizes {
+ a := rndNat1(i)
+ b := rndNat1(j)
+ // the test requires b >= 2
+ if len(b) == 1 && b[0] == 1 {
+ b[0] = 2
+ }
+ // choose a remainder c < b
+ c := rndNat1(len(b))
+ if len(c) == len(b) && c[len(c)-1] >= b[len(b)-1] {
+ c[len(c)-1] = 0
+ c = c.norm()
+ }
+ // compute x = a*b+c
+ x := nat(nil).mul(a, b)
+ x = x.add(x, c)
+
+ var q, r nat
+ q, r = q.div(r, x, b)
+ if q.cmp(a) != 0 {
+ t.Fatalf("wrong quotient: got %s; want %s for %s/%s", q.utoa(10), a.utoa(10), x.utoa(10), b.utoa(10))
+ }
+ if r.cmp(c) != 0 {
+ t.Fatalf("wrong remainder: got %s; want %s for %s/%s", r.utoa(10), c.utoa(10), x.utoa(10), b.utoa(10))
+ }
+ }
+ }
+}
+
+// TestIssue37499 triggers the edge case of divBasic where
+// the inaccurate estimate of the first word's quotient
+// happens at the very beginning of the loop.
+func TestIssue37499(t *testing.T) {
+ // Choose u and v such that v is slightly larger than u >> N.
+ // This tricks divBasic into choosing 1 as the first word
+ // of the quotient. This works in both 32-bit and 64-bit settings.
+ u := natFromString("0x2b6c385a05be027f5c22005b63c42a1165b79ff510e1706b39f8489c1d28e57bb5ba4ef9fd9387a3e344402c0a453381")
+ v := natFromString("0x2b6c385a05be027f5c22005b63c42a1165b79ff510e1706c")
+
+ q := nat(nil).make(8)
+ q.divBasic(u, v)
+ q = q.norm()
+ if s := string(q.utoa(16)); s != "fffffffffffffffffffffffffffffffffffffffffffffffb" {
+ t.Fatalf("incorrect quotient: %s", s)
+ }
+}
+
+// TestIssue42552 triggers an edge case of recursive division
+// where the first division loop is never entered, and correcting
+// the remainder takes exactly two iterations in the final loop.
+func TestIssue42552(t *testing.T) {
+ u := natFromString("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")
+ v := natFromString("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")
+ q := nat(nil).make(16)
+ q.div(q, u, v)
+}
diff --git a/src/math/big/natconv.go b/src/math/big/natconv.go
new file mode 100644
index 0000000..42d1ccc
--- /dev/null
+++ b/src/math/big/natconv.go
@@ -0,0 +1,512 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements nat-to-string conversion functions.
+
+package big
+
+import (
+ "errors"
+ "fmt"
+ "io"
+ "math"
+ "math/bits"
+ "sync"
+)
+
+const digits = "0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ"
+
+// Note: MaxBase = len(digits), but it must remain an untyped rune constant
+// for API compatibility.
+
+// MaxBase is the largest number base accepted for string conversions.
+const MaxBase = 10 + ('z' - 'a' + 1) + ('Z' - 'A' + 1)
+const maxBaseSmall = 10 + ('z' - 'a' + 1)
+
+// maxPow returns (b**n, n) such that b**n is the largest power b**n <= _M.
+// For instance maxPow(10) == (1e19, 19) for 19 decimal digits in a 64bit Word.
+// In other words, at most n digits in base b fit into a Word.
+// TODO(gri) replace this with a table, generated at build time.
+func maxPow(b Word) (p Word, n int) {
+ p, n = b, 1 // assuming b <= _M
+ for max := _M / b; p <= max; {
+ // p == b**n && p <= max
+ p *= b
+ n++
+ }
+ // p == b**n && p <= _M
+ return
+}
+
+// pow returns x**n for n > 0, and 1 otherwise.
+func pow(x Word, n int) (p Word) {
+ // n == sum of bi * 2**i, for 0 <= i < imax, and bi is 0 or 1
+ // thus x**n == product of x**(2**i) for all i where bi == 1
+ // (Russian Peasant Method for exponentiation)
+ p = 1
+ for n > 0 {
+ if n&1 != 0 {
+ p *= x
+ }
+ x *= x
+ n >>= 1
+ }
+ return
+}
+
+// scan errors
+var (
+ errNoDigits = errors.New("number has no digits")
+ errInvalSep = errors.New("'_' must separate successive digits")
+)
+
+// scan scans the number corresponding to the longest possible prefix
+// from r representing an unsigned number in a given conversion base.
+// scan returns the corresponding natural number res, the actual base b,
+// a digit count, and a read or syntax error err, if any.
+//
+// For base 0, an underscore character ``_'' may appear between a base
+// prefix and an adjacent digit, and between successive digits; such
+// underscores do not change the value of the number, or the returned
+// digit count. Incorrect placement of underscores is reported as an
+// error if there are no other errors. If base != 0, underscores are
+// not recognized and thus terminate scanning like any other character
+// that is not a valid radix point or digit.
+//
+// number = mantissa | prefix pmantissa .
+// prefix = "0" [ "b" | "B" | "o" | "O" | "x" | "X" ] .
+// mantissa = digits "." [ digits ] | digits | "." digits .
+// pmantissa = [ "_" ] digits "." [ digits ] | [ "_" ] digits | "." digits .
+// digits = digit { [ "_" ] digit } .
+// digit = "0" ... "9" | "a" ... "z" | "A" ... "Z" .
+//
+// Unless fracOk is set, the base argument must be 0 or a value between
+// 2 and MaxBase. If fracOk is set, the base argument must be one of
+// 0, 2, 8, 10, or 16. Providing an invalid base argument leads to a run-
+// time panic.
+//
+// For base 0, the number prefix determines the actual base: A prefix of
+// ``0b'' or ``0B'' selects base 2, ``0o'' or ``0O'' selects base 8, and
+// ``0x'' or ``0X'' selects base 16. If fracOk is false, a ``0'' prefix
+// (immediately followed by digits) selects base 8 as well. Otherwise,
+// the selected base is 10 and no prefix is accepted.
+//
+// If fracOk is set, a period followed by a fractional part is permitted.
+// The result value is computed as if there were no period present; and
+// the count value is used to determine the fractional part.
+//
+// For bases <= 36, lower and upper case letters are considered the same:
+// The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
+// For bases > 36, the upper case letters 'A' to 'Z' represent the digit
+// values 36 to 61.
+//
+// A result digit count > 0 corresponds to the number of (non-prefix) digits
+// parsed. A digit count <= 0 indicates the presence of a period (if fracOk
+// is set, only), and -count is the number of fractional digits found.
+// In this case, the actual value of the scanned number is res * b**count.
+//
+func (z nat) scan(r io.ByteScanner, base int, fracOk bool) (res nat, b, count int, err error) {
+ // reject invalid bases
+ baseOk := base == 0 ||
+ !fracOk && 2 <= base && base <= MaxBase ||
+ fracOk && (base == 2 || base == 8 || base == 10 || base == 16)
+ if !baseOk {
+ panic(fmt.Sprintf("invalid number base %d", base))
+ }
+
+ // prev encodes the previously seen char: it is one
+ // of '_', '0' (a digit), or '.' (anything else). A
+ // valid separator '_' may only occur after a digit
+ // and if base == 0.
+ prev := '.'
+ invalSep := false
+
+ // one char look-ahead
+ ch, err := r.ReadByte()
+
+ // determine actual base
+ b, prefix := base, 0
+ if base == 0 {
+ // actual base is 10 unless there's a base prefix
+ b = 10
+ if err == nil && ch == '0' {
+ prev = '0'
+ count = 1
+ ch, err = r.ReadByte()
+ if err == nil {
+ // possibly one of 0b, 0B, 0o, 0O, 0x, 0X
+ switch ch {
+ case 'b', 'B':
+ b, prefix = 2, 'b'
+ case 'o', 'O':
+ b, prefix = 8, 'o'
+ case 'x', 'X':
+ b, prefix = 16, 'x'
+ default:
+ if !fracOk {
+ b, prefix = 8, '0'
+ }
+ }
+ if prefix != 0 {
+ count = 0 // prefix is not counted
+ if prefix != '0' {
+ ch, err = r.ReadByte()
+ }
+ }
+ }
+ }
+ }
+
+ // convert string
+ // Algorithm: Collect digits in groups of at most n digits in di
+ // and then use mulAddWW for every such group to add them to the
+ // result.
+ z = z[:0]
+ b1 := Word(b)
+ bn, n := maxPow(b1) // at most n digits in base b1 fit into Word
+ di := Word(0) // 0 <= di < b1**i < bn
+ i := 0 // 0 <= i < n
+ dp := -1 // position of decimal point
+ for err == nil {
+ if ch == '.' && fracOk {
+ fracOk = false
+ if prev == '_' {
+ invalSep = true
+ }
+ prev = '.'
+ dp = count
+ } else if ch == '_' && base == 0 {
+ if prev != '0' {
+ invalSep = true
+ }
+ prev = '_'
+ } else {
+ // convert rune into digit value d1
+ var d1 Word
+ switch {
+ case '0' <= ch && ch <= '9':
+ d1 = Word(ch - '0')
+ case 'a' <= ch && ch <= 'z':
+ d1 = Word(ch - 'a' + 10)
+ case 'A' <= ch && ch <= 'Z':
+ if b <= maxBaseSmall {
+ d1 = Word(ch - 'A' + 10)
+ } else {
+ d1 = Word(ch - 'A' + maxBaseSmall)
+ }
+ default:
+ d1 = MaxBase + 1
+ }
+ if d1 >= b1 {
+ r.UnreadByte() // ch does not belong to number anymore
+ break
+ }
+ prev = '0'
+ count++
+
+ // collect d1 in di
+ di = di*b1 + d1
+ i++
+
+ // if di is "full", add it to the result
+ if i == n {
+ z = z.mulAddWW(z, bn, di)
+ di = 0
+ i = 0
+ }
+ }
+
+ ch, err = r.ReadByte()
+ }
+
+ if err == io.EOF {
+ err = nil
+ }
+
+ // other errors take precedence over invalid separators
+ if err == nil && (invalSep || prev == '_') {
+ err = errInvalSep
+ }
+
+ if count == 0 {
+ // no digits found
+ if prefix == '0' {
+ // there was only the octal prefix 0 (possibly followed by separators and digits > 7);
+ // interpret as decimal 0
+ return z[:0], 10, 1, err
+ }
+ err = errNoDigits // fall through; result will be 0
+ }
+
+ // add remaining digits to result
+ if i > 0 {
+ z = z.mulAddWW(z, pow(b1, i), di)
+ }
+ res = z.norm()
+
+ // adjust count for fraction, if any
+ if dp >= 0 {
+ // 0 <= dp <= count
+ count = dp - count
+ }
+
+ return
+}
+
+// utoa converts x to an ASCII representation in the given base;
+// base must be between 2 and MaxBase, inclusive.
+func (x nat) utoa(base int) []byte {
+ return x.itoa(false, base)
+}
+
+// itoa is like utoa but it prepends a '-' if neg && x != 0.
+func (x nat) itoa(neg bool, base int) []byte {
+ if base < 2 || base > MaxBase {
+ panic("invalid base")
+ }
+
+ // x == 0
+ if len(x) == 0 {
+ return []byte("0")
+ }
+ // len(x) > 0
+
+ // allocate buffer for conversion
+ i := int(float64(x.bitLen())/math.Log2(float64(base))) + 1 // off by 1 at most
+ if neg {
+ i++
+ }
+ s := make([]byte, i)
+
+ // convert power of two and non power of two bases separately
+ if b := Word(base); b == b&-b {
+ // shift is base b digit size in bits
+ shift := uint(bits.TrailingZeros(uint(b))) // shift > 0 because b >= 2
+ mask := Word(1<<shift - 1)
+ w := x[0] // current word
+ nbits := uint(_W) // number of unprocessed bits in w
+
+ // convert less-significant words (include leading zeros)
+ for k := 1; k < len(x); k++ {
+ // convert full digits
+ for nbits >= shift {
+ i--
+ s[i] = digits[w&mask]
+ w >>= shift
+ nbits -= shift
+ }
+
+ // convert any partial leading digit and advance to next word
+ if nbits == 0 {
+ // no partial digit remaining, just advance
+ w = x[k]
+ nbits = _W
+ } else {
+ // partial digit in current word w (== x[k-1]) and next word x[k]
+ w |= x[k] << nbits
+ i--
+ s[i] = digits[w&mask]
+
+ // advance
+ w = x[k] >> (shift - nbits)
+ nbits = _W - (shift - nbits)
+ }
+ }
+
+ // convert digits of most-significant word w (omit leading zeros)
+ for w != 0 {
+ i--
+ s[i] = digits[w&mask]
+ w >>= shift
+ }
+
+ } else {
+ bb, ndigits := maxPow(b)
+
+ // construct table of successive squares of bb*leafSize to use in subdivisions
+ // result (table != nil) <=> (len(x) > leafSize > 0)
+ table := divisors(len(x), b, ndigits, bb)
+
+ // preserve x, create local copy for use by convertWords
+ q := nat(nil).set(x)
+
+ // convert q to string s in base b
+ q.convertWords(s, b, ndigits, bb, table)
+
+ // strip leading zeros
+ // (x != 0; thus s must contain at least one non-zero digit
+ // and the loop will terminate)
+ i = 0
+ for s[i] == '0' {
+ i++
+ }
+ }
+
+ if neg {
+ i--
+ s[i] = '-'
+ }
+
+ return s[i:]
+}
+
+// Convert words of q to base b digits in s. If q is large, it is recursively "split in half"
+// by nat/nat division using tabulated divisors. Otherwise, it is converted iteratively using
+// repeated nat/Word division.
+//
+// The iterative method processes n Words by n divW() calls, each of which visits every Word in the
+// incrementally shortened q for a total of n + (n-1) + (n-2) ... + 2 + 1, or n(n+1)/2 divW()'s.
+// Recursive conversion divides q by its approximate square root, yielding two parts, each half
+// the size of q. Using the iterative method on both halves means 2 * (n/2)(n/2 + 1)/2 divW()'s
+// plus the expensive long div(). Asymptotically, the ratio is favorable at 1/2 the divW()'s, and
+// is made better by splitting the subblocks recursively. Best is to split blocks until one more
+// split would take longer (because of the nat/nat div()) than the twice as many divW()'s of the
+// iterative approach. This threshold is represented by leafSize. Benchmarking of leafSize in the
+// range 2..64 shows that values of 8 and 16 work well, with a 4x speedup at medium lengths and
+// ~30x for 20000 digits. Use nat_test.go's BenchmarkLeafSize tests to optimize leafSize for
+// specific hardware.
+//
+func (q nat) convertWords(s []byte, b Word, ndigits int, bb Word, table []divisor) {
+ // split larger blocks recursively
+ if table != nil {
+ // len(q) > leafSize > 0
+ var r nat
+ index := len(table) - 1
+ for len(q) > leafSize {
+ // find divisor close to sqrt(q) if possible, but in any case < q
+ maxLength := q.bitLen() // ~= log2 q, or at of least largest possible q of this bit length
+ minLength := maxLength >> 1 // ~= log2 sqrt(q)
+ for index > 0 && table[index-1].nbits > minLength {
+ index-- // desired
+ }
+ if table[index].nbits >= maxLength && table[index].bbb.cmp(q) >= 0 {
+ index--
+ if index < 0 {
+ panic("internal inconsistency")
+ }
+ }
+
+ // split q into the two digit number (q'*bbb + r) to form independent subblocks
+ q, r = q.div(r, q, table[index].bbb)
+
+ // convert subblocks and collect results in s[:h] and s[h:]
+ h := len(s) - table[index].ndigits
+ r.convertWords(s[h:], b, ndigits, bb, table[0:index])
+ s = s[:h] // == q.convertWords(s, b, ndigits, bb, table[0:index+1])
+ }
+ }
+
+ // having split any large blocks now process the remaining (small) block iteratively
+ i := len(s)
+ var r Word
+ if b == 10 {
+ // hard-coding for 10 here speeds this up by 1.25x (allows for / and % by constants)
+ for len(q) > 0 {
+ // extract least significant, base bb "digit"
+ q, r = q.divW(q, bb)
+ for j := 0; j < ndigits && i > 0; j++ {
+ i--
+ // avoid % computation since r%10 == r - int(r/10)*10;
+ // this appears to be faster for BenchmarkString10000Base10
+ // and smaller strings (but a bit slower for larger ones)
+ t := r / 10
+ s[i] = '0' + byte(r-t*10)
+ r = t
+ }
+ }
+ } else {
+ for len(q) > 0 {
+ // extract least significant, base bb "digit"
+ q, r = q.divW(q, bb)
+ for j := 0; j < ndigits && i > 0; j++ {
+ i--
+ s[i] = digits[r%b]
+ r /= b
+ }
+ }
+ }
+
+ // prepend high-order zeros
+ for i > 0 { // while need more leading zeros
+ i--
+ s[i] = '0'
+ }
+}
+
+// Split blocks greater than leafSize Words (or set to 0 to disable recursive conversion)
+// Benchmark and configure leafSize using: go test -bench="Leaf"
+// 8 and 16 effective on 3.0 GHz Xeon "Clovertown" CPU (128 byte cache lines)
+// 8 and 16 effective on 2.66 GHz Core 2 Duo "Penryn" CPU
+var leafSize int = 8 // number of Word-size binary values treat as a monolithic block
+
+type divisor struct {
+ bbb nat // divisor
+ nbits int // bit length of divisor (discounting leading zeros) ~= log2(bbb)
+ ndigits int // digit length of divisor in terms of output base digits
+}
+
+var cacheBase10 struct {
+ sync.Mutex
+ table [64]divisor // cached divisors for base 10
+}
+
+// expWW computes x**y
+func (z nat) expWW(x, y Word) nat {
+ return z.expNN(nat(nil).setWord(x), nat(nil).setWord(y), nil)
+}
+
+// construct table of powers of bb*leafSize to use in subdivisions
+func divisors(m int, b Word, ndigits int, bb Word) []divisor {
+ // only compute table when recursive conversion is enabled and x is large
+ if leafSize == 0 || m <= leafSize {
+ return nil
+ }
+
+ // determine k where (bb**leafSize)**(2**k) >= sqrt(x)
+ k := 1
+ for words := leafSize; words < m>>1 && k < len(cacheBase10.table); words <<= 1 {
+ k++
+ }
+
+ // reuse and extend existing table of divisors or create new table as appropriate
+ var table []divisor // for b == 10, table overlaps with cacheBase10.table
+ if b == 10 {
+ cacheBase10.Lock()
+ table = cacheBase10.table[0:k] // reuse old table for this conversion
+ } else {
+ table = make([]divisor, k) // create new table for this conversion
+ }
+
+ // extend table
+ if table[k-1].ndigits == 0 {
+ // add new entries as needed
+ var larger nat
+ for i := 0; i < k; i++ {
+ if table[i].ndigits == 0 {
+ if i == 0 {
+ table[0].bbb = nat(nil).expWW(bb, Word(leafSize))
+ table[0].ndigits = ndigits * leafSize
+ } else {
+ table[i].bbb = nat(nil).sqr(table[i-1].bbb)
+ table[i].ndigits = 2 * table[i-1].ndigits
+ }
+
+ // optimization: exploit aggregated extra bits in macro blocks
+ larger = nat(nil).set(table[i].bbb)
+ for mulAddVWW(larger, larger, b, 0) == 0 {
+ table[i].bbb = table[i].bbb.set(larger)
+ table[i].ndigits++
+ }
+
+ table[i].nbits = table[i].bbb.bitLen()
+ }
+ }
+ }
+
+ if b == 10 {
+ cacheBase10.Unlock()
+ }
+
+ return table
+}
diff --git a/src/math/big/natconv_test.go b/src/math/big/natconv_test.go
new file mode 100644
index 0000000..d390272
--- /dev/null
+++ b/src/math/big/natconv_test.go
@@ -0,0 +1,463 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "fmt"
+ "io"
+ "math/bits"
+ "strings"
+ "testing"
+)
+
+func TestMaxBase(t *testing.T) {
+ if MaxBase != len(digits) {
+ t.Fatalf("%d != %d", MaxBase, len(digits))
+ }
+}
+
+// log2 computes the integer binary logarithm of x.
+// The result is the integer n for which 2^n <= x < 2^(n+1).
+// If x == 0, the result is -1.
+func log2(x Word) int {
+ return bits.Len(uint(x)) - 1
+}
+
+func itoa(x nat, base int) []byte {
+ // special cases
+ switch {
+ case base < 2:
+ panic("illegal base")
+ case len(x) == 0:
+ return []byte("0")
+ }
+
+ // allocate buffer for conversion
+ i := x.bitLen()/log2(Word(base)) + 1 // +1: round up
+ s := make([]byte, i)
+
+ // don't destroy x
+ q := nat(nil).set(x)
+
+ // convert
+ for len(q) > 0 {
+ i--
+ var r Word
+ q, r = q.divW(q, Word(base))
+ s[i] = digits[r]
+ }
+
+ return s[i:]
+}
+
+var strTests = []struct {
+ x nat // nat value to be converted
+ b int // conversion base
+ s string // expected result
+}{
+ {nil, 2, "0"},
+ {nat{1}, 2, "1"},
+ {nat{0xc5}, 2, "11000101"},
+ {nat{03271}, 8, "3271"},
+ {nat{10}, 10, "10"},
+ {nat{1234567890}, 10, "1234567890"},
+ {nat{0xdeadbeef}, 16, "deadbeef"},
+ {nat{0x229be7}, 17, "1a2b3c"},
+ {nat{0x309663e6}, 32, "o9cov6"},
+ {nat{0x309663e6}, 62, "TakXI"},
+}
+
+func TestString(t *testing.T) {
+ // test invalid base explicitly
+ var panicStr string
+ func() {
+ defer func() {
+ panicStr = recover().(string)
+ }()
+ natOne.utoa(1)
+ }()
+ if panicStr != "invalid base" {
+ t.Errorf("expected panic for invalid base")
+ }
+
+ for _, a := range strTests {
+ s := string(a.x.utoa(a.b))
+ if s != a.s {
+ t.Errorf("string%+v\n\tgot s = %s; want %s", a, s, a.s)
+ }
+
+ x, b, _, err := nat(nil).scan(strings.NewReader(a.s), a.b, false)
+ if x.cmp(a.x) != 0 {
+ t.Errorf("scan%+v\n\tgot z = %v; want %v", a, x, a.x)
+ }
+ if b != a.b {
+ t.Errorf("scan%+v\n\tgot b = %d; want %d", a, b, a.b)
+ }
+ if err != nil {
+ t.Errorf("scan%+v\n\tgot error = %s", a, err)
+ }
+ }
+}
+
+var natScanTests = []struct {
+ s string // string to be scanned
+ base int // input base
+ frac bool // fraction ok
+ x nat // expected nat
+ b int // expected base
+ count int // expected digit count
+ err error // expected error
+ next rune // next character (or 0, if at EOF)
+}{
+ // invalid: no digits
+ {"", 0, false, nil, 10, 0, errNoDigits, 0},
+ {"_", 0, false, nil, 10, 0, errNoDigits, 0},
+ {"?", 0, false, nil, 10, 0, errNoDigits, '?'},
+ {"?", 10, false, nil, 10, 0, errNoDigits, '?'},
+ {"", 10, false, nil, 10, 0, errNoDigits, 0},
+ {"", 36, false, nil, 36, 0, errNoDigits, 0},
+ {"", 62, false, nil, 62, 0, errNoDigits, 0},
+ {"0b", 0, false, nil, 2, 0, errNoDigits, 0},
+ {"0o", 0, false, nil, 8, 0, errNoDigits, 0},
+ {"0x", 0, false, nil, 16, 0, errNoDigits, 0},
+ {"0x_", 0, false, nil, 16, 0, errNoDigits, 0},
+ {"0b2", 0, false, nil, 2, 0, errNoDigits, '2'},
+ {"0B2", 0, false, nil, 2, 0, errNoDigits, '2'},
+ {"0o8", 0, false, nil, 8, 0, errNoDigits, '8'},
+ {"0O8", 0, false, nil, 8, 0, errNoDigits, '8'},
+ {"0xg", 0, false, nil, 16, 0, errNoDigits, 'g'},
+ {"0Xg", 0, false, nil, 16, 0, errNoDigits, 'g'},
+ {"345", 2, false, nil, 2, 0, errNoDigits, '3'},
+
+ // invalid: incorrect use of decimal point
+ {"._", 0, true, nil, 10, 0, errNoDigits, 0},
+ {".0", 0, false, nil, 10, 0, errNoDigits, '.'},
+ {".0", 10, false, nil, 10, 0, errNoDigits, '.'},
+ {".", 0, true, nil, 10, 0, errNoDigits, 0},
+ {"0x.", 0, true, nil, 16, 0, errNoDigits, 0},
+ {"0x.g", 0, true, nil, 16, 0, errNoDigits, 'g'},
+ {"0x.0", 0, false, nil, 16, 0, errNoDigits, '.'},
+
+ // invalid: incorrect use of separators
+ {"_0", 0, false, nil, 10, 1, errInvalSep, 0},
+ {"0_", 0, false, nil, 10, 1, errInvalSep, 0},
+ {"0__0", 0, false, nil, 8, 1, errInvalSep, 0},
+ {"0x___0", 0, false, nil, 16, 1, errInvalSep, 0},
+ {"0_x", 0, false, nil, 10, 1, errInvalSep, 'x'},
+ {"0_8", 0, false, nil, 10, 1, errInvalSep, '8'},
+ {"123_.", 0, true, nat{123}, 10, 0, errInvalSep, 0},
+ {"._123", 0, true, nat{123}, 10, -3, errInvalSep, 0},
+ {"0b__1000", 0, false, nat{0x8}, 2, 4, errInvalSep, 0},
+ {"0o60___0", 0, false, nat{0600}, 8, 3, errInvalSep, 0},
+ {"0466_", 0, false, nat{0466}, 8, 3, errInvalSep, 0},
+ {"01234567_8", 0, false, nat{01234567}, 8, 7, errInvalSep, '8'},
+ {"1_.", 0, true, nat{1}, 10, 0, errInvalSep, 0},
+ {"0._1", 0, true, nat{1}, 10, -1, errInvalSep, 0},
+ {"2.7_", 0, true, nat{27}, 10, -1, errInvalSep, 0},
+ {"0x1.0_", 0, true, nat{0x10}, 16, -1, errInvalSep, 0},
+
+ // valid: separators are not accepted for base != 0
+ {"0_", 10, false, nil, 10, 1, nil, '_'},
+ {"1__0", 10, false, nat{1}, 10, 1, nil, '_'},
+ {"0__8", 10, false, nil, 10, 1, nil, '_'},
+ {"xy_z_", 36, false, nat{33*36 + 34}, 36, 2, nil, '_'},
+
+ // valid, no decimal point
+ {"0", 0, false, nil, 10, 1, nil, 0},
+ {"0", 36, false, nil, 36, 1, nil, 0},
+ {"0", 62, false, nil, 62, 1, nil, 0},
+ {"1", 0, false, nat{1}, 10, 1, nil, 0},
+ {"1", 10, false, nat{1}, 10, 1, nil, 0},
+ {"0 ", 0, false, nil, 10, 1, nil, ' '},
+ {"00 ", 0, false, nil, 8, 1, nil, ' '}, // octal 0
+ {"0b1", 0, false, nat{1}, 2, 1, nil, 0},
+ {"0B11000101", 0, false, nat{0xc5}, 2, 8, nil, 0},
+ {"0B110001012", 0, false, nat{0xc5}, 2, 8, nil, '2'},
+ {"07", 0, false, nat{7}, 8, 1, nil, 0},
+ {"08", 0, false, nil, 10, 1, nil, '8'},
+ {"08", 10, false, nat{8}, 10, 2, nil, 0},
+ {"018", 0, false, nat{1}, 8, 1, nil, '8'},
+ {"0o7", 0, false, nat{7}, 8, 1, nil, 0},
+ {"0o18", 0, false, nat{1}, 8, 1, nil, '8'},
+ {"0O17", 0, false, nat{017}, 8, 2, nil, 0},
+ {"03271", 0, false, nat{03271}, 8, 4, nil, 0},
+ {"10ab", 0, false, nat{10}, 10, 2, nil, 'a'},
+ {"1234567890", 0, false, nat{1234567890}, 10, 10, nil, 0},
+ {"A", 36, false, nat{10}, 36, 1, nil, 0},
+ {"A", 37, false, nat{36}, 37, 1, nil, 0},
+ {"xyz", 36, false, nat{(33*36+34)*36 + 35}, 36, 3, nil, 0},
+ {"XYZ?", 36, false, nat{(33*36+34)*36 + 35}, 36, 3, nil, '?'},
+ {"XYZ?", 62, false, nat{(59*62+60)*62 + 61}, 62, 3, nil, '?'},
+ {"0x", 16, false, nil, 16, 1, nil, 'x'},
+ {"0xdeadbeef", 0, false, nat{0xdeadbeef}, 16, 8, nil, 0},
+ {"0XDEADBEEF", 0, false, nat{0xdeadbeef}, 16, 8, nil, 0},
+
+ // valid, with decimal point
+ {"0.", 0, false, nil, 10, 1, nil, '.'},
+ {"0.", 10, true, nil, 10, 0, nil, 0},
+ {"0.1.2", 10, true, nat{1}, 10, -1, nil, '.'},
+ {".000", 10, true, nil, 10, -3, nil, 0},
+ {"12.3", 10, true, nat{123}, 10, -1, nil, 0},
+ {"012.345", 10, true, nat{12345}, 10, -3, nil, 0},
+ {"0.1", 0, true, nat{1}, 10, -1, nil, 0},
+ {"0.1", 2, true, nat{1}, 2, -1, nil, 0},
+ {"0.12", 2, true, nat{1}, 2, -1, nil, '2'},
+ {"0b0.1", 0, true, nat{1}, 2, -1, nil, 0},
+ {"0B0.12", 0, true, nat{1}, 2, -1, nil, '2'},
+ {"0o0.7", 0, true, nat{7}, 8, -1, nil, 0},
+ {"0O0.78", 0, true, nat{7}, 8, -1, nil, '8'},
+ {"0xdead.beef", 0, true, nat{0xdeadbeef}, 16, -4, nil, 0},
+
+ // valid, with separators
+ {"1_000", 0, false, nat{1000}, 10, 4, nil, 0},
+ {"0_466", 0, false, nat{0466}, 8, 3, nil, 0},
+ {"0o_600", 0, false, nat{0600}, 8, 3, nil, 0},
+ {"0x_f0_0d", 0, false, nat{0xf00d}, 16, 4, nil, 0},
+ {"0b1000_0001", 0, false, nat{0x81}, 2, 8, nil, 0},
+ {"1_000.000_1", 0, true, nat{10000001}, 10, -4, nil, 0},
+ {"0x_f00d.1e", 0, true, nat{0xf00d1e}, 16, -2, nil, 0},
+ {"0x_f00d.1E2", 0, true, nat{0xf00d1e2}, 16, -3, nil, 0},
+ {"0x_f00d.1eg", 0, true, nat{0xf00d1e}, 16, -2, nil, 'g'},
+}
+
+func TestScanBase(t *testing.T) {
+ for _, a := range natScanTests {
+ r := strings.NewReader(a.s)
+ x, b, count, err := nat(nil).scan(r, a.base, a.frac)
+ if err != a.err {
+ t.Errorf("scan%+v\n\tgot error = %v; want %v", a, err, a.err)
+ }
+ if x.cmp(a.x) != 0 {
+ t.Errorf("scan%+v\n\tgot z = %v; want %v", a, x, a.x)
+ }
+ if b != a.b {
+ t.Errorf("scan%+v\n\tgot b = %d; want %d", a, b, a.base)
+ }
+ if count != a.count {
+ t.Errorf("scan%+v\n\tgot count = %d; want %d", a, count, a.count)
+ }
+ next, _, err := r.ReadRune()
+ if err == io.EOF {
+ next = 0
+ err = nil
+ }
+ if err == nil && next != a.next {
+ t.Errorf("scan%+v\n\tgot next = %q; want %q", a, next, a.next)
+ }
+ }
+}
+
+var pi = "3" +
+ "14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651" +
+ "32823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461" +
+ "28475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920" +
+ "96282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179" +
+ "31051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798" +
+ "60943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901" +
+ "22495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837" +
+ "29780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083" +
+ "81420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909" +
+ "21642019893809525720106548586327886593615338182796823030195203530185296899577362259941389124972177528347913151" +
+ "55748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035" +
+ "63707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104" +
+ "75216205696602405803815019351125338243003558764024749647326391419927260426992279678235478163600934172164121992" +
+ "45863150302861829745557067498385054945885869269956909272107975093029553211653449872027559602364806654991198818" +
+ "34797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548" +
+ "16136115735255213347574184946843852332390739414333454776241686251898356948556209921922218427255025425688767179" +
+ "04946016534668049886272327917860857843838279679766814541009538837863609506800642251252051173929848960841284886" +
+ "26945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645" +
+ "99581339047802759009946576407895126946839835259570982582262052248940772671947826848260147699090264013639443745" +
+ "53050682034962524517493996514314298091906592509372216964615157098583874105978859597729754989301617539284681382" +
+ "68683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244" +
+ "13654976278079771569143599770012961608944169486855584840635342207222582848864815845602850601684273945226746767" +
+ "88952521385225499546667278239864565961163548862305774564980355936345681743241125150760694794510965960940252288" +
+ "79710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821" +
+ "68299894872265880485756401427047755513237964145152374623436454285844479526586782105114135473573952311342716610" +
+ "21359695362314429524849371871101457654035902799344037420073105785390621983874478084784896833214457138687519435" +
+ "06430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675" +
+ "14269123974894090718649423196156794520809514655022523160388193014209376213785595663893778708303906979207734672" +
+ "21825625996615014215030680384477345492026054146659252014974428507325186660021324340881907104863317346496514539" +
+ "05796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007" +
+ "23055876317635942187312514712053292819182618612586732157919841484882916447060957527069572209175671167229109816" +
+ "90915280173506712748583222871835209353965725121083579151369882091444210067510334671103141267111369908658516398" +
+ "31501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064" +
+ "20467525907091548141654985946163718027098199430992448895757128289059232332609729971208443357326548938239119325" +
+ "97463667305836041428138830320382490375898524374417029132765618093773444030707469211201913020330380197621101100" +
+ "44929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915" +
+ "44110104468232527162010526522721116603966655730925471105578537634668206531098965269186205647693125705863566201" +
+ "85581007293606598764861179104533488503461136576867532494416680396265797877185560845529654126654085306143444318" +
+ "58676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797" +
+ "27082668306343285878569830523580893306575740679545716377525420211495576158140025012622859413021647155097925923" +
+ "09907965473761255176567513575178296664547791745011299614890304639947132962107340437518957359614589019389713111" +
+ "79042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043" +
+ "06332175182979866223717215916077166925474873898665494945011465406284336639379003976926567214638530673609657120" +
+ "91807638327166416274888800786925602902284721040317211860820419000422966171196377921337575114959501566049631862" +
+ "94726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939" +
+ "78054193414473774418426312986080998886874132604721569516239658645730216315981931951673538129741677294786724229" +
+ "24654366800980676928238280689964004824354037014163149658979409243237896907069779422362508221688957383798623001" +
+ "59377647165122893578601588161755782973523344604281512627203734314653197777416031990665541876397929334419521541" +
+ "34189948544473456738316249934191318148092777710386387734317720754565453220777092120190516609628049092636019759" +
+ "88281613323166636528619326686336062735676303544776280350450777235547105859548702790814356240145171806246436267" +
+ "94561275318134078330336254232783944975382437205835311477119926063813346776879695970309833913077109870408591337"
+
+// Test case for BenchmarkScanPi.
+func TestScanPi(t *testing.T) {
+ var x nat
+ z, _, _, err := x.scan(strings.NewReader(pi), 10, false)
+ if err != nil {
+ t.Errorf("scanning pi: %s", err)
+ }
+ if s := string(z.utoa(10)); s != pi {
+ t.Errorf("scanning pi: got %s", s)
+ }
+}
+
+func TestScanPiParallel(t *testing.T) {
+ const n = 2
+ c := make(chan int)
+ for i := 0; i < n; i++ {
+ go func() {
+ TestScanPi(t)
+ c <- 0
+ }()
+ }
+ for i := 0; i < n; i++ {
+ <-c
+ }
+}
+
+func BenchmarkScanPi(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ var x nat
+ x.scan(strings.NewReader(pi), 10, false)
+ }
+}
+
+func BenchmarkStringPiParallel(b *testing.B) {
+ var x nat
+ x, _, _, _ = x.scan(strings.NewReader(pi), 0, false)
+ if string(x.utoa(10)) != pi {
+ panic("benchmark incorrect: conversion failed")
+ }
+ b.RunParallel(func(pb *testing.PB) {
+ for pb.Next() {
+ x.utoa(10)
+ }
+ })
+}
+
+func BenchmarkScan(b *testing.B) {
+ const x = 10
+ for _, base := range []int{2, 8, 10, 16} {
+ for _, y := range []Word{10, 100, 1000, 10000, 100000} {
+ if isRaceBuilder && y > 1000 {
+ continue
+ }
+ b.Run(fmt.Sprintf("%d/Base%d", y, base), func(b *testing.B) {
+ b.StopTimer()
+ var z nat
+ z = z.expWW(x, y)
+
+ s := z.utoa(base)
+ if t := itoa(z, base); !bytes.Equal(s, t) {
+ b.Fatalf("scanning: got %s; want %s", s, t)
+ }
+ b.StartTimer()
+
+ for i := 0; i < b.N; i++ {
+ z.scan(bytes.NewReader(s), base, false)
+ }
+ })
+ }
+ }
+}
+
+func BenchmarkString(b *testing.B) {
+ const x = 10
+ for _, base := range []int{2, 8, 10, 16} {
+ for _, y := range []Word{10, 100, 1000, 10000, 100000} {
+ if isRaceBuilder && y > 1000 {
+ continue
+ }
+ b.Run(fmt.Sprintf("%d/Base%d", y, base), func(b *testing.B) {
+ b.StopTimer()
+ var z nat
+ z = z.expWW(x, y)
+ z.utoa(base) // warm divisor cache
+ b.StartTimer()
+
+ for i := 0; i < b.N; i++ {
+ _ = z.utoa(base)
+ }
+ })
+ }
+ }
+}
+
+func BenchmarkLeafSize(b *testing.B) {
+ for n := 0; n <= 16; n++ {
+ b.Run(fmt.Sprint(n), func(b *testing.B) { LeafSizeHelper(b, 10, n) })
+ }
+ // Try some large lengths
+ for _, n := range []int{32, 64} {
+ b.Run(fmt.Sprint(n), func(b *testing.B) { LeafSizeHelper(b, 10, n) })
+ }
+}
+
+func LeafSizeHelper(b *testing.B, base, size int) {
+ b.StopTimer()
+ originalLeafSize := leafSize
+ resetTable(cacheBase10.table[:])
+ leafSize = size
+ b.StartTimer()
+
+ for d := 1; d <= 10000; d *= 10 {
+ b.StopTimer()
+ var z nat
+ z = z.expWW(Word(base), Word(d)) // build target number
+ _ = z.utoa(base) // warm divisor cache
+ b.StartTimer()
+
+ for i := 0; i < b.N; i++ {
+ _ = z.utoa(base)
+ }
+ }
+
+ b.StopTimer()
+ resetTable(cacheBase10.table[:])
+ leafSize = originalLeafSize
+ b.StartTimer()
+}
+
+func resetTable(table []divisor) {
+ if table != nil && table[0].bbb != nil {
+ for i := 0; i < len(table); i++ {
+ table[i].bbb = nil
+ table[i].nbits = 0
+ table[i].ndigits = 0
+ }
+ }
+}
+
+func TestStringPowers(t *testing.T) {
+ var p Word
+ for b := 2; b <= 16; b++ {
+ for p = 0; p <= 512; p++ {
+ if testing.Short() && p > 10 {
+ break
+ }
+ x := nat(nil).expWW(Word(b), p)
+ xs := x.utoa(b)
+ xs2 := itoa(x, b)
+ if !bytes.Equal(xs, xs2) {
+ t.Errorf("failed at %d ** %d in base %d: %s != %s", b, p, b, xs, xs2)
+ }
+ }
+ if b >= 3 && testing.Short() {
+ break
+ }
+ }
+}
diff --git a/src/math/big/prime.go b/src/math/big/prime.go
new file mode 100644
index 0000000..d9a5f1e
--- /dev/null
+++ b/src/math/big/prime.go
@@ -0,0 +1,320 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import "math/rand"
+
+// ProbablyPrime reports whether x is probably prime,
+// applying the Miller-Rabin test with n pseudorandomly chosen bases
+// as well as a Baillie-PSW test.
+//
+// If x is prime, ProbablyPrime returns true.
+// If x is chosen randomly and not prime, ProbablyPrime probably returns false.
+// The probability of returning true for a randomly chosen non-prime is at most ¼ⁿ.
+//
+// ProbablyPrime is 100% accurate for inputs less than 2⁶⁴.
+// See Menezes et al., Handbook of Applied Cryptography, 1997, pp. 145-149,
+// and FIPS 186-4 Appendix F for further discussion of the error probabilities.
+//
+// ProbablyPrime is not suitable for judging primes that an adversary may
+// have crafted to fool the test.
+//
+// As of Go 1.8, ProbablyPrime(0) is allowed and applies only a Baillie-PSW test.
+// Before Go 1.8, ProbablyPrime applied only the Miller-Rabin tests, and ProbablyPrime(0) panicked.
+func (x *Int) ProbablyPrime(n int) bool {
+ // Note regarding the doc comment above:
+ // It would be more precise to say that the Baillie-PSW test uses the
+ // extra strong Lucas test as its Lucas test, but since no one knows
+ // how to tell any of the Lucas tests apart inside a Baillie-PSW test
+ // (they all work equally well empirically), that detail need not be
+ // documented or implicitly guaranteed.
+ // The comment does avoid saying "the" Baillie-PSW test
+ // because of this general ambiguity.
+
+ if n < 0 {
+ panic("negative n for ProbablyPrime")
+ }
+ if x.neg || len(x.abs) == 0 {
+ return false
+ }
+
+ // primeBitMask records the primes < 64.
+ const primeBitMask uint64 = 1<<2 | 1<<3 | 1<<5 | 1<<7 |
+ 1<<11 | 1<<13 | 1<<17 | 1<<19 | 1<<23 | 1<<29 | 1<<31 |
+ 1<<37 | 1<<41 | 1<<43 | 1<<47 | 1<<53 | 1<<59 | 1<<61
+
+ w := x.abs[0]
+ if len(x.abs) == 1 && w < 64 {
+ return primeBitMask&(1<<w) != 0
+ }
+
+ if w&1 == 0 {
+ return false // x is even
+ }
+
+ const primesA = 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 37
+ const primesB = 29 * 31 * 41 * 43 * 47 * 53
+
+ var rA, rB uint32
+ switch _W {
+ case 32:
+ rA = uint32(x.abs.modW(primesA))
+ rB = uint32(x.abs.modW(primesB))
+ case 64:
+ r := x.abs.modW((primesA * primesB) & _M)
+ rA = uint32(r % primesA)
+ rB = uint32(r % primesB)
+ default:
+ panic("math/big: invalid word size")
+ }
+
+ if rA%3 == 0 || rA%5 == 0 || rA%7 == 0 || rA%11 == 0 || rA%13 == 0 || rA%17 == 0 || rA%19 == 0 || rA%23 == 0 || rA%37 == 0 ||
+ rB%29 == 0 || rB%31 == 0 || rB%41 == 0 || rB%43 == 0 || rB%47 == 0 || rB%53 == 0 {
+ return false
+ }
+
+ return x.abs.probablyPrimeMillerRabin(n+1, true) && x.abs.probablyPrimeLucas()
+}
+
+// probablyPrimeMillerRabin reports whether n passes reps rounds of the
+// Miller-Rabin primality test, using pseudo-randomly chosen bases.
+// If force2 is true, one of the rounds is forced to use base 2.
+// See Handbook of Applied Cryptography, p. 139, Algorithm 4.24.
+// The number n is known to be non-zero.
+func (n nat) probablyPrimeMillerRabin(reps int, force2 bool) bool {
+ nm1 := nat(nil).sub(n, natOne)
+ // determine q, k such that nm1 = q << k
+ k := nm1.trailingZeroBits()
+ q := nat(nil).shr(nm1, k)
+
+ nm3 := nat(nil).sub(nm1, natTwo)
+ rand := rand.New(rand.NewSource(int64(n[0])))
+
+ var x, y, quotient nat
+ nm3Len := nm3.bitLen()
+
+NextRandom:
+ for i := 0; i < reps; i++ {
+ if i == reps-1 && force2 {
+ x = x.set(natTwo)
+ } else {
+ x = x.random(rand, nm3, nm3Len)
+ x = x.add(x, natTwo)
+ }
+ y = y.expNN(x, q, n)
+ if y.cmp(natOne) == 0 || y.cmp(nm1) == 0 {
+ continue
+ }
+ for j := uint(1); j < k; j++ {
+ y = y.sqr(y)
+ quotient, y = quotient.div(y, y, n)
+ if y.cmp(nm1) == 0 {
+ continue NextRandom
+ }
+ if y.cmp(natOne) == 0 {
+ return false
+ }
+ }
+ return false
+ }
+
+ return true
+}
+
+// probablyPrimeLucas reports whether n passes the "almost extra strong" Lucas probable prime test,
+// using Baillie-OEIS parameter selection. This corresponds to "AESLPSP" on Jacobsen's tables (link below).
+// The combination of this test and a Miller-Rabin/Fermat test with base 2 gives a Baillie-PSW test.
+//
+// References:
+//
+// Baillie and Wagstaff, "Lucas Pseudoprimes", Mathematics of Computation 35(152),
+// October 1980, pp. 1391-1417, especially page 1401.
+// https://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583518-6/S0025-5718-1980-0583518-6.pdf
+//
+// Grantham, "Frobenius Pseudoprimes", Mathematics of Computation 70(234),
+// March 2000, pp. 873-891.
+// https://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01197-2/S0025-5718-00-01197-2.pdf
+//
+// Baillie, "Extra strong Lucas pseudoprimes", OEIS A217719, https://oeis.org/A217719.
+//
+// Jacobsen, "Pseudoprime Statistics, Tables, and Data", http://ntheory.org/pseudoprimes.html.
+//
+// Nicely, "The Baillie-PSW Primality Test", http://www.trnicely.net/misc/bpsw.html.
+// (Note that Nicely's definition of the "extra strong" test gives the wrong Jacobi condition,
+// as pointed out by Jacobsen.)
+//
+// Crandall and Pomerance, Prime Numbers: A Computational Perspective, 2nd ed.
+// Springer, 2005.
+func (n nat) probablyPrimeLucas() bool {
+ // Discard 0, 1.
+ if len(n) == 0 || n.cmp(natOne) == 0 {
+ return false
+ }
+ // Two is the only even prime.
+ // Already checked by caller, but here to allow testing in isolation.
+ if n[0]&1 == 0 {
+ return n.cmp(natTwo) == 0
+ }
+
+ // Baillie-OEIS "method C" for choosing D, P, Q,
+ // as in https://oeis.org/A217719/a217719.txt:
+ // try increasing P ≥ 3 such that D = P² - 4 (so Q = 1)
+ // until Jacobi(D, n) = -1.
+ // The search is expected to succeed for non-square n after just a few trials.
+ // After more than expected failures, check whether n is square
+ // (which would cause Jacobi(D, n) = 1 for all D not dividing n).
+ p := Word(3)
+ d := nat{1}
+ t1 := nat(nil) // temp
+ intD := &Int{abs: d}
+ intN := &Int{abs: n}
+ for ; ; p++ {
+ if p > 10000 {
+ // This is widely believed to be impossible.
+ // If we get a report, we'll want the exact number n.
+ panic("math/big: internal error: cannot find (D/n) = -1 for " + intN.String())
+ }
+ d[0] = p*p - 4
+ j := Jacobi(intD, intN)
+ if j == -1 {
+ break
+ }
+ if j == 0 {
+ // d = p²-4 = (p-2)(p+2).
+ // If (d/n) == 0 then d shares a prime factor with n.
+ // Since the loop proceeds in increasing p and starts with p-2==1,
+ // the shared prime factor must be p+2.
+ // If p+2 == n, then n is prime; otherwise p+2 is a proper factor of n.
+ return len(n) == 1 && n[0] == p+2
+ }
+ if p == 40 {
+ // We'll never find (d/n) = -1 if n is a square.
+ // If n is a non-square we expect to find a d in just a few attempts on average.
+ // After 40 attempts, take a moment to check if n is indeed a square.
+ t1 = t1.sqrt(n)
+ t1 = t1.sqr(t1)
+ if t1.cmp(n) == 0 {
+ return false
+ }
+ }
+ }
+
+ // Grantham definition of "extra strong Lucas pseudoprime", after Thm 2.3 on p. 876
+ // (D, P, Q above have become Δ, b, 1):
+ //
+ // Let U_n = U_n(b, 1), V_n = V_n(b, 1), and Δ = b²-4.
+ // An extra strong Lucas pseudoprime to base b is a composite n = 2^r s + Jacobi(Δ, n),
+ // where s is odd and gcd(n, 2*Δ) = 1, such that either (i) U_s ≡ 0 mod n and V_s ≡ ±2 mod n,
+ // or (ii) V_{2^t s} ≡ 0 mod n for some 0 ≤ t < r-1.
+ //
+ // We know gcd(n, Δ) = 1 or else we'd have found Jacobi(d, n) == 0 above.
+ // We know gcd(n, 2) = 1 because n is odd.
+ //
+ // Arrange s = (n - Jacobi(Δ, n)) / 2^r = (n+1) / 2^r.
+ s := nat(nil).add(n, natOne)
+ r := int(s.trailingZeroBits())
+ s = s.shr(s, uint(r))
+ nm2 := nat(nil).sub(n, natTwo) // n-2
+
+ // We apply the "almost extra strong" test, which checks the above conditions
+ // except for U_s ≡ 0 mod n, which allows us to avoid computing any U_k values.
+ // Jacobsen points out that maybe we should just do the full extra strong test:
+ // "It is also possible to recover U_n using Crandall and Pomerance equation 3.13:
+ // U_n = D^-1 (2V_{n+1} - PV_n) allowing us to run the full extra-strong test
+ // at the cost of a single modular inversion. This computation is easy and fast in GMP,
+ // so we can get the full extra-strong test at essentially the same performance as the
+ // almost extra strong test."
+
+ // Compute Lucas sequence V_s(b, 1), where:
+ //
+ // V(0) = 2
+ // V(1) = P
+ // V(k) = P V(k-1) - Q V(k-2).
+ //
+ // (Remember that due to method C above, P = b, Q = 1.)
+ //
+ // In general V(k) = α^k + β^k, where α and β are roots of x² - Px + Q.
+ // Crandall and Pomerance (p.147) observe that for 0 ≤ j ≤ k,
+ //
+ // V(j+k) = V(j)V(k) - V(k-j).
+ //
+ // So in particular, to quickly double the subscript:
+ //
+ // V(2k) = V(k)² - 2
+ // V(2k+1) = V(k) V(k+1) - P
+ //
+ // We can therefore start with k=0 and build up to k=s in log₂(s) steps.
+ natP := nat(nil).setWord(p)
+ vk := nat(nil).setWord(2)
+ vk1 := nat(nil).setWord(p)
+ t2 := nat(nil) // temp
+ for i := int(s.bitLen()); i >= 0; i-- {
+ if s.bit(uint(i)) != 0 {
+ // k' = 2k+1
+ // V(k') = V(2k+1) = V(k) V(k+1) - P.
+ t1 = t1.mul(vk, vk1)
+ t1 = t1.add(t1, n)
+ t1 = t1.sub(t1, natP)
+ t2, vk = t2.div(vk, t1, n)
+ // V(k'+1) = V(2k+2) = V(k+1)² - 2.
+ t1 = t1.sqr(vk1)
+ t1 = t1.add(t1, nm2)
+ t2, vk1 = t2.div(vk1, t1, n)
+ } else {
+ // k' = 2k
+ // V(k'+1) = V(2k+1) = V(k) V(k+1) - P.
+ t1 = t1.mul(vk, vk1)
+ t1 = t1.add(t1, n)
+ t1 = t1.sub(t1, natP)
+ t2, vk1 = t2.div(vk1, t1, n)
+ // V(k') = V(2k) = V(k)² - 2
+ t1 = t1.sqr(vk)
+ t1 = t1.add(t1, nm2)
+ t2, vk = t2.div(vk, t1, n)
+ }
+ }
+
+ // Now k=s, so vk = V(s). Check V(s) ≡ ±2 (mod n).
+ if vk.cmp(natTwo) == 0 || vk.cmp(nm2) == 0 {
+ // Check U(s) ≡ 0.
+ // As suggested by Jacobsen, apply Crandall and Pomerance equation 3.13:
+ //
+ // U(k) = D⁻¹ (2 V(k+1) - P V(k))
+ //
+ // Since we are checking for U(k) == 0 it suffices to check 2 V(k+1) == P V(k) mod n,
+ // or P V(k) - 2 V(k+1) == 0 mod n.
+ t1 := t1.mul(vk, natP)
+ t2 := t2.shl(vk1, 1)
+ if t1.cmp(t2) < 0 {
+ t1, t2 = t2, t1
+ }
+ t1 = t1.sub(t1, t2)
+ t3 := vk1 // steal vk1, no longer needed below
+ vk1 = nil
+ _ = vk1
+ t2, t3 = t2.div(t3, t1, n)
+ if len(t3) == 0 {
+ return true
+ }
+ }
+
+ // Check V(2^t s) ≡ 0 mod n for some 0 ≤ t < r-1.
+ for t := 0; t < r-1; t++ {
+ if len(vk) == 0 { // vk == 0
+ return true
+ }
+ // Optimization: V(k) = 2 is a fixed point for V(k') = V(k)² - 2,
+ // so if V(k) = 2, we can stop: we will never find a future V(k) == 0.
+ if len(vk) == 1 && vk[0] == 2 { // vk == 2
+ return false
+ }
+ // k' = 2k
+ // V(k') = V(2k) = V(k)² - 2
+ t1 = t1.sqr(vk)
+ t1 = t1.sub(t1, natTwo)
+ t2, vk = t2.div(vk, t1, n)
+ }
+ return false
+}
diff --git a/src/math/big/prime_test.go b/src/math/big/prime_test.go
new file mode 100644
index 0000000..8596e33
--- /dev/null
+++ b/src/math/big/prime_test.go
@@ -0,0 +1,222 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "fmt"
+ "strings"
+ "testing"
+ "unicode"
+)
+
+var primes = []string{
+ "2",
+ "3",
+ "5",
+ "7",
+ "11",
+
+ "13756265695458089029",
+ "13496181268022124907",
+ "10953742525620032441",
+ "17908251027575790097",
+
+ // https://golang.org/issue/638
+ "18699199384836356663",
+
+ "98920366548084643601728869055592650835572950932266967461790948584315647051443",
+ "94560208308847015747498523884063394671606671904944666360068158221458669711639",
+
+ // https://primes.utm.edu/lists/small/small3.html
+ "449417999055441493994709297093108513015373787049558499205492347871729927573118262811508386655998299074566974373711472560655026288668094291699357843464363003144674940345912431129144354948751003607115263071543163",
+ "230975859993204150666423538988557839555560243929065415434980904258310530753006723857139742334640122533598517597674807096648905501653461687601339782814316124971547968912893214002992086353183070342498989426570593",
+ "5521712099665906221540423207019333379125265462121169655563495403888449493493629943498064604536961775110765377745550377067893607246020694972959780839151452457728855382113555867743022746090187341871655890805971735385789993",
+ "203956878356401977405765866929034577280193993314348263094772646453283062722701277632936616063144088173312372882677123879538709400158306567338328279154499698366071906766440037074217117805690872792848149112022286332144876183376326512083574821647933992961249917319836219304274280243803104015000563790123",
+
+ // ECC primes: https://tools.ietf.org/html/draft-ladd-safecurves-02
+ "3618502788666131106986593281521497120414687020801267626233049500247285301239", // Curve1174: 2^251-9
+ "57896044618658097711785492504343953926634992332820282019728792003956564819949", // Curve25519: 2^255-19
+ "9850501549098619803069760025035903451269934817616361666987073351061430442874302652853566563721228910201656997576599", // E-382: 2^382-105
+ "42307582002575910332922579714097346549017899709713998034217522897561970639123926132812109468141778230245837569601494931472367", // Curve41417: 2^414-17
+ "6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", // E-521: 2^521-1
+}
+
+var composites = []string{
+ "0",
+ "1",
+ "21284175091214687912771199898307297748211672914763848041968395774954376176754",
+ "6084766654921918907427900243509372380954290099172559290432744450051395395951",
+ "84594350493221918389213352992032324280367711247940675652888030554255915464401",
+ "82793403787388584738507275144194252681",
+
+ // Arnault, "Rabin-Miller Primality Test: Composite Numbers Which Pass It",
+ // Mathematics of Computation, 64(209) (January 1995), pp. 335-361.
+ "1195068768795265792518361315725116351898245581", // strong pseudoprime to prime bases 2 through 29
+ // strong pseudoprime to all prime bases up to 200
+ `
+ 80383745745363949125707961434194210813883768828755814583748891752229
+ 74273765333652186502336163960045457915042023603208766569966760987284
+ 0439654082329287387918508691668573282677617710293896977394701670823
+ 0428687109997439976544144845341155872450633409279022275296229414984
+ 2306881685404326457534018329786111298960644845216191652872597534901`,
+
+ // Extra-strong Lucas pseudoprimes. https://oeis.org/A217719
+ "989",
+ "3239",
+ "5777",
+ "10877",
+ "27971",
+ "29681",
+ "30739",
+ "31631",
+ "39059",
+ "72389",
+ "73919",
+ "75077",
+ "100127",
+ "113573",
+ "125249",
+ "137549",
+ "137801",
+ "153931",
+ "155819",
+ "161027",
+ "162133",
+ "189419",
+ "218321",
+ "231703",
+ "249331",
+ "370229",
+ "429479",
+ "430127",
+ "459191",
+ "473891",
+ "480689",
+ "600059",
+ "621781",
+ "632249",
+ "635627",
+
+ "3673744903",
+ "3281593591",
+ "2385076987",
+ "2738053141",
+ "2009621503",
+ "1502682721",
+ "255866131",
+ "117987841",
+ "587861",
+
+ "6368689",
+ "8725753",
+ "80579735209",
+ "105919633",
+}
+
+func cutSpace(r rune) rune {
+ if unicode.IsSpace(r) {
+ return -1
+ }
+ return r
+}
+
+func TestProbablyPrime(t *testing.T) {
+ nreps := 20
+ if testing.Short() {
+ nreps = 1
+ }
+ for i, s := range primes {
+ p, _ := new(Int).SetString(s, 10)
+ if !p.ProbablyPrime(nreps) || nreps != 1 && !p.ProbablyPrime(1) || !p.ProbablyPrime(0) {
+ t.Errorf("#%d prime found to be non-prime (%s)", i, s)
+ }
+ }
+
+ for i, s := range composites {
+ s = strings.Map(cutSpace, s)
+ c, _ := new(Int).SetString(s, 10)
+ if c.ProbablyPrime(nreps) || nreps != 1 && c.ProbablyPrime(1) || c.ProbablyPrime(0) {
+ t.Errorf("#%d composite found to be prime (%s)", i, s)
+ }
+ }
+
+ // check that ProbablyPrime panics if n <= 0
+ c := NewInt(11) // a prime
+ for _, n := range []int{-1, 0, 1} {
+ func() {
+ defer func() {
+ if n < 0 && recover() == nil {
+ t.Fatalf("expected panic from ProbablyPrime(%d)", n)
+ }
+ }()
+ if !c.ProbablyPrime(n) {
+ t.Fatalf("%v should be a prime", c)
+ }
+ }()
+ }
+}
+
+func BenchmarkProbablyPrime(b *testing.B) {
+ p, _ := new(Int).SetString("203956878356401977405765866929034577280193993314348263094772646453283062722701277632936616063144088173312372882677123879538709400158306567338328279154499698366071906766440037074217117805690872792848149112022286332144876183376326512083574821647933992961249917319836219304274280243803104015000563790123", 10)
+ for _, n := range []int{0, 1, 5, 10, 20} {
+ b.Run(fmt.Sprintf("n=%d", n), func(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ p.ProbablyPrime(n)
+ }
+ })
+ }
+
+ b.Run("Lucas", func(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ p.abs.probablyPrimeLucas()
+ }
+ })
+ b.Run("MillerRabinBase2", func(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ p.abs.probablyPrimeMillerRabin(1, true)
+ }
+ })
+}
+
+func TestMillerRabinPseudoprimes(t *testing.T) {
+ testPseudoprimes(t, "probablyPrimeMillerRabin",
+ func(n nat) bool { return n.probablyPrimeMillerRabin(1, true) && !n.probablyPrimeLucas() },
+ // https://oeis.org/A001262
+ []int{2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, 52633, 65281, 74665, 80581, 85489, 88357, 90751})
+}
+
+func TestLucasPseudoprimes(t *testing.T) {
+ testPseudoprimes(t, "probablyPrimeLucas",
+ func(n nat) bool { return n.probablyPrimeLucas() && !n.probablyPrimeMillerRabin(1, true) },
+ // https://oeis.org/A217719
+ []int{989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059, 72389, 73919, 75077})
+}
+
+func testPseudoprimes(t *testing.T, name string, cond func(nat) bool, want []int) {
+ n := nat{1}
+ for i := 3; i < 100000; i += 2 {
+ if testing.Short() {
+ if len(want) == 0 {
+ break
+ }
+ if i < want[0]-2 {
+ i = want[0] - 2
+ }
+ }
+ n[0] = Word(i)
+ pseudo := cond(n)
+ if pseudo && (len(want) == 0 || i != want[0]) {
+ t.Errorf("%s(%v, base=2) = true, want false", name, i)
+ } else if !pseudo && len(want) >= 1 && i == want[0] {
+ t.Errorf("%s(%v, base=2) = false, want true", name, i)
+ }
+ if len(want) > 0 && i == want[0] {
+ want = want[1:]
+ }
+ }
+ if len(want) > 0 {
+ t.Fatalf("forgot to test %v", want)
+ }
+}
diff --git a/src/math/big/rat.go b/src/math/big/rat.go
new file mode 100644
index 0000000..d35cd4c
--- /dev/null
+++ b/src/math/big/rat.go
@@ -0,0 +1,544 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements multi-precision rational numbers.
+
+package big
+
+import (
+ "fmt"
+ "math"
+)
+
+// A Rat represents a quotient a/b of arbitrary precision.
+// The zero value for a Rat represents the value 0.
+//
+// Operations always take pointer arguments (*Rat) rather
+// than Rat values, and each unique Rat value requires
+// its own unique *Rat pointer. To "copy" a Rat value,
+// an existing (or newly allocated) Rat must be set to
+// a new value using the Rat.Set method; shallow copies
+// of Rats are not supported and may lead to errors.
+type Rat struct {
+ // To make zero values for Rat work w/o initialization,
+ // a zero value of b (len(b) == 0) acts like b == 1. At
+ // the earliest opportunity (when an assignment to the Rat
+ // is made), such uninitialized denominators are set to 1.
+ // a.neg determines the sign of the Rat, b.neg is ignored.
+ a, b Int
+}
+
+// NewRat creates a new Rat with numerator a and denominator b.
+func NewRat(a, b int64) *Rat {
+ return new(Rat).SetFrac64(a, b)
+}
+
+// SetFloat64 sets z to exactly f and returns z.
+// If f is not finite, SetFloat returns nil.
+func (z *Rat) SetFloat64(f float64) *Rat {
+ const expMask = 1<<11 - 1
+ bits := math.Float64bits(f)
+ mantissa := bits & (1<<52 - 1)
+ exp := int((bits >> 52) & expMask)
+ switch exp {
+ case expMask: // non-finite
+ return nil
+ case 0: // denormal
+ exp -= 1022
+ default: // normal
+ mantissa |= 1 << 52
+ exp -= 1023
+ }
+
+ shift := 52 - exp
+
+ // Optimization (?): partially pre-normalise.
+ for mantissa&1 == 0 && shift > 0 {
+ mantissa >>= 1
+ shift--
+ }
+
+ z.a.SetUint64(mantissa)
+ z.a.neg = f < 0
+ z.b.Set(intOne)
+ if shift > 0 {
+ z.b.Lsh(&z.b, uint(shift))
+ } else {
+ z.a.Lsh(&z.a, uint(-shift))
+ }
+ return z.norm()
+}
+
+// quotToFloat32 returns the non-negative float32 value
+// nearest to the quotient a/b, using round-to-even in
+// halfway cases. It does not mutate its arguments.
+// Preconditions: b is non-zero; a and b have no common factors.
+func quotToFloat32(a, b nat) (f float32, exact bool) {
+ const (
+ // float size in bits
+ Fsize = 32
+
+ // mantissa
+ Msize = 23
+ Msize1 = Msize + 1 // incl. implicit 1
+ Msize2 = Msize1 + 1
+
+ // exponent
+ Esize = Fsize - Msize1
+ Ebias = 1<<(Esize-1) - 1
+ Emin = 1 - Ebias
+ Emax = Ebias
+ )
+
+ // TODO(adonovan): specialize common degenerate cases: 1.0, integers.
+ alen := a.bitLen()
+ if alen == 0 {
+ return 0, true
+ }
+ blen := b.bitLen()
+ if blen == 0 {
+ panic("division by zero")
+ }
+
+ // 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
+ // (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
+ // This is 2 or 3 more than the float32 mantissa field width of Msize:
+ // - the optional extra bit is shifted away in step 3 below.
+ // - the high-order 1 is omitted in "normal" representation;
+ // - the low-order 1 will be used during rounding then discarded.
+ exp := alen - blen
+ var a2, b2 nat
+ a2 = a2.set(a)
+ b2 = b2.set(b)
+ if shift := Msize2 - exp; shift > 0 {
+ a2 = a2.shl(a2, uint(shift))
+ } else if shift < 0 {
+ b2 = b2.shl(b2, uint(-shift))
+ }
+
+ // 2. Compute quotient and remainder (q, r). NB: due to the
+ // extra shift, the low-order bit of q is logically the
+ // high-order bit of r.
+ var q nat
+ q, r := q.div(a2, a2, b2) // (recycle a2)
+ mantissa := low32(q)
+ haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
+
+ // 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
+ // (in effect---we accomplish this incrementally).
+ if mantissa>>Msize2 == 1 {
+ if mantissa&1 == 1 {
+ haveRem = true
+ }
+ mantissa >>= 1
+ exp++
+ }
+ if mantissa>>Msize1 != 1 {
+ panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
+ }
+
+ // 4. Rounding.
+ if Emin-Msize <= exp && exp <= Emin {
+ // Denormal case; lose 'shift' bits of precision.
+ shift := uint(Emin - (exp - 1)) // [1..Esize1)
+ lostbits := mantissa & (1<<shift - 1)
+ haveRem = haveRem || lostbits != 0
+ mantissa >>= shift
+ exp = 2 - Ebias // == exp + shift
+ }
+ // Round q using round-half-to-even.
+ exact = !haveRem
+ if mantissa&1 != 0 {
+ exact = false
+ if haveRem || mantissa&2 != 0 {
+ if mantissa++; mantissa >= 1<<Msize2 {
+ // Complete rollover 11...1 => 100...0, so shift is safe
+ mantissa >>= 1
+ exp++
+ }
+ }
+ }
+ mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<<Msize1.
+
+ f = float32(math.Ldexp(float64(mantissa), exp-Msize1))
+ if math.IsInf(float64(f), 0) {
+ exact = false
+ }
+ return
+}
+
+// quotToFloat64 returns the non-negative float64 value
+// nearest to the quotient a/b, using round-to-even in
+// halfway cases. It does not mutate its arguments.
+// Preconditions: b is non-zero; a and b have no common factors.
+func quotToFloat64(a, b nat) (f float64, exact bool) {
+ const (
+ // float size in bits
+ Fsize = 64
+
+ // mantissa
+ Msize = 52
+ Msize1 = Msize + 1 // incl. implicit 1
+ Msize2 = Msize1 + 1
+
+ // exponent
+ Esize = Fsize - Msize1
+ Ebias = 1<<(Esize-1) - 1
+ Emin = 1 - Ebias
+ Emax = Ebias
+ )
+
+ // TODO(adonovan): specialize common degenerate cases: 1.0, integers.
+ alen := a.bitLen()
+ if alen == 0 {
+ return 0, true
+ }
+ blen := b.bitLen()
+ if blen == 0 {
+ panic("division by zero")
+ }
+
+ // 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
+ // (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
+ // This is 2 or 3 more than the float64 mantissa field width of Msize:
+ // - the optional extra bit is shifted away in step 3 below.
+ // - the high-order 1 is omitted in "normal" representation;
+ // - the low-order 1 will be used during rounding then discarded.
+ exp := alen - blen
+ var a2, b2 nat
+ a2 = a2.set(a)
+ b2 = b2.set(b)
+ if shift := Msize2 - exp; shift > 0 {
+ a2 = a2.shl(a2, uint(shift))
+ } else if shift < 0 {
+ b2 = b2.shl(b2, uint(-shift))
+ }
+
+ // 2. Compute quotient and remainder (q, r). NB: due to the
+ // extra shift, the low-order bit of q is logically the
+ // high-order bit of r.
+ var q nat
+ q, r := q.div(a2, a2, b2) // (recycle a2)
+ mantissa := low64(q)
+ haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
+
+ // 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
+ // (in effect---we accomplish this incrementally).
+ if mantissa>>Msize2 == 1 {
+ if mantissa&1 == 1 {
+ haveRem = true
+ }
+ mantissa >>= 1
+ exp++
+ }
+ if mantissa>>Msize1 != 1 {
+ panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
+ }
+
+ // 4. Rounding.
+ if Emin-Msize <= exp && exp <= Emin {
+ // Denormal case; lose 'shift' bits of precision.
+ shift := uint(Emin - (exp - 1)) // [1..Esize1)
+ lostbits := mantissa & (1<<shift - 1)
+ haveRem = haveRem || lostbits != 0
+ mantissa >>= shift
+ exp = 2 - Ebias // == exp + shift
+ }
+ // Round q using round-half-to-even.
+ exact = !haveRem
+ if mantissa&1 != 0 {
+ exact = false
+ if haveRem || mantissa&2 != 0 {
+ if mantissa++; mantissa >= 1<<Msize2 {
+ // Complete rollover 11...1 => 100...0, so shift is safe
+ mantissa >>= 1
+ exp++
+ }
+ }
+ }
+ mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<<Msize1.
+
+ f = math.Ldexp(float64(mantissa), exp-Msize1)
+ if math.IsInf(f, 0) {
+ exact = false
+ }
+ return
+}
+
+// Float32 returns the nearest float32 value for x and a bool indicating
+// whether f represents x exactly. If the magnitude of x is too large to
+// be represented by a float32, f is an infinity and exact is false.
+// The sign of f always matches the sign of x, even if f == 0.
+func (x *Rat) Float32() (f float32, exact bool) {
+ b := x.b.abs
+ if len(b) == 0 {
+ b = natOne
+ }
+ f, exact = quotToFloat32(x.a.abs, b)
+ if x.a.neg {
+ f = -f
+ }
+ return
+}
+
+// Float64 returns the nearest float64 value for x and a bool indicating
+// whether f represents x exactly. If the magnitude of x is too large to
+// be represented by a float64, f is an infinity and exact is false.
+// The sign of f always matches the sign of x, even if f == 0.
+func (x *Rat) Float64() (f float64, exact bool) {
+ b := x.b.abs
+ if len(b) == 0 {
+ b = natOne
+ }
+ f, exact = quotToFloat64(x.a.abs, b)
+ if x.a.neg {
+ f = -f
+ }
+ return
+}
+
+// SetFrac sets z to a/b and returns z.
+// If b == 0, SetFrac panics.
+func (z *Rat) SetFrac(a, b *Int) *Rat {
+ z.a.neg = a.neg != b.neg
+ babs := b.abs
+ if len(babs) == 0 {
+ panic("division by zero")
+ }
+ if &z.a == b || alias(z.a.abs, babs) {
+ babs = nat(nil).set(babs) // make a copy
+ }
+ z.a.abs = z.a.abs.set(a.abs)
+ z.b.abs = z.b.abs.set(babs)
+ return z.norm()
+}
+
+// SetFrac64 sets z to a/b and returns z.
+// If b == 0, SetFrac64 panics.
+func (z *Rat) SetFrac64(a, b int64) *Rat {
+ if b == 0 {
+ panic("division by zero")
+ }
+ z.a.SetInt64(a)
+ if b < 0 {
+ b = -b
+ z.a.neg = !z.a.neg
+ }
+ z.b.abs = z.b.abs.setUint64(uint64(b))
+ return z.norm()
+}
+
+// SetInt sets z to x (by making a copy of x) and returns z.
+func (z *Rat) SetInt(x *Int) *Rat {
+ z.a.Set(x)
+ z.b.abs = z.b.abs.setWord(1)
+ return z
+}
+
+// SetInt64 sets z to x and returns z.
+func (z *Rat) SetInt64(x int64) *Rat {
+ z.a.SetInt64(x)
+ z.b.abs = z.b.abs.setWord(1)
+ return z
+}
+
+// SetUint64 sets z to x and returns z.
+func (z *Rat) SetUint64(x uint64) *Rat {
+ z.a.SetUint64(x)
+ z.b.abs = z.b.abs.setWord(1)
+ return z
+}
+
+// Set sets z to x (by making a copy of x) and returns z.
+func (z *Rat) Set(x *Rat) *Rat {
+ if z != x {
+ z.a.Set(&x.a)
+ z.b.Set(&x.b)
+ }
+ if len(z.b.abs) == 0 {
+ z.b.abs = z.b.abs.setWord(1)
+ }
+ return z
+}
+
+// Abs sets z to |x| (the absolute value of x) and returns z.
+func (z *Rat) Abs(x *Rat) *Rat {
+ z.Set(x)
+ z.a.neg = false
+ return z
+}
+
+// Neg sets z to -x and returns z.
+func (z *Rat) Neg(x *Rat) *Rat {
+ z.Set(x)
+ z.a.neg = len(z.a.abs) > 0 && !z.a.neg // 0 has no sign
+ return z
+}
+
+// Inv sets z to 1/x and returns z.
+// If x == 0, Inv panics.
+func (z *Rat) Inv(x *Rat) *Rat {
+ if len(x.a.abs) == 0 {
+ panic("division by zero")
+ }
+ z.Set(x)
+ z.a.abs, z.b.abs = z.b.abs, z.a.abs
+ return z
+}
+
+// Sign returns:
+//
+// -1 if x < 0
+// 0 if x == 0
+// +1 if x > 0
+//
+func (x *Rat) Sign() int {
+ return x.a.Sign()
+}
+
+// IsInt reports whether the denominator of x is 1.
+func (x *Rat) IsInt() bool {
+ return len(x.b.abs) == 0 || x.b.abs.cmp(natOne) == 0
+}
+
+// Num returns the numerator of x; it may be <= 0.
+// The result is a reference to x's numerator; it
+// may change if a new value is assigned to x, and vice versa.
+// The sign of the numerator corresponds to the sign of x.
+func (x *Rat) Num() *Int {
+ return &x.a
+}
+
+// Denom returns the denominator of x; it is always > 0.
+// The result is a reference to x's denominator, unless
+// x is an uninitialized (zero value) Rat, in which case
+// the result is a new Int of value 1. (To initialize x,
+// any operation that sets x will do, including x.Set(x).)
+// If the result is a reference to x's denominator it
+// may change if a new value is assigned to x, and vice versa.
+func (x *Rat) Denom() *Int {
+ x.b.neg = false // the result is always >= 0
+ if len(x.b.abs) == 0 {
+ // Note: If this proves problematic, we could
+ // panic instead and require the Rat to
+ // be explicitly initialized.
+ return &Int{abs: nat{1}}
+ }
+ return &x.b
+}
+
+func (z *Rat) norm() *Rat {
+ switch {
+ case len(z.a.abs) == 0:
+ // z == 0; normalize sign and denominator
+ z.a.neg = false
+ fallthrough
+ case len(z.b.abs) == 0:
+ // z is integer; normalize denominator
+ z.b.abs = z.b.abs.setWord(1)
+ default:
+ // z is fraction; normalize numerator and denominator
+ neg := z.a.neg
+ z.a.neg = false
+ z.b.neg = false
+ if f := NewInt(0).lehmerGCD(nil, nil, &z.a, &z.b); f.Cmp(intOne) != 0 {
+ z.a.abs, _ = z.a.abs.div(nil, z.a.abs, f.abs)
+ z.b.abs, _ = z.b.abs.div(nil, z.b.abs, f.abs)
+ }
+ z.a.neg = neg
+ }
+ return z
+}
+
+// mulDenom sets z to the denominator product x*y (by taking into
+// account that 0 values for x or y must be interpreted as 1) and
+// returns z.
+func mulDenom(z, x, y nat) nat {
+ switch {
+ case len(x) == 0 && len(y) == 0:
+ return z.setWord(1)
+ case len(x) == 0:
+ return z.set(y)
+ case len(y) == 0:
+ return z.set(x)
+ }
+ return z.mul(x, y)
+}
+
+// scaleDenom sets z to the product x*f.
+// If f == 0 (zero value of denominator), z is set to (a copy of) x.
+func (z *Int) scaleDenom(x *Int, f nat) {
+ if len(f) == 0 {
+ z.Set(x)
+ return
+ }
+ z.abs = z.abs.mul(x.abs, f)
+ z.neg = x.neg
+}
+
+// Cmp compares x and y and returns:
+//
+// -1 if x < y
+// 0 if x == y
+// +1 if x > y
+//
+func (x *Rat) Cmp(y *Rat) int {
+ var a, b Int
+ a.scaleDenom(&x.a, y.b.abs)
+ b.scaleDenom(&y.a, x.b.abs)
+ return a.Cmp(&b)
+}
+
+// Add sets z to the sum x+y and returns z.
+func (z *Rat) Add(x, y *Rat) *Rat {
+ var a1, a2 Int
+ a1.scaleDenom(&x.a, y.b.abs)
+ a2.scaleDenom(&y.a, x.b.abs)
+ z.a.Add(&a1, &a2)
+ z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+ return z.norm()
+}
+
+// Sub sets z to the difference x-y and returns z.
+func (z *Rat) Sub(x, y *Rat) *Rat {
+ var a1, a2 Int
+ a1.scaleDenom(&x.a, y.b.abs)
+ a2.scaleDenom(&y.a, x.b.abs)
+ z.a.Sub(&a1, &a2)
+ z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+ return z.norm()
+}
+
+// Mul sets z to the product x*y and returns z.
+func (z *Rat) Mul(x, y *Rat) *Rat {
+ if x == y {
+ // a squared Rat is positive and can't be reduced (no need to call norm())
+ z.a.neg = false
+ z.a.abs = z.a.abs.sqr(x.a.abs)
+ if len(x.b.abs) == 0 {
+ z.b.abs = z.b.abs.setWord(1)
+ } else {
+ z.b.abs = z.b.abs.sqr(x.b.abs)
+ }
+ return z
+ }
+ z.a.Mul(&x.a, &y.a)
+ z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+ return z.norm()
+}
+
+// Quo sets z to the quotient x/y and returns z.
+// If y == 0, Quo panics.
+func (z *Rat) Quo(x, y *Rat) *Rat {
+ if len(y.a.abs) == 0 {
+ panic("division by zero")
+ }
+ var a, b Int
+ a.scaleDenom(&x.a, y.b.abs)
+ b.scaleDenom(&y.a, x.b.abs)
+ z.a.abs = a.abs
+ z.b.abs = b.abs
+ z.a.neg = a.neg != b.neg
+ return z.norm()
+}
diff --git a/src/math/big/rat_test.go b/src/math/big/rat_test.go
new file mode 100644
index 0000000..02569c1
--- /dev/null
+++ b/src/math/big/rat_test.go
@@ -0,0 +1,728 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "math"
+ "testing"
+)
+
+func TestZeroRat(t *testing.T) {
+ var x, y, z Rat
+ y.SetFrac64(0, 42)
+
+ if x.Cmp(&y) != 0 {
+ t.Errorf("x and y should be both equal and zero")
+ }
+
+ if s := x.String(); s != "0/1" {
+ t.Errorf("got x = %s, want 0/1", s)
+ }
+
+ if s := x.RatString(); s != "0" {
+ t.Errorf("got x = %s, want 0", s)
+ }
+
+ z.Add(&x, &y)
+ if s := z.RatString(); s != "0" {
+ t.Errorf("got x+y = %s, want 0", s)
+ }
+
+ z.Sub(&x, &y)
+ if s := z.RatString(); s != "0" {
+ t.Errorf("got x-y = %s, want 0", s)
+ }
+
+ z.Mul(&x, &y)
+ if s := z.RatString(); s != "0" {
+ t.Errorf("got x*y = %s, want 0", s)
+ }
+
+ // check for division by zero
+ defer func() {
+ if s := recover(); s == nil || s.(string) != "division by zero" {
+ panic(s)
+ }
+ }()
+ z.Quo(&x, &y)
+}
+
+func TestRatSign(t *testing.T) {
+ zero := NewRat(0, 1)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ s := x.Sign()
+ e := x.Cmp(zero)
+ if s != e {
+ t.Errorf("got %d; want %d for z = %v", s, e, &x)
+ }
+ }
+}
+
+var ratCmpTests = []struct {
+ rat1, rat2 string
+ out int
+}{
+ {"0", "0/1", 0},
+ {"1/1", "1", 0},
+ {"-1", "-2/2", 0},
+ {"1", "0", 1},
+ {"0/1", "1/1", -1},
+ {"-5/1434770811533343057144", "-5/1434770811533343057145", -1},
+ {"49832350382626108453/8964749413", "49832350382626108454/8964749413", -1},
+ {"-37414950961700930/7204075375675961", "37414950961700930/7204075375675961", -1},
+ {"37414950961700930/7204075375675961", "74829901923401860/14408150751351922", 0},
+}
+
+func TestRatCmp(t *testing.T) {
+ for i, test := range ratCmpTests {
+ x, _ := new(Rat).SetString(test.rat1)
+ y, _ := new(Rat).SetString(test.rat2)
+
+ out := x.Cmp(y)
+ if out != test.out {
+ t.Errorf("#%d got out = %v; want %v", i, out, test.out)
+ }
+ }
+}
+
+func TestIsInt(t *testing.T) {
+ one := NewInt(1)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ i := x.IsInt()
+ e := x.Denom().Cmp(one) == 0
+ if i != e {
+ t.Errorf("got IsInt(%v) == %v; want %v", x, i, e)
+ }
+ }
+}
+
+func TestRatAbs(t *testing.T) {
+ zero := new(Rat)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ e := new(Rat).Set(x)
+ if e.Cmp(zero) < 0 {
+ e.Sub(zero, e)
+ }
+ z := new(Rat).Abs(x)
+ if z.Cmp(e) != 0 {
+ t.Errorf("got Abs(%v) = %v; want %v", x, z, e)
+ }
+ }
+}
+
+func TestRatNeg(t *testing.T) {
+ zero := new(Rat)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ e := new(Rat).Sub(zero, x)
+ z := new(Rat).Neg(x)
+ if z.Cmp(e) != 0 {
+ t.Errorf("got Neg(%v) = %v; want %v", x, z, e)
+ }
+ }
+}
+
+func TestRatInv(t *testing.T) {
+ zero := new(Rat)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ if x.Cmp(zero) == 0 {
+ continue // avoid division by zero
+ }
+ e := new(Rat).SetFrac(x.Denom(), x.Num())
+ z := new(Rat).Inv(x)
+ if z.Cmp(e) != 0 {
+ t.Errorf("got Inv(%v) = %v; want %v", x, z, e)
+ }
+ }
+}
+
+type ratBinFun func(z, x, y *Rat) *Rat
+type ratBinArg struct {
+ x, y, z string
+}
+
+func testRatBin(t *testing.T, i int, name string, f ratBinFun, a ratBinArg) {
+ x, _ := new(Rat).SetString(a.x)
+ y, _ := new(Rat).SetString(a.y)
+ z, _ := new(Rat).SetString(a.z)
+ out := f(new(Rat), x, y)
+
+ if out.Cmp(z) != 0 {
+ t.Errorf("%s #%d got %s want %s", name, i, out, z)
+ }
+}
+
+var ratBinTests = []struct {
+ x, y string
+ sum, prod string
+}{
+ {"0", "0", "0", "0"},
+ {"0", "1", "1", "0"},
+ {"-1", "0", "-1", "0"},
+ {"-1", "1", "0", "-1"},
+ {"1", "1", "2", "1"},
+ {"1/2", "1/2", "1", "1/4"},
+ {"1/4", "1/3", "7/12", "1/12"},
+ {"2/5", "-14/3", "-64/15", "-28/15"},
+ {"4707/49292519774798173060", "-3367/70976135186689855734", "84058377121001851123459/1749296273614329067191168098769082663020", "-1760941/388732505247628681598037355282018369560"},
+ {"-61204110018146728334/3", "-31052192278051565633/2", "-215564796870448153567/6", "950260896245257153059642991192710872711/3"},
+ {"-854857841473707320655/4237645934602118692642972629634714039", "-18/31750379913563777419", "-27/133467566250814981", "15387441146526731771790/134546868362786310073779084329032722548987800600710485341"},
+ {"618575745270541348005638912139/19198433543745179392300736", "-19948846211000086/637313996471", "27674141753240653/30123979153216", "-6169936206128396568797607742807090270137721977/6117715203873571641674006593837351328"},
+ {"-3/26206484091896184128", "5/2848423294177090248", "15310893822118706237/9330894968229805033368778458685147968", "-5/24882386581946146755650075889827061248"},
+ {"26946729/330400702820", "41563965/225583428284", "1238218672302860271/4658307703098666660055", "224002580204097/14906584649915733312176"},
+ {"-8259900599013409474/7", "-84829337473700364773/56707961321161574960", "-468402123685491748914621885145127724451/396955729248131024720", "350340947706464153265156004876107029701/198477864624065512360"},
+ {"575775209696864/1320203974639986246357", "29/712593081308", "410331716733912717985762465/940768218243776489278275419794956", "808/45524274987585732633"},
+ {"1786597389946320496771/2066653520653241", "6269770/1992362624741777", "3559549865190272133656109052308126637/4117523232840525481453983149257", "8967230/3296219033"},
+ {"-36459180403360509753/32150500941194292113930", "9381566963714/9633539", "301622077145533298008420642898530153/309723104686531919656937098270", "-3784609207827/3426986245"},
+}
+
+func TestRatBin(t *testing.T) {
+ for i, test := range ratBinTests {
+ arg := ratBinArg{test.x, test.y, test.sum}
+ testRatBin(t, i, "Add", (*Rat).Add, arg)
+
+ arg = ratBinArg{test.y, test.x, test.sum}
+ testRatBin(t, i, "Add symmetric", (*Rat).Add, arg)
+
+ arg = ratBinArg{test.sum, test.x, test.y}
+ testRatBin(t, i, "Sub", (*Rat).Sub, arg)
+
+ arg = ratBinArg{test.sum, test.y, test.x}
+ testRatBin(t, i, "Sub symmetric", (*Rat).Sub, arg)
+
+ arg = ratBinArg{test.x, test.y, test.prod}
+ testRatBin(t, i, "Mul", (*Rat).Mul, arg)
+
+ arg = ratBinArg{test.y, test.x, test.prod}
+ testRatBin(t, i, "Mul symmetric", (*Rat).Mul, arg)
+
+ if test.x != "0" {
+ arg = ratBinArg{test.prod, test.x, test.y}
+ testRatBin(t, i, "Quo", (*Rat).Quo, arg)
+ }
+
+ if test.y != "0" {
+ arg = ratBinArg{test.prod, test.y, test.x}
+ testRatBin(t, i, "Quo symmetric", (*Rat).Quo, arg)
+ }
+ }
+}
+
+func TestIssue820(t *testing.T) {
+ x := NewRat(3, 1)
+ y := NewRat(2, 1)
+ z := y.Quo(x, y)
+ q := NewRat(3, 2)
+ if z.Cmp(q) != 0 {
+ t.Errorf("got %s want %s", z, q)
+ }
+
+ y = NewRat(3, 1)
+ x = NewRat(2, 1)
+ z = y.Quo(x, y)
+ q = NewRat(2, 3)
+ if z.Cmp(q) != 0 {
+ t.Errorf("got %s want %s", z, q)
+ }
+
+ x = NewRat(3, 1)
+ z = x.Quo(x, x)
+ q = NewRat(3, 3)
+ if z.Cmp(q) != 0 {
+ t.Errorf("got %s want %s", z, q)
+ }
+}
+
+var setFrac64Tests = []struct {
+ a, b int64
+ out string
+}{
+ {0, 1, "0"},
+ {0, -1, "0"},
+ {1, 1, "1"},
+ {-1, 1, "-1"},
+ {1, -1, "-1"},
+ {-1, -1, "1"},
+ {-9223372036854775808, -9223372036854775808, "1"},
+}
+
+func TestRatSetFrac64Rat(t *testing.T) {
+ for i, test := range setFrac64Tests {
+ x := new(Rat).SetFrac64(test.a, test.b)
+ if x.RatString() != test.out {
+ t.Errorf("#%d got %s want %s", i, x.RatString(), test.out)
+ }
+ }
+}
+
+func TestIssue2379(t *testing.T) {
+ // 1) no aliasing
+ q := NewRat(3, 2)
+ x := new(Rat)
+ x.SetFrac(NewInt(3), NewInt(2))
+ if x.Cmp(q) != 0 {
+ t.Errorf("1) got %s want %s", x, q)
+ }
+
+ // 2) aliasing of numerator
+ x = NewRat(2, 3)
+ x.SetFrac(NewInt(3), x.Num())
+ if x.Cmp(q) != 0 {
+ t.Errorf("2) got %s want %s", x, q)
+ }
+
+ // 3) aliasing of denominator
+ x = NewRat(2, 3)
+ x.SetFrac(x.Denom(), NewInt(2))
+ if x.Cmp(q) != 0 {
+ t.Errorf("3) got %s want %s", x, q)
+ }
+
+ // 4) aliasing of numerator and denominator
+ x = NewRat(2, 3)
+ x.SetFrac(x.Denom(), x.Num())
+ if x.Cmp(q) != 0 {
+ t.Errorf("4) got %s want %s", x, q)
+ }
+
+ // 5) numerator and denominator are the same
+ q = NewRat(1, 1)
+ x = new(Rat)
+ n := NewInt(7)
+ x.SetFrac(n, n)
+ if x.Cmp(q) != 0 {
+ t.Errorf("5) got %s want %s", x, q)
+ }
+}
+
+func TestIssue3521(t *testing.T) {
+ a := new(Int)
+ b := new(Int)
+ a.SetString("64375784358435883458348587", 0)
+ b.SetString("4789759874531", 0)
+
+ // 0) a raw zero value has 1 as denominator
+ zero := new(Rat)
+ one := NewInt(1)
+ if zero.Denom().Cmp(one) != 0 {
+ t.Errorf("0) got %s want %s", zero.Denom(), one)
+ }
+
+ // 1a) the denominator of an (uninitialized) zero value is not shared with the value
+ s := &zero.b
+ d := zero.Denom()
+ if d == s {
+ t.Errorf("1a) got %s (%p) == %s (%p) want different *Int values", d, d, s, s)
+ }
+
+ // 1b) the denominator of an (uninitialized) value is a new 1 each time
+ d1 := zero.Denom()
+ d2 := zero.Denom()
+ if d1 == d2 {
+ t.Errorf("1b) got %s (%p) == %s (%p) want different *Int values", d1, d1, d2, d2)
+ }
+
+ // 1c) the denominator of an initialized zero value is shared with the value
+ x := new(Rat)
+ x.Set(x) // initialize x (any operation that sets x explicitly will do)
+ s = &x.b
+ d = x.Denom()
+ if d != s {
+ t.Errorf("1c) got %s (%p) != %s (%p) want identical *Int values", d, d, s, s)
+ }
+
+ // 1d) a zero value remains zero independent of denominator
+ x.Denom().Set(new(Int).Neg(b))
+ if x.Cmp(zero) != 0 {
+ t.Errorf("1d) got %s want %s", x, zero)
+ }
+
+ // 1e) a zero value may have a denominator != 0 and != 1
+ x.Num().Set(a)
+ qab := new(Rat).SetFrac(a, b)
+ if x.Cmp(qab) != 0 {
+ t.Errorf("1e) got %s want %s", x, qab)
+ }
+
+ // 2a) an integral value becomes a fraction depending on denominator
+ x.SetFrac64(10, 2)
+ x.Denom().SetInt64(3)
+ q53 := NewRat(5, 3)
+ if x.Cmp(q53) != 0 {
+ t.Errorf("2a) got %s want %s", x, q53)
+ }
+
+ // 2b) an integral value becomes a fraction depending on denominator
+ x = NewRat(10, 2)
+ x.Denom().SetInt64(3)
+ if x.Cmp(q53) != 0 {
+ t.Errorf("2b) got %s want %s", x, q53)
+ }
+
+ // 3) changing the numerator/denominator of a Rat changes the Rat
+ x.SetFrac(a, b)
+ a = x.Num()
+ b = x.Denom()
+ a.SetInt64(5)
+ b.SetInt64(3)
+ if x.Cmp(q53) != 0 {
+ t.Errorf("3) got %s want %s", x, q53)
+ }
+}
+
+func TestFloat32Distribution(t *testing.T) {
+ // Generate a distribution of (sign, mantissa, exp) values
+ // broader than the float32 range, and check Rat.Float32()
+ // always picks the closest float32 approximation.
+ var add = []int64{
+ 0,
+ 1,
+ 3,
+ 5,
+ 7,
+ 9,
+ 11,
+ }
+ var winc, einc = uint64(5), 15 // quick test (~60ms on x86-64)
+ if *long {
+ winc, einc = uint64(1), 1 // soak test (~1.5s on x86-64)
+ }
+
+ for _, sign := range "+-" {
+ for _, a := range add {
+ for wid := uint64(0); wid < 30; wid += winc {
+ b := 1<<wid + a
+ if sign == '-' {
+ b = -b
+ }
+ for exp := -150; exp < 150; exp += einc {
+ num, den := NewInt(b), NewInt(1)
+ if exp > 0 {
+ num.Lsh(num, uint(exp))
+ } else {
+ den.Lsh(den, uint(-exp))
+ }
+ r := new(Rat).SetFrac(num, den)
+ f, _ := r.Float32()
+
+ if !checkIsBestApprox32(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was mantissa %#x, exp %d; f = %g (%b); f ~ %g; r = %v)",
+ b, exp, f, f, math.Ldexp(float64(b), exp), r)
+ }
+
+ checkNonLossyRoundtrip32(t, f)
+ }
+ }
+ }
+ }
+}
+
+func TestFloat64Distribution(t *testing.T) {
+ // Generate a distribution of (sign, mantissa, exp) values
+ // broader than the float64 range, and check Rat.Float64()
+ // always picks the closest float64 approximation.
+ var add = []int64{
+ 0,
+ 1,
+ 3,
+ 5,
+ 7,
+ 9,
+ 11,
+ }
+ var winc, einc = uint64(10), 500 // quick test (~12ms on x86-64)
+ if *long {
+ winc, einc = uint64(1), 1 // soak test (~75s on x86-64)
+ }
+
+ for _, sign := range "+-" {
+ for _, a := range add {
+ for wid := uint64(0); wid < 60; wid += winc {
+ b := 1<<wid + a
+ if sign == '-' {
+ b = -b
+ }
+ for exp := -1100; exp < 1100; exp += einc {
+ num, den := NewInt(b), NewInt(1)
+ if exp > 0 {
+ num.Lsh(num, uint(exp))
+ } else {
+ den.Lsh(den, uint(-exp))
+ }
+ r := new(Rat).SetFrac(num, den)
+ f, _ := r.Float64()
+
+ if !checkIsBestApprox64(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was mantissa %#x, exp %d; f = %g (%b); f ~ %g; r = %v)",
+ b, exp, f, f, math.Ldexp(float64(b), exp), r)
+ }
+
+ checkNonLossyRoundtrip64(t, f)
+ }
+ }
+ }
+ }
+}
+
+// TestSetFloat64NonFinite checks that SetFloat64 of a non-finite value
+// returns nil.
+func TestSetFloat64NonFinite(t *testing.T) {
+ for _, f := range []float64{math.NaN(), math.Inf(+1), math.Inf(-1)} {
+ var r Rat
+ if r2 := r.SetFloat64(f); r2 != nil {
+ t.Errorf("SetFloat64(%g) was %v, want nil", f, r2)
+ }
+ }
+}
+
+// checkNonLossyRoundtrip32 checks that a float->Rat->float roundtrip is
+// non-lossy for finite f.
+func checkNonLossyRoundtrip32(t *testing.T, f float32) {
+ if !isFinite(float64(f)) {
+ return
+ }
+ r := new(Rat).SetFloat64(float64(f))
+ if r == nil {
+ t.Errorf("Rat.SetFloat64(float64(%g) (%b)) == nil", f, f)
+ return
+ }
+ f2, exact := r.Float32()
+ if f != f2 || !exact {
+ t.Errorf("Rat.SetFloat64(float64(%g)).Float32() = %g (%b), %v, want %g (%b), %v; delta = %b",
+ f, f2, f2, exact, f, f, true, f2-f)
+ }
+}
+
+// checkNonLossyRoundtrip64 checks that a float->Rat->float roundtrip is
+// non-lossy for finite f.
+func checkNonLossyRoundtrip64(t *testing.T, f float64) {
+ if !isFinite(f) {
+ return
+ }
+ r := new(Rat).SetFloat64(f)
+ if r == nil {
+ t.Errorf("Rat.SetFloat64(%g (%b)) == nil", f, f)
+ return
+ }
+ f2, exact := r.Float64()
+ if f != f2 || !exact {
+ t.Errorf("Rat.SetFloat64(%g).Float64() = %g (%b), %v, want %g (%b), %v; delta = %b",
+ f, f2, f2, exact, f, f, true, f2-f)
+ }
+}
+
+// delta returns the absolute difference between r and f.
+func delta(r *Rat, f float64) *Rat {
+ d := new(Rat).Sub(r, new(Rat).SetFloat64(f))
+ return d.Abs(d)
+}
+
+// checkIsBestApprox32 checks that f is the best possible float32
+// approximation of r.
+// Returns true on success.
+func checkIsBestApprox32(t *testing.T, f float32, r *Rat) bool {
+ if math.Abs(float64(f)) >= math.MaxFloat32 {
+ // Cannot check +Inf, -Inf, nor the float next to them (MaxFloat32).
+ // But we have tests for these special cases.
+ return true
+ }
+
+ // r must be strictly between f0 and f1, the floats bracketing f.
+ f0 := math.Nextafter32(f, float32(math.Inf(-1)))
+ f1 := math.Nextafter32(f, float32(math.Inf(+1)))
+
+ // For f to be correct, r must be closer to f than to f0 or f1.
+ df := delta(r, float64(f))
+ df0 := delta(r, float64(f0))
+ df1 := delta(r, float64(f1))
+ if df.Cmp(df0) > 0 {
+ t.Errorf("Rat(%v).Float32() = %g (%b), but previous float32 %g (%b) is closer", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) > 0 {
+ t.Errorf("Rat(%v).Float32() = %g (%b), but next float32 %g (%b) is closer", r, f, f, f1, f1)
+ return false
+ }
+ if df.Cmp(df0) == 0 && !isEven32(f) {
+ t.Errorf("Rat(%v).Float32() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) == 0 && !isEven32(f) {
+ t.Errorf("Rat(%v).Float32() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f1, f1)
+ return false
+ }
+ return true
+}
+
+// checkIsBestApprox64 checks that f is the best possible float64
+// approximation of r.
+// Returns true on success.
+func checkIsBestApprox64(t *testing.T, f float64, r *Rat) bool {
+ if math.Abs(f) >= math.MaxFloat64 {
+ // Cannot check +Inf, -Inf, nor the float next to them (MaxFloat64).
+ // But we have tests for these special cases.
+ return true
+ }
+
+ // r must be strictly between f0 and f1, the floats bracketing f.
+ f0 := math.Nextafter(f, math.Inf(-1))
+ f1 := math.Nextafter(f, math.Inf(+1))
+
+ // For f to be correct, r must be closer to f than to f0 or f1.
+ df := delta(r, f)
+ df0 := delta(r, f0)
+ df1 := delta(r, f1)
+ if df.Cmp(df0) > 0 {
+ t.Errorf("Rat(%v).Float64() = %g (%b), but previous float64 %g (%b) is closer", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) > 0 {
+ t.Errorf("Rat(%v).Float64() = %g (%b), but next float64 %g (%b) is closer", r, f, f, f1, f1)
+ return false
+ }
+ if df.Cmp(df0) == 0 && !isEven64(f) {
+ t.Errorf("Rat(%v).Float64() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) == 0 && !isEven64(f) {
+ t.Errorf("Rat(%v).Float64() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f1, f1)
+ return false
+ }
+ return true
+}
+
+func isEven32(f float32) bool { return math.Float32bits(f)&1 == 0 }
+func isEven64(f float64) bool { return math.Float64bits(f)&1 == 0 }
+
+func TestIsFinite(t *testing.T) {
+ finites := []float64{
+ 1.0 / 3,
+ 4891559871276714924261e+222,
+ math.MaxFloat64,
+ math.SmallestNonzeroFloat64,
+ -math.MaxFloat64,
+ -math.SmallestNonzeroFloat64,
+ }
+ for _, f := range finites {
+ if !isFinite(f) {
+ t.Errorf("!IsFinite(%g (%b))", f, f)
+ }
+ }
+ nonfinites := []float64{
+ math.NaN(),
+ math.Inf(-1),
+ math.Inf(+1),
+ }
+ for _, f := range nonfinites {
+ if isFinite(f) {
+ t.Errorf("IsFinite(%g, (%b))", f, f)
+ }
+ }
+}
+
+func TestRatSetInt64(t *testing.T) {
+ var testCases = []int64{
+ 0,
+ 1,
+ -1,
+ 12345,
+ -98765,
+ math.MaxInt64,
+ math.MinInt64,
+ }
+ var r = new(Rat)
+ for i, want := range testCases {
+ r.SetInt64(want)
+ if !r.IsInt() {
+ t.Errorf("#%d: Rat.SetInt64(%d) is not an integer", i, want)
+ }
+ num := r.Num()
+ if !num.IsInt64() {
+ t.Errorf("#%d: Rat.SetInt64(%d) numerator is not an int64", i, want)
+ }
+ got := num.Int64()
+ if got != want {
+ t.Errorf("#%d: Rat.SetInt64(%d) = %d, but expected %d", i, want, got, want)
+ }
+ }
+}
+
+func TestRatSetUint64(t *testing.T) {
+ var testCases = []uint64{
+ 0,
+ 1,
+ 12345,
+ ^uint64(0),
+ }
+ var r = new(Rat)
+ for i, want := range testCases {
+ r.SetUint64(want)
+ if !r.IsInt() {
+ t.Errorf("#%d: Rat.SetUint64(%d) is not an integer", i, want)
+ }
+ num := r.Num()
+ if !num.IsUint64() {
+ t.Errorf("#%d: Rat.SetUint64(%d) numerator is not a uint64", i, want)
+ }
+ got := num.Uint64()
+ if got != want {
+ t.Errorf("#%d: Rat.SetUint64(%d) = %d, but expected %d", i, want, got, want)
+ }
+ }
+}
+
+func BenchmarkRatCmp(b *testing.B) {
+ x, y := NewRat(4, 1), NewRat(7, 2)
+ for i := 0; i < b.N; i++ {
+ x.Cmp(y)
+ }
+}
+
+// TestIssue34919 verifies that a Rat's denominator is not modified
+// when simply accessing the Rat value.
+func TestIssue34919(t *testing.T) {
+ for _, acc := range []struct {
+ name string
+ f func(*Rat)
+ }{
+ {"Float32", func(x *Rat) { x.Float32() }},
+ {"Float64", func(x *Rat) { x.Float64() }},
+ {"Inv", func(x *Rat) { new(Rat).Inv(x) }},
+ {"Sign", func(x *Rat) { x.Sign() }},
+ {"IsInt", func(x *Rat) { x.IsInt() }},
+ {"Num", func(x *Rat) { x.Num() }},
+ // {"Denom", func(x *Rat) { x.Denom() }}, TODO(gri) should we change the API? See issue #33792.
+ } {
+ // A denominator of length 0 is interpreted as 1. Make sure that
+ // "materialization" of the denominator doesn't lead to setting
+ // the underlying array element 0 to 1.
+ r := &Rat{Int{abs: nat{991}}, Int{abs: make(nat, 0, 1)}}
+ acc.f(r)
+ if d := r.b.abs[:1][0]; d != 0 {
+ t.Errorf("%s modified denominator: got %d, want 0", acc.name, d)
+ }
+ }
+}
diff --git a/src/math/big/ratconv.go b/src/math/big/ratconv.go
new file mode 100644
index 0000000..ac3c8bd
--- /dev/null
+++ b/src/math/big/ratconv.go
@@ -0,0 +1,375 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements rat-to-string conversion functions.
+
+package big
+
+import (
+ "errors"
+ "fmt"
+ "io"
+ "strconv"
+ "strings"
+)
+
+func ratTok(ch rune) bool {
+ return strings.ContainsRune("+-/0123456789.eE", ch)
+}
+
+var ratZero Rat
+var _ fmt.Scanner = &ratZero // *Rat must implement fmt.Scanner
+
+// Scan is a support routine for fmt.Scanner. It accepts the formats
+// 'e', 'E', 'f', 'F', 'g', 'G', and 'v'. All formats are equivalent.
+func (z *Rat) Scan(s fmt.ScanState, ch rune) error {
+ tok, err := s.Token(true, ratTok)
+ if err != nil {
+ return err
+ }
+ if !strings.ContainsRune("efgEFGv", ch) {
+ return errors.New("Rat.Scan: invalid verb")
+ }
+ if _, ok := z.SetString(string(tok)); !ok {
+ return errors.New("Rat.Scan: invalid syntax")
+ }
+ return nil
+}
+
+// SetString sets z to the value of s and returns z and a boolean indicating
+// success. s can be given as a (possibly signed) fraction "a/b", or as a
+// floating-point number optionally followed by an exponent.
+// If a fraction is provided, both the dividend and the divisor may be a
+// decimal integer or independently use a prefix of ``0b'', ``0'' or ``0o'',
+// or ``0x'' (or their upper-case variants) to denote a binary, octal, or
+// hexadecimal integer, respectively. The divisor may not be signed.
+// If a floating-point number is provided, it may be in decimal form or
+// use any of the same prefixes as above but for ``0'' to denote a non-decimal
+// mantissa. A leading ``0'' is considered a decimal leading 0; it does not
+// indicate octal representation in this case.
+// An optional base-10 ``e'' or base-2 ``p'' (or their upper-case variants)
+// exponent may be provided as well, except for hexadecimal floats which
+// only accept an (optional) ``p'' exponent (because an ``e'' or ``E'' cannot
+// be distinguished from a mantissa digit). If the exponent's absolute value
+// is too large, the operation may fail.
+// The entire string, not just a prefix, must be valid for success. If the
+// operation failed, the value of z is undefined but the returned value is nil.
+func (z *Rat) SetString(s string) (*Rat, bool) {
+ if len(s) == 0 {
+ return nil, false
+ }
+ // len(s) > 0
+
+ // parse fraction a/b, if any
+ if sep := strings.Index(s, "/"); sep >= 0 {
+ if _, ok := z.a.SetString(s[:sep], 0); !ok {
+ return nil, false
+ }
+ r := strings.NewReader(s[sep+1:])
+ var err error
+ if z.b.abs, _, _, err = z.b.abs.scan(r, 0, false); err != nil {
+ return nil, false
+ }
+ // entire string must have been consumed
+ if _, err = r.ReadByte(); err != io.EOF {
+ return nil, false
+ }
+ if len(z.b.abs) == 0 {
+ return nil, false
+ }
+ return z.norm(), true
+ }
+
+ // parse floating-point number
+ r := strings.NewReader(s)
+
+ // sign
+ neg, err := scanSign(r)
+ if err != nil {
+ return nil, false
+ }
+
+ // mantissa
+ var base int
+ var fcount int // fractional digit count; valid if <= 0
+ z.a.abs, base, fcount, err = z.a.abs.scan(r, 0, true)
+ if err != nil {
+ return nil, false
+ }
+
+ // exponent
+ var exp int64
+ var ebase int
+ exp, ebase, err = scanExponent(r, true, true)
+ if err != nil {
+ return nil, false
+ }
+
+ // there should be no unread characters left
+ if _, err = r.ReadByte(); err != io.EOF {
+ return nil, false
+ }
+
+ // special-case 0 (see also issue #16176)
+ if len(z.a.abs) == 0 {
+ return z, true
+ }
+ // len(z.a.abs) > 0
+
+ // The mantissa may have a radix point (fcount <= 0) and there
+ // may be a nonzero exponent exp. The radix point amounts to a
+ // division by base**(-fcount), which equals a multiplication by
+ // base**fcount. An exponent means multiplication by ebase**exp.
+ // Multiplications are commutative, so we can apply them in any
+ // order. We only have powers of 2 and 10, and we split powers
+ // of 10 into the product of the same powers of 2 and 5. This
+ // may reduce the size of shift/multiplication factors or
+ // divisors required to create the final fraction, depending
+ // on the actual floating-point value.
+
+ // determine binary or decimal exponent contribution of radix point
+ var exp2, exp5 int64
+ if fcount < 0 {
+ // The mantissa has a radix point ddd.dddd; and
+ // -fcount is the number of digits to the right
+ // of '.'. Adjust relevant exponent accordingly.
+ d := int64(fcount)
+ switch base {
+ case 10:
+ exp5 = d
+ fallthrough // 10**e == 5**e * 2**e
+ case 2:
+ exp2 = d
+ case 8:
+ exp2 = d * 3 // octal digits are 3 bits each
+ case 16:
+ exp2 = d * 4 // hexadecimal digits are 4 bits each
+ default:
+ panic("unexpected mantissa base")
+ }
+ // fcount consumed - not needed anymore
+ }
+
+ // take actual exponent into account
+ switch ebase {
+ case 10:
+ exp5 += exp
+ fallthrough // see fallthrough above
+ case 2:
+ exp2 += exp
+ default:
+ panic("unexpected exponent base")
+ }
+ // exp consumed - not needed anymore
+
+ // apply exp5 contributions
+ // (start with exp5 so the numbers to multiply are smaller)
+ if exp5 != 0 {
+ n := exp5
+ if n < 0 {
+ n = -n
+ }
+ if n > 1e6 {
+ return nil, false // avoid excessively large exponents
+ }
+ pow5 := z.b.abs.expNN(natFive, nat(nil).setWord(Word(n)), nil) // use underlying array of z.b.abs
+ if exp5 > 0 {
+ z.a.abs = z.a.abs.mul(z.a.abs, pow5)
+ z.b.abs = z.b.abs.setWord(1)
+ } else {
+ z.b.abs = pow5
+ }
+ } else {
+ z.b.abs = z.b.abs.setWord(1)
+ }
+
+ // apply exp2 contributions
+ if exp2 < -1e7 || exp2 > 1e7 {
+ return nil, false // avoid excessively large exponents
+ }
+ if exp2 > 0 {
+ z.a.abs = z.a.abs.shl(z.a.abs, uint(exp2))
+ } else if exp2 < 0 {
+ z.b.abs = z.b.abs.shl(z.b.abs, uint(-exp2))
+ }
+
+ z.a.neg = neg && len(z.a.abs) > 0 // 0 has no sign
+
+ return z.norm(), true
+}
+
+// scanExponent scans the longest possible prefix of r representing a base 10
+// (``e'', ``E'') or a base 2 (``p'', ``P'') exponent, if any. It returns the
+// exponent, the exponent base (10 or 2), or a read or syntax error, if any.
+//
+// If sepOk is set, an underscore character ``_'' may appear between successive
+// exponent digits; such underscores do not change the value of the exponent.
+// Incorrect placement of underscores is reported as an error if there are no
+// other errors. If sepOk is not set, underscores are not recognized and thus
+// terminate scanning like any other character that is not a valid digit.
+//
+// exponent = ( "e" | "E" | "p" | "P" ) [ sign ] digits .
+// sign = "+" | "-" .
+// digits = digit { [ '_' ] digit } .
+// digit = "0" ... "9" .
+//
+// A base 2 exponent is only permitted if base2ok is set.
+func scanExponent(r io.ByteScanner, base2ok, sepOk bool) (exp int64, base int, err error) {
+ // one char look-ahead
+ ch, err := r.ReadByte()
+ if err != nil {
+ if err == io.EOF {
+ err = nil
+ }
+ return 0, 10, err
+ }
+
+ // exponent char
+ switch ch {
+ case 'e', 'E':
+ base = 10
+ case 'p', 'P':
+ if base2ok {
+ base = 2
+ break // ok
+ }
+ fallthrough // binary exponent not permitted
+ default:
+ r.UnreadByte() // ch does not belong to exponent anymore
+ return 0, 10, nil
+ }
+
+ // sign
+ var digits []byte
+ ch, err = r.ReadByte()
+ if err == nil && (ch == '+' || ch == '-') {
+ if ch == '-' {
+ digits = append(digits, '-')
+ }
+ ch, err = r.ReadByte()
+ }
+
+ // prev encodes the previously seen char: it is one
+ // of '_', '0' (a digit), or '.' (anything else). A
+ // valid separator '_' may only occur after a digit.
+ prev := '.'
+ invalSep := false
+
+ // exponent value
+ hasDigits := false
+ for err == nil {
+ if '0' <= ch && ch <= '9' {
+ digits = append(digits, ch)
+ prev = '0'
+ hasDigits = true
+ } else if ch == '_' && sepOk {
+ if prev != '0' {
+ invalSep = true
+ }
+ prev = '_'
+ } else {
+ r.UnreadByte() // ch does not belong to number anymore
+ break
+ }
+ ch, err = r.ReadByte()
+ }
+
+ if err == io.EOF {
+ err = nil
+ }
+ if err == nil && !hasDigits {
+ err = errNoDigits
+ }
+ if err == nil {
+ exp, err = strconv.ParseInt(string(digits), 10, 64)
+ }
+ // other errors take precedence over invalid separators
+ if err == nil && (invalSep || prev == '_') {
+ err = errInvalSep
+ }
+
+ return
+}
+
+// String returns a string representation of x in the form "a/b" (even if b == 1).
+func (x *Rat) String() string {
+ return string(x.marshal())
+}
+
+// marshal implements String returning a slice of bytes
+func (x *Rat) marshal() []byte {
+ var buf []byte
+ buf = x.a.Append(buf, 10)
+ buf = append(buf, '/')
+ if len(x.b.abs) != 0 {
+ buf = x.b.Append(buf, 10)
+ } else {
+ buf = append(buf, '1')
+ }
+ return buf
+}
+
+// RatString returns a string representation of x in the form "a/b" if b != 1,
+// and in the form "a" if b == 1.
+func (x *Rat) RatString() string {
+ if x.IsInt() {
+ return x.a.String()
+ }
+ return x.String()
+}
+
+// FloatString returns a string representation of x in decimal form with prec
+// digits of precision after the radix point. The last digit is rounded to
+// nearest, with halves rounded away from zero.
+func (x *Rat) FloatString(prec int) string {
+ var buf []byte
+
+ if x.IsInt() {
+ buf = x.a.Append(buf, 10)
+ if prec > 0 {
+ buf = append(buf, '.')
+ for i := prec; i > 0; i-- {
+ buf = append(buf, '0')
+ }
+ }
+ return string(buf)
+ }
+ // x.b.abs != 0
+
+ q, r := nat(nil).div(nat(nil), x.a.abs, x.b.abs)
+
+ p := natOne
+ if prec > 0 {
+ p = nat(nil).expNN(natTen, nat(nil).setUint64(uint64(prec)), nil)
+ }
+
+ r = r.mul(r, p)
+ r, r2 := r.div(nat(nil), r, x.b.abs)
+
+ // see if we need to round up
+ r2 = r2.add(r2, r2)
+ if x.b.abs.cmp(r2) <= 0 {
+ r = r.add(r, natOne)
+ if r.cmp(p) >= 0 {
+ q = nat(nil).add(q, natOne)
+ r = nat(nil).sub(r, p)
+ }
+ }
+
+ if x.a.neg {
+ buf = append(buf, '-')
+ }
+ buf = append(buf, q.utoa(10)...) // itoa ignores sign if q == 0
+
+ if prec > 0 {
+ buf = append(buf, '.')
+ rs := r.utoa(10)
+ for i := prec - len(rs); i > 0; i-- {
+ buf = append(buf, '0')
+ }
+ buf = append(buf, rs...)
+ }
+
+ return string(buf)
+}
diff --git a/src/math/big/ratconv_test.go b/src/math/big/ratconv_test.go
new file mode 100644
index 0000000..15d206c
--- /dev/null
+++ b/src/math/big/ratconv_test.go
@@ -0,0 +1,616 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "fmt"
+ "io"
+ "math"
+ "strconv"
+ "strings"
+ "testing"
+)
+
+var exponentTests = []struct {
+ s string // string to be scanned
+ base2ok bool // true if 'p'/'P' exponents are accepted
+ sepOk bool // true if '_' separators are accepted
+ x int64 // expected exponent
+ b int // expected exponent base
+ err error // expected error
+ next rune // next character (or 0, if at EOF)
+}{
+ // valid, without separators
+ {"", false, false, 0, 10, nil, 0},
+ {"1", false, false, 0, 10, nil, '1'},
+ {"e0", false, false, 0, 10, nil, 0},
+ {"E1", false, false, 1, 10, nil, 0},
+ {"e+10", false, false, 10, 10, nil, 0},
+ {"e-10", false, false, -10, 10, nil, 0},
+ {"e123456789a", false, false, 123456789, 10, nil, 'a'},
+ {"p", false, false, 0, 10, nil, 'p'},
+ {"P+100", false, false, 0, 10, nil, 'P'},
+ {"p0", true, false, 0, 2, nil, 0},
+ {"P-123", true, false, -123, 2, nil, 0},
+ {"p+0a", true, false, 0, 2, nil, 'a'},
+ {"p+123__", true, false, 123, 2, nil, '_'}, // '_' is not part of the number anymore
+
+ // valid, with separators
+ {"e+1_0", false, true, 10, 10, nil, 0},
+ {"e-1_0", false, true, -10, 10, nil, 0},
+ {"e123_456_789a", false, true, 123456789, 10, nil, 'a'},
+ {"P+1_00", false, true, 0, 10, nil, 'P'},
+ {"p-1_2_3", true, true, -123, 2, nil, 0},
+
+ // invalid: no digits
+ {"e", false, false, 0, 10, errNoDigits, 0},
+ {"ef", false, false, 0, 10, errNoDigits, 'f'},
+ {"e+", false, false, 0, 10, errNoDigits, 0},
+ {"E-x", false, false, 0, 10, errNoDigits, 'x'},
+ {"p", true, false, 0, 2, errNoDigits, 0},
+ {"P-", true, false, 0, 2, errNoDigits, 0},
+ {"p+e", true, false, 0, 2, errNoDigits, 'e'},
+ {"e+_x", false, true, 0, 10, errNoDigits, 'x'},
+
+ // invalid: incorrect use of separator
+ {"e0_", false, true, 0, 10, errInvalSep, 0},
+ {"e_0", false, true, 0, 10, errInvalSep, 0},
+ {"e-1_2__3", false, true, -123, 10, errInvalSep, 0},
+}
+
+func TestScanExponent(t *testing.T) {
+ for _, a := range exponentTests {
+ r := strings.NewReader(a.s)
+ x, b, err := scanExponent(r, a.base2ok, a.sepOk)
+ if err != a.err {
+ t.Errorf("scanExponent%+v\n\tgot error = %v; want %v", a, err, a.err)
+ }
+ if x != a.x {
+ t.Errorf("scanExponent%+v\n\tgot z = %v; want %v", a, x, a.x)
+ }
+ if b != a.b {
+ t.Errorf("scanExponent%+v\n\tgot b = %d; want %d", a, b, a.b)
+ }
+ next, _, err := r.ReadRune()
+ if err == io.EOF {
+ next = 0
+ err = nil
+ }
+ if err == nil && next != a.next {
+ t.Errorf("scanExponent%+v\n\tgot next = %q; want %q", a, next, a.next)
+ }
+ }
+}
+
+type StringTest struct {
+ in, out string
+ ok bool
+}
+
+var setStringTests = []StringTest{
+ // invalid
+ {in: "1e"},
+ {in: "1.e"},
+ {in: "1e+14e-5"},
+ {in: "1e4.5"},
+ {in: "r"},
+ {in: "a/b"},
+ {in: "a.b"},
+ {in: "1/0"},
+ {in: "4/3/2"}, // issue 17001
+ {in: "4/3/"},
+ {in: "4/3."},
+ {in: "4/"},
+
+ // valid
+ {"0", "0", true},
+ {"-0", "0", true},
+ {"1", "1", true},
+ {"-1", "-1", true},
+ {"1.", "1", true},
+ {"1e0", "1", true},
+ {"1.e1", "10", true},
+ {"-0.1", "-1/10", true},
+ {"-.1", "-1/10", true},
+ {"2/4", "1/2", true},
+ {".25", "1/4", true},
+ {"-1/5", "-1/5", true},
+ {"8129567.7690E14", "812956776900000000000", true},
+ {"78189e+4", "781890000", true},
+ {"553019.8935e+8", "55301989350000", true},
+ {"98765432109876543210987654321e-10", "98765432109876543210987654321/10000000000", true},
+ {"9877861857500000E-7", "3951144743/4", true},
+ {"2169378.417e-3", "2169378417/1000000", true},
+ {"884243222337379604041632732738665534", "884243222337379604041632732738665534", true},
+ {"53/70893980658822810696", "53/70893980658822810696", true},
+ {"106/141787961317645621392", "53/70893980658822810696", true},
+ {"204211327800791583.81095", "4084226556015831676219/20000", true},
+ {"0e9999999999", "0", true}, // issue #16176
+}
+
+// These are not supported by fmt.Fscanf.
+var setStringTests2 = []StringTest{
+ // invalid
+ {in: "4/3x"},
+ {in: "0/-1"},
+ {in: "-1/-1"},
+
+ // invalid with separators
+ // (smoke tests only - a comprehensive set of tests is in natconv_test.go)
+ {in: "10_/1"},
+ {in: "_10/1"},
+ {in: "1/1__0"},
+
+ // valid
+ {"0b1000/3", "8/3", true},
+ {"0B1000/0x8", "1", true},
+ {"-010/1", "-8", true}, // 0-prefix indicates octal in this case
+ {"-010.0", "-10", true},
+ {"-0o10/1", "-8", true},
+ {"0x10/1", "16", true},
+ {"0x10/0x20", "1/2", true},
+
+ {"0010", "10", true}, // 0-prefix is ignored in this case (not a fraction)
+ {"0x10.0", "16", true},
+ {"0x1.8", "3/2", true},
+ {"0X1.8p4", "24", true},
+ {"0x1.1E2", "2289/2048", true}, // E is part of hex mantissa, not exponent
+ {"0b1.1E2", "150", true},
+ {"0B1.1P3", "12", true},
+ {"0o10e-2", "2/25", true},
+ {"0O10p-3", "1", true},
+
+ // valid with separators
+ // (smoke tests only - a comprehensive set of tests is in natconv_test.go)
+ {"0b_1000/3", "8/3", true},
+ {"0B_10_00/0x8", "1", true},
+ {"0xdead/0B1101_1110_1010_1101", "1", true},
+ {"0B1101_1110_1010_1101/0XD_E_A_D", "1", true},
+ {"1_000.0", "1000", true},
+
+ {"0x_10.0", "16", true},
+ {"0x1_0.0", "16", true},
+ {"0x1.8_0", "3/2", true},
+ {"0X1.8p0_4", "24", true},
+ {"0b1.1_0E2", "150", true},
+ {"0o1_0e-2", "2/25", true},
+ {"0O_10p-3", "1", true},
+}
+
+func TestRatSetString(t *testing.T) {
+ var tests []StringTest
+ tests = append(tests, setStringTests...)
+ tests = append(tests, setStringTests2...)
+
+ for i, test := range tests {
+ x, ok := new(Rat).SetString(test.in)
+
+ if ok {
+ if !test.ok {
+ t.Errorf("#%d SetString(%q) expected failure", i, test.in)
+ } else if x.RatString() != test.out {
+ t.Errorf("#%d SetString(%q) got %s want %s", i, test.in, x.RatString(), test.out)
+ }
+ } else {
+ if test.ok {
+ t.Errorf("#%d SetString(%q) expected success", i, test.in)
+ } else if x != nil {
+ t.Errorf("#%d SetString(%q) got %p want nil", i, test.in, x)
+ }
+ }
+ }
+}
+
+func TestRatScan(t *testing.T) {
+ var buf bytes.Buffer
+ for i, test := range setStringTests {
+ x := new(Rat)
+ buf.Reset()
+ buf.WriteString(test.in)
+
+ _, err := fmt.Fscanf(&buf, "%v", x)
+ if err == nil != test.ok {
+ if test.ok {
+ t.Errorf("#%d (%s) error: %s", i, test.in, err)
+ } else {
+ t.Errorf("#%d (%s) expected error", i, test.in)
+ }
+ continue
+ }
+ if err == nil && x.RatString() != test.out {
+ t.Errorf("#%d got %s want %s", i, x.RatString(), test.out)
+ }
+ }
+}
+
+var floatStringTests = []struct {
+ in string
+ prec int
+ out string
+}{
+ {"0", 0, "0"},
+ {"0", 4, "0.0000"},
+ {"1", 0, "1"},
+ {"1", 2, "1.00"},
+ {"-1", 0, "-1"},
+ {"0.05", 1, "0.1"},
+ {"-0.05", 1, "-0.1"},
+ {".25", 2, "0.25"},
+ {".25", 1, "0.3"},
+ {".25", 3, "0.250"},
+ {"-1/3", 3, "-0.333"},
+ {"-2/3", 4, "-0.6667"},
+ {"0.96", 1, "1.0"},
+ {"0.999", 2, "1.00"},
+ {"0.9", 0, "1"},
+ {".25", -1, "0"},
+ {".55", -1, "1"},
+}
+
+func TestFloatString(t *testing.T) {
+ for i, test := range floatStringTests {
+ x, _ := new(Rat).SetString(test.in)
+
+ if x.FloatString(test.prec) != test.out {
+ t.Errorf("#%d got %s want %s", i, x.FloatString(test.prec), test.out)
+ }
+ }
+}
+
+// Test inputs to Rat.SetString. The prefix "long:" causes the test
+// to be skipped except in -long mode. (The threshold is about 500us.)
+var float64inputs = []string{
+ // Constants plundered from strconv/testfp.txt.
+
+ // Table 1: Stress Inputs for Conversion to 53-bit Binary, < 1/2 ULP
+ "5e+125",
+ "69e+267",
+ "999e-026",
+ "7861e-034",
+ "75569e-254",
+ "928609e-261",
+ "9210917e+080",
+ "84863171e+114",
+ "653777767e+273",
+ "5232604057e-298",
+ "27235667517e-109",
+ "653532977297e-123",
+ "3142213164987e-294",
+ "46202199371337e-072",
+ "231010996856685e-073",
+ "9324754620109615e+212",
+ "78459735791271921e+049",
+ "272104041512242479e+200",
+ "6802601037806061975e+198",
+ "20505426358836677347e-221",
+ "836168422905420598437e-234",
+ "4891559871276714924261e+222",
+
+ // Table 2: Stress Inputs for Conversion to 53-bit Binary, > 1/2 ULP
+ "9e-265",
+ "85e-037",
+ "623e+100",
+ "3571e+263",
+ "81661e+153",
+ "920657e-023",
+ "4603285e-024",
+ "87575437e-309",
+ "245540327e+122",
+ "6138508175e+120",
+ "83356057653e+193",
+ "619534293513e+124",
+ "2335141086879e+218",
+ "36167929443327e-159",
+ "609610927149051e-255",
+ "3743626360493413e-165",
+ "94080055902682397e-242",
+ "899810892172646163e+283",
+ "7120190517612959703e+120",
+ "25188282901709339043e-252",
+ "308984926168550152811e-052",
+ "6372891218502368041059e+064",
+
+ // Table 14: Stress Inputs for Conversion to 24-bit Binary, <1/2 ULP
+ "5e-20",
+ "67e+14",
+ "985e+15",
+ "7693e-42",
+ "55895e-16",
+ "996622e-44",
+ "7038531e-32",
+ "60419369e-46",
+ "702990899e-20",
+ "6930161142e-48",
+ "25933168707e+13",
+ "596428896559e+20",
+
+ // Table 15: Stress Inputs for Conversion to 24-bit Binary, >1/2 ULP
+ "3e-23",
+ "57e+18",
+ "789e-35",
+ "2539e-18",
+ "76173e+28",
+ "887745e-11",
+ "5382571e-37",
+ "82381273e-35",
+ "750486563e-38",
+ "3752432815e-39",
+ "75224575729e-45",
+ "459926601011e+15",
+
+ // Constants plundered from strconv/atof_test.go.
+
+ "0",
+ "1",
+ "+1",
+ "1e23",
+ "1E23",
+ "100000000000000000000000",
+ "1e-100",
+ "123456700",
+ "99999999999999974834176",
+ "100000000000000000000001",
+ "100000000000000008388608",
+ "100000000000000016777215",
+ "100000000000000016777216",
+ "-1",
+ "-0.1",
+ "-0", // NB: exception made for this input
+ "1e-20",
+ "625e-3",
+
+ // largest float64
+ "1.7976931348623157e308",
+ "-1.7976931348623157e308",
+ // next float64 - too large
+ "1.7976931348623159e308",
+ "-1.7976931348623159e308",
+ // the border is ...158079
+ // borderline - okay
+ "1.7976931348623158e308",
+ "-1.7976931348623158e308",
+ // borderline - too large
+ "1.797693134862315808e308",
+ "-1.797693134862315808e308",
+
+ // a little too large
+ "1e308",
+ "2e308",
+ "1e309",
+
+ // way too large
+ "1e310",
+ "-1e310",
+ "1e400",
+ "-1e400",
+ "long:1e400000",
+ "long:-1e400000",
+
+ // denormalized
+ "1e-305",
+ "1e-306",
+ "1e-307",
+ "1e-308",
+ "1e-309",
+ "1e-310",
+ "1e-322",
+ // smallest denormal
+ "5e-324",
+ "4e-324",
+ "3e-324",
+ // too small
+ "2e-324",
+ // way too small
+ "1e-350",
+ "long:1e-400000",
+ // way too small, negative
+ "-1e-350",
+ "long:-1e-400000",
+
+ // try to overflow exponent
+ // [Disabled: too slow and memory-hungry with rationals.]
+ // "1e-4294967296",
+ // "1e+4294967296",
+ // "1e-18446744073709551616",
+ // "1e+18446744073709551616",
+
+ // https://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
+ "2.2250738585072012e-308",
+ // https://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
+ "2.2250738585072011e-308",
+
+ // A very large number (initially wrongly parsed by the fast algorithm).
+ "4.630813248087435e+307",
+
+ // A different kind of very large number.
+ "22.222222222222222",
+ "long:2." + strings.Repeat("2", 4000) + "e+1",
+
+ // Exactly halfway between 1 and math.Nextafter(1, 2).
+ // Round to even (down).
+ "1.00000000000000011102230246251565404236316680908203125",
+ // Slightly lower; still round down.
+ "1.00000000000000011102230246251565404236316680908203124",
+ // Slightly higher; round up.
+ "1.00000000000000011102230246251565404236316680908203126",
+ // Slightly higher, but you have to read all the way to the end.
+ "long:1.00000000000000011102230246251565404236316680908203125" + strings.Repeat("0", 10000) + "1",
+
+ // Smallest denormal, 2^(-1022-52)
+ "4.940656458412465441765687928682213723651e-324",
+ // Half of smallest denormal, 2^(-1022-53)
+ "2.470328229206232720882843964341106861825e-324",
+ // A little more than the exact half of smallest denormal
+ // 2^-1075 + 2^-1100. (Rounds to 1p-1074.)
+ "2.470328302827751011111470718709768633275e-324",
+ // The exact halfway between smallest normal and largest denormal:
+ // 2^-1022 - 2^-1075. (Rounds to 2^-1022.)
+ "2.225073858507201136057409796709131975935e-308",
+
+ "1152921504606846975", // 1<<60 - 1
+ "-1152921504606846975", // -(1<<60 - 1)
+ "1152921504606846977", // 1<<60 + 1
+ "-1152921504606846977", // -(1<<60 + 1)
+
+ "1/3",
+}
+
+// isFinite reports whether f represents a finite rational value.
+// It is equivalent to !math.IsNan(f) && !math.IsInf(f, 0).
+func isFinite(f float64) bool {
+ return math.Abs(f) <= math.MaxFloat64
+}
+
+func TestFloat32SpecialCases(t *testing.T) {
+ for _, input := range float64inputs {
+ if strings.HasPrefix(input, "long:") {
+ if !*long {
+ continue
+ }
+ input = input[len("long:"):]
+ }
+
+ r, ok := new(Rat).SetString(input)
+ if !ok {
+ t.Errorf("Rat.SetString(%q) failed", input)
+ continue
+ }
+ f, exact := r.Float32()
+
+ // 1. Check string -> Rat -> float32 conversions are
+ // consistent with strconv.ParseFloat.
+ // Skip this check if the input uses "a/b" rational syntax.
+ if !strings.Contains(input, "/") {
+ e64, _ := strconv.ParseFloat(input, 32)
+ e := float32(e64)
+
+ // Careful: negative Rats too small for
+ // float64 become -0, but Rat obviously cannot
+ // preserve the sign from SetString("-0").
+ switch {
+ case math.Float32bits(e) == math.Float32bits(f):
+ // Ok: bitwise equal.
+ case f == 0 && r.Num().BitLen() == 0:
+ // Ok: Rat(0) is equivalent to both +/- float64(0).
+ default:
+ t.Errorf("strconv.ParseFloat(%q) = %g (%b), want %g (%b); delta = %g", input, e, e, f, f, f-e)
+ }
+ }
+
+ if !isFinite(float64(f)) {
+ continue
+ }
+
+ // 2. Check f is best approximation to r.
+ if !checkIsBestApprox32(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was %q)", input)
+ }
+
+ // 3. Check f->R->f roundtrip is non-lossy.
+ checkNonLossyRoundtrip32(t, f)
+
+ // 4. Check exactness using slow algorithm.
+ if wasExact := new(Rat).SetFloat64(float64(f)).Cmp(r) == 0; wasExact != exact {
+ t.Errorf("Rat.SetString(%q).Float32().exact = %t, want %t", input, exact, wasExact)
+ }
+ }
+}
+
+func TestFloat64SpecialCases(t *testing.T) {
+ for _, input := range float64inputs {
+ if strings.HasPrefix(input, "long:") {
+ if !*long {
+ continue
+ }
+ input = input[len("long:"):]
+ }
+
+ r, ok := new(Rat).SetString(input)
+ if !ok {
+ t.Errorf("Rat.SetString(%q) failed", input)
+ continue
+ }
+ f, exact := r.Float64()
+
+ // 1. Check string -> Rat -> float64 conversions are
+ // consistent with strconv.ParseFloat.
+ // Skip this check if the input uses "a/b" rational syntax.
+ if !strings.Contains(input, "/") {
+ e, _ := strconv.ParseFloat(input, 64)
+
+ // Careful: negative Rats too small for
+ // float64 become -0, but Rat obviously cannot
+ // preserve the sign from SetString("-0").
+ switch {
+ case math.Float64bits(e) == math.Float64bits(f):
+ // Ok: bitwise equal.
+ case f == 0 && r.Num().BitLen() == 0:
+ // Ok: Rat(0) is equivalent to both +/- float64(0).
+ default:
+ t.Errorf("strconv.ParseFloat(%q) = %g (%b), want %g (%b); delta = %g", input, e, e, f, f, f-e)
+ }
+ }
+
+ if !isFinite(f) {
+ continue
+ }
+
+ // 2. Check f is best approximation to r.
+ if !checkIsBestApprox64(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was %q)", input)
+ }
+
+ // 3. Check f->R->f roundtrip is non-lossy.
+ checkNonLossyRoundtrip64(t, f)
+
+ // 4. Check exactness using slow algorithm.
+ if wasExact := new(Rat).SetFloat64(f).Cmp(r) == 0; wasExact != exact {
+ t.Errorf("Rat.SetString(%q).Float64().exact = %t, want %t", input, exact, wasExact)
+ }
+ }
+}
+
+func TestIssue31184(t *testing.T) {
+ var x Rat
+ for _, want := range []string{
+ "-213.090",
+ "8.192",
+ "16.000",
+ } {
+ x.SetString(want)
+ got := x.FloatString(3)
+ if got != want {
+ t.Errorf("got %s, want %s", got, want)
+ }
+ }
+}
+
+func TestIssue45910(t *testing.T) {
+ var x Rat
+ for _, test := range []struct {
+ input string
+ want bool
+ }{
+ {"1e-1000001", false},
+ {"1e-1000000", true},
+ {"1e+1000000", true},
+ {"1e+1000001", false},
+
+ {"0p1000000000000", true},
+ {"1p-10000001", false},
+ {"1p-10000000", true},
+ {"1p+10000000", true},
+ {"1p+10000001", false},
+ {"1.770p02041010010011001001", false}, // test case from issue
+ } {
+ _, got := x.SetString(test.input)
+ if got != test.want {
+ t.Errorf("SetString(%s) got ok = %v; want %v", test.input, got, test.want)
+ }
+ }
+}
diff --git a/src/math/big/ratmarsh.go b/src/math/big/ratmarsh.go
new file mode 100644
index 0000000..fbc7b60
--- /dev/null
+++ b/src/math/big/ratmarsh.go
@@ -0,0 +1,75 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements encoding/decoding of Rats.
+
+package big
+
+import (
+ "encoding/binary"
+ "errors"
+ "fmt"
+)
+
+// Gob codec version. Permits backward-compatible changes to the encoding.
+const ratGobVersion byte = 1
+
+// GobEncode implements the gob.GobEncoder interface.
+func (x *Rat) GobEncode() ([]byte, error) {
+ if x == nil {
+ return nil, nil
+ }
+ buf := make([]byte, 1+4+(len(x.a.abs)+len(x.b.abs))*_S) // extra bytes for version and sign bit (1), and numerator length (4)
+ i := x.b.abs.bytes(buf)
+ j := x.a.abs.bytes(buf[:i])
+ n := i - j
+ if int(uint32(n)) != n {
+ // this should never happen
+ return nil, errors.New("Rat.GobEncode: numerator too large")
+ }
+ binary.BigEndian.PutUint32(buf[j-4:j], uint32(n))
+ j -= 1 + 4
+ b := ratGobVersion << 1 // make space for sign bit
+ if x.a.neg {
+ b |= 1
+ }
+ buf[j] = b
+ return buf[j:], nil
+}
+
+// GobDecode implements the gob.GobDecoder interface.
+func (z *Rat) GobDecode(buf []byte) error {
+ if len(buf) == 0 {
+ // Other side sent a nil or default value.
+ *z = Rat{}
+ return nil
+ }
+ b := buf[0]
+ if b>>1 != ratGobVersion {
+ return fmt.Errorf("Rat.GobDecode: encoding version %d not supported", b>>1)
+ }
+ const j = 1 + 4
+ i := j + binary.BigEndian.Uint32(buf[j-4:j])
+ z.a.neg = b&1 != 0
+ z.a.abs = z.a.abs.setBytes(buf[j:i])
+ z.b.abs = z.b.abs.setBytes(buf[i:])
+ return nil
+}
+
+// MarshalText implements the encoding.TextMarshaler interface.
+func (x *Rat) MarshalText() (text []byte, err error) {
+ if x.IsInt() {
+ return x.a.MarshalText()
+ }
+ return x.marshal(), nil
+}
+
+// UnmarshalText implements the encoding.TextUnmarshaler interface.
+func (z *Rat) UnmarshalText(text []byte) error {
+ // TODO(gri): get rid of the []byte/string conversion
+ if _, ok := z.SetString(string(text)); !ok {
+ return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Rat", text)
+ }
+ return nil
+}
diff --git a/src/math/big/ratmarsh_test.go b/src/math/big/ratmarsh_test.go
new file mode 100644
index 0000000..351d109
--- /dev/null
+++ b/src/math/big/ratmarsh_test.go
@@ -0,0 +1,125 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "encoding/gob"
+ "encoding/json"
+ "encoding/xml"
+ "testing"
+)
+
+func TestRatGobEncoding(t *testing.T) {
+ var medium bytes.Buffer
+ enc := gob.NewEncoder(&medium)
+ dec := gob.NewDecoder(&medium)
+ for _, test := range encodingTests {
+ medium.Reset() // empty buffer for each test case (in case of failures)
+ var tx Rat
+ tx.SetString(test + ".14159265")
+ if err := enc.Encode(&tx); err != nil {
+ t.Errorf("encoding of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Rat
+ if err := dec.Decode(&rx); err != nil {
+ t.Errorf("decoding of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("transmission of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+// Sending a nil Rat pointer (inside a slice) on a round trip through gob should yield a zero.
+// TODO: top-level nils.
+func TestGobEncodingNilRatInSlice(t *testing.T) {
+ buf := new(bytes.Buffer)
+ enc := gob.NewEncoder(buf)
+ dec := gob.NewDecoder(buf)
+
+ var in = make([]*Rat, 1)
+ err := enc.Encode(&in)
+ if err != nil {
+ t.Errorf("gob encode failed: %q", err)
+ }
+ var out []*Rat
+ err = dec.Decode(&out)
+ if err != nil {
+ t.Fatalf("gob decode failed: %q", err)
+ }
+ if len(out) != 1 {
+ t.Fatalf("wrong len; want 1 got %d", len(out))
+ }
+ var zero Rat
+ if out[0].Cmp(&zero) != 0 {
+ t.Fatalf("transmission of (*Int)(nil) failed: got %s want 0", out)
+ }
+}
+
+var ratNums = []string{
+ "-141592653589793238462643383279502884197169399375105820974944592307816406286",
+ "-1415926535897932384626433832795028841971",
+ "-141592653589793",
+ "-1",
+ "0",
+ "1",
+ "141592653589793",
+ "1415926535897932384626433832795028841971",
+ "141592653589793238462643383279502884197169399375105820974944592307816406286",
+}
+
+var ratDenoms = []string{
+ "1",
+ "718281828459045",
+ "7182818284590452353602874713526624977572",
+ "718281828459045235360287471352662497757247093699959574966967627724076630353",
+}
+
+func TestRatJSONEncoding(t *testing.T) {
+ for _, num := range ratNums {
+ for _, denom := range ratDenoms {
+ var tx Rat
+ tx.SetString(num + "/" + denom)
+ b, err := json.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Rat
+ if err := json.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+ }
+}
+
+func TestRatXMLEncoding(t *testing.T) {
+ for _, num := range ratNums {
+ for _, denom := range ratDenoms {
+ var tx Rat
+ tx.SetString(num + "/" + denom)
+ b, err := xml.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Rat
+ if err := xml.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("XML encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+ }
+}
diff --git a/src/math/big/roundingmode_string.go b/src/math/big/roundingmode_string.go
new file mode 100644
index 0000000..c7629eb
--- /dev/null
+++ b/src/math/big/roundingmode_string.go
@@ -0,0 +1,16 @@
+// Code generated by "stringer -type=RoundingMode"; DO NOT EDIT.
+
+package big
+
+import "strconv"
+
+const _RoundingMode_name = "ToNearestEvenToNearestAwayToZeroAwayFromZeroToNegativeInfToPositiveInf"
+
+var _RoundingMode_index = [...]uint8{0, 13, 26, 32, 44, 57, 70}
+
+func (i RoundingMode) String() string {
+ if i >= RoundingMode(len(_RoundingMode_index)-1) {
+ return "RoundingMode(" + strconv.FormatInt(int64(i), 10) + ")"
+ }
+ return _RoundingMode_name[_RoundingMode_index[i]:_RoundingMode_index[i+1]]
+}
diff --git a/src/math/big/sqrt.go b/src/math/big/sqrt.go
new file mode 100644
index 0000000..0d50164
--- /dev/null
+++ b/src/math/big/sqrt.go
@@ -0,0 +1,128 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "math"
+ "sync"
+)
+
+var threeOnce struct {
+ sync.Once
+ v *Float
+}
+
+func three() *Float {
+ threeOnce.Do(func() {
+ threeOnce.v = NewFloat(3.0)
+ })
+ return threeOnce.v
+}
+
+// Sqrt sets z to the rounded square root of x, and returns it.
+//
+// If z's precision is 0, it is changed to x's precision before the
+// operation. Rounding is performed according to z's precision and
+// rounding mode, but z's accuracy is not computed. Specifically, the
+// result of z.Acc() is undefined.
+//
+// The function panics if z < 0. The value of z is undefined in that
+// case.
+func (z *Float) Sqrt(x *Float) *Float {
+ if debugFloat {
+ x.validate()
+ }
+
+ if z.prec == 0 {
+ z.prec = x.prec
+ }
+
+ if x.Sign() == -1 {
+ // following IEEE754-2008 (section 7.2)
+ panic(ErrNaN{"square root of negative operand"})
+ }
+
+ // handle ±0 and +∞
+ if x.form != finite {
+ z.acc = Exact
+ z.form = x.form
+ z.neg = x.neg // IEEE754-2008 requires √±0 = ±0
+ return z
+ }
+
+ // MantExp sets the argument's precision to the receiver's, and
+ // when z.prec > x.prec this will lower z.prec. Restore it after
+ // the MantExp call.
+ prec := z.prec
+ b := x.MantExp(z)
+ z.prec = prec
+
+ // Compute √(z·2**b) as
+ // √( z)·2**(½b) if b is even
+ // √(2z)·2**(⌊½b⌋) if b > 0 is odd
+ // √(½z)·2**(⌈½b⌉) if b < 0 is odd
+ switch b % 2 {
+ case 0:
+ // nothing to do
+ case 1:
+ z.exp++
+ case -1:
+ z.exp--
+ }
+ // 0.25 <= z < 2.0
+
+ // Solving 1/x² - z = 0 avoids Quo calls and is faster, especially
+ // for high precisions.
+ z.sqrtInverse(z)
+
+ // re-attach halved exponent
+ return z.SetMantExp(z, b/2)
+}
+
+// Compute √x (to z.prec precision) by solving
+// 1/t² - x = 0
+// for t (using Newton's method), and then inverting.
+func (z *Float) sqrtInverse(x *Float) {
+ // let
+ // f(t) = 1/t² - x
+ // then
+ // g(t) = f(t)/f'(t) = -½t(1 - xt²)
+ // and the next guess is given by
+ // t2 = t - g(t) = ½t(3 - xt²)
+ u := newFloat(z.prec)
+ v := newFloat(z.prec)
+ three := three()
+ ng := func(t *Float) *Float {
+ u.prec = t.prec
+ v.prec = t.prec
+ u.Mul(t, t) // u = t²
+ u.Mul(x, u) // = xt²
+ v.Sub(three, u) // v = 3 - xt²
+ u.Mul(t, v) // u = t(3 - xt²)
+ u.exp-- // = ½t(3 - xt²)
+ return t.Set(u)
+ }
+
+ xf, _ := x.Float64()
+ sqi := newFloat(z.prec)
+ sqi.SetFloat64(1 / math.Sqrt(xf))
+ for prec := z.prec + 32; sqi.prec < prec; {
+ sqi.prec *= 2
+ sqi = ng(sqi)
+ }
+ // sqi = 1/√x
+
+ // x/√x = √x
+ z.Mul(x, sqi)
+}
+
+// newFloat returns a new *Float with space for twice the given
+// precision.
+func newFloat(prec2 uint32) *Float {
+ z := new(Float)
+ // nat.make ensures the slice length is > 0
+ z.mant = z.mant.make(int(prec2/_W) * 2)
+ return z
+}
diff --git a/src/math/big/sqrt_test.go b/src/math/big/sqrt_test.go
new file mode 100644
index 0000000..d314711
--- /dev/null
+++ b/src/math/big/sqrt_test.go
@@ -0,0 +1,126 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "fmt"
+ "math"
+ "math/rand"
+ "testing"
+)
+
+// TestFloatSqrt64 tests that Float.Sqrt of numbers with 53bit mantissa
+// behaves like float math.Sqrt.
+func TestFloatSqrt64(t *testing.T) {
+ for i := 0; i < 1e5; i++ {
+ if i == 1e2 && testing.Short() {
+ break
+ }
+ r := rand.Float64()
+
+ got := new(Float).SetPrec(53)
+ got.Sqrt(NewFloat(r))
+ want := NewFloat(math.Sqrt(r))
+ if got.Cmp(want) != 0 {
+ t.Fatalf("Sqrt(%g) =\n got %g;\nwant %g", r, got, want)
+ }
+ }
+}
+
+func TestFloatSqrt(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ want string
+ }{
+ // Test values were generated on Wolfram Alpha using query
+ // 'sqrt(N) to 350 digits'
+ // 350 decimal digits give up to 1000 binary digits.
+ {"0.03125", "0.17677669529663688110021109052621225982120898442211850914708496724884155980776337985629844179095519659187673077886403712811560450698134215158051518713749197892665283324093819909447499381264409775757143376369499645074628431682460775184106467733011114982619404115381053858929018135497032545349940642599871090667456829147610370507757690729404938184321879"},
+ {"0.125", "0.35355339059327376220042218105242451964241796884423701829416993449768311961552675971259688358191039318375346155772807425623120901396268430316103037427498395785330566648187639818894998762528819551514286752738999290149256863364921550368212935466022229965238808230762107717858036270994065090699881285199742181334913658295220741015515381458809876368643757"},
+ {"0.5", "0.70710678118654752440084436210484903928483593768847403658833986899536623923105351942519376716382078636750692311545614851246241802792536860632206074854996791570661133296375279637789997525057639103028573505477998580298513726729843100736425870932044459930477616461524215435716072541988130181399762570399484362669827316590441482031030762917619752737287514"},
+ {"2.0", "1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727350138462309122970249248360558507372126441214970999358314132226659275055927557999505011527820605714701095599716059702745345968620147285174186408891986095523292304843087143214508397626036279952514079896872533965463318088296406206152583523950547457503"},
+ {"3.0", "1.7320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485756756261414154067030299699450949989524788116555120943736485280932319023055820679748201010846749232650153123432669033228866506722546689218379712270471316603678615880190499865373798593894676503475065760507566183481296061009476021871903250831458295239598"},
+ {"4.0", "2.0"},
+
+ {"1p512", "1p256"},
+ {"4p1024", "2p512"},
+ {"9p2048", "3p1024"},
+
+ {"1p-1024", "1p-512"},
+ {"4p-2048", "2p-1024"},
+ {"9p-4096", "3p-2048"},
+ } {
+ for _, prec := range []uint{24, 53, 64, 65, 100, 128, 129, 200, 256, 400, 600, 800, 1000} {
+ x := new(Float).SetPrec(prec)
+ x.Parse(test.x, 10)
+
+ got := new(Float).SetPrec(prec).Sqrt(x)
+ want := new(Float).SetPrec(prec)
+ want.Parse(test.want, 10)
+ if got.Cmp(want) != 0 {
+ t.Errorf("prec = %d, Sqrt(%v) =\ngot %g;\nwant %g",
+ prec, test.x, got, want)
+ }
+
+ // Square test.
+ // If got holds the square root of x to precision p, then
+ // got = √x + k
+ // for some k such that |k| < 2**(-p). Thus,
+ // got² = (√x + k)² = x + 2k√n + k²
+ // and the error must satisfy
+ // err = |got² - x| ≈ | 2k√n | < 2**(-p+1)*√n
+ // Ignoring the k² term for simplicity.
+
+ // err = |got² - x|
+ // (but do intermediate steps with 32 guard digits to
+ // avoid introducing spurious rounding-related errors)
+ sq := new(Float).SetPrec(prec+32).Mul(got, got)
+ diff := new(Float).Sub(sq, x)
+ err := diff.Abs(diff).SetPrec(prec)
+
+ // maxErr = 2**(-p+1)*√x
+ one := new(Float).SetPrec(prec).SetInt64(1)
+ maxErr := new(Float).Mul(new(Float).SetMantExp(one, -int(prec)+1), got)
+
+ if err.Cmp(maxErr) >= 0 {
+ t.Errorf("prec = %d, Sqrt(%v) =\ngot err %g;\nwant maxErr %g",
+ prec, test.x, err, maxErr)
+ }
+ }
+ }
+}
+
+func TestFloatSqrtSpecial(t *testing.T) {
+ for _, test := range []struct {
+ x *Float
+ want *Float
+ }{
+ {NewFloat(+0), NewFloat(+0)},
+ {NewFloat(-0), NewFloat(-0)},
+ {NewFloat(math.Inf(+1)), NewFloat(math.Inf(+1))},
+ } {
+ got := new(Float).Sqrt(test.x)
+ if got.neg != test.want.neg || got.form != test.want.form {
+ t.Errorf("Sqrt(%v) = %v (neg: %v); want %v (neg: %v)",
+ test.x, got, got.neg, test.want, test.want.neg)
+ }
+ }
+
+}
+
+// Benchmarks
+
+func BenchmarkFloatSqrt(b *testing.B) {
+ for _, prec := range []uint{64, 128, 256, 1e3, 1e4, 1e5, 1e6} {
+ x := NewFloat(2)
+ z := new(Float).SetPrec(prec)
+ b.Run(fmt.Sprintf("%v", prec), func(b *testing.B) {
+ b.ReportAllocs()
+ for n := 0; n < b.N; n++ {
+ z.Sqrt(x)
+ }
+ })
+ }
+}
diff --git a/src/math/bits.go b/src/math/bits.go
new file mode 100644
index 0000000..77bcdbe
--- /dev/null
+++ b/src/math/bits.go
@@ -0,0 +1,62 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+const (
+ uvnan = 0x7FF8000000000001
+ uvinf = 0x7FF0000000000000
+ uvneginf = 0xFFF0000000000000
+ uvone = 0x3FF0000000000000
+ mask = 0x7FF
+ shift = 64 - 11 - 1
+ bias = 1023
+ signMask = 1 << 63
+ fracMask = 1<<shift - 1
+)
+
+// Inf returns positive infinity if sign >= 0, negative infinity if sign < 0.
+func Inf(sign int) float64 {
+ var v uint64
+ if sign >= 0 {
+ v = uvinf
+ } else {
+ v = uvneginf
+ }
+ return Float64frombits(v)
+}
+
+// NaN returns an IEEE 754 ``not-a-number'' value.
+func NaN() float64 { return Float64frombits(uvnan) }
+
+// IsNaN reports whether f is an IEEE 754 ``not-a-number'' value.
+func IsNaN(f float64) (is bool) {
+ // IEEE 754 says that only NaNs satisfy f != f.
+ // To avoid the floating-point hardware, could use:
+ // x := Float64bits(f);
+ // return uint32(x>>shift)&mask == mask && x != uvinf && x != uvneginf
+ return f != f
+}
+
+// IsInf reports whether f is an infinity, according to sign.
+// If sign > 0, IsInf reports whether f is positive infinity.
+// If sign < 0, IsInf reports whether f is negative infinity.
+// If sign == 0, IsInf reports whether f is either infinity.
+func IsInf(f float64, sign int) bool {
+ // Test for infinity by comparing against maximum float.
+ // To avoid the floating-point hardware, could use:
+ // x := Float64bits(f);
+ // return sign >= 0 && x == uvinf || sign <= 0 && x == uvneginf;
+ return sign >= 0 && f > MaxFloat64 || sign <= 0 && f < -MaxFloat64
+}
+
+// normalize returns a normal number y and exponent exp
+// satisfying x == y × 2**exp. It assumes x is finite and non-zero.
+func normalize(x float64) (y float64, exp int) {
+ const SmallestNormal = 2.2250738585072014e-308 // 2**-1022
+ if Abs(x) < SmallestNormal {
+ return x * (1 << 52), -52
+ }
+ return x, 0
+}
diff --git a/src/math/bits/bits.go b/src/math/bits/bits.go
new file mode 100644
index 0000000..879ef2d
--- /dev/null
+++ b/src/math/bits/bits.go
@@ -0,0 +1,588 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+//go:generate go run make_tables.go
+
+// Package bits implements bit counting and manipulation
+// functions for the predeclared unsigned integer types.
+package bits
+
+const uintSize = 32 << (^uint(0) >> 32 & 1) // 32 or 64
+
+// UintSize is the size of a uint in bits.
+const UintSize = uintSize
+
+// --- LeadingZeros ---
+
+// LeadingZeros returns the number of leading zero bits in x; the result is UintSize for x == 0.
+func LeadingZeros(x uint) int { return UintSize - Len(x) }
+
+// LeadingZeros8 returns the number of leading zero bits in x; the result is 8 for x == 0.
+func LeadingZeros8(x uint8) int { return 8 - Len8(x) }
+
+// LeadingZeros16 returns the number of leading zero bits in x; the result is 16 for x == 0.
+func LeadingZeros16(x uint16) int { return 16 - Len16(x) }
+
+// LeadingZeros32 returns the number of leading zero bits in x; the result is 32 for x == 0.
+func LeadingZeros32(x uint32) int { return 32 - Len32(x) }
+
+// LeadingZeros64 returns the number of leading zero bits in x; the result is 64 for x == 0.
+func LeadingZeros64(x uint64) int { return 64 - Len64(x) }
+
+// --- TrailingZeros ---
+
+// See http://supertech.csail.mit.edu/papers/debruijn.pdf
+const deBruijn32 = 0x077CB531
+
+var deBruijn32tab = [32]byte{
+ 0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8,
+ 31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9,
+}
+
+const deBruijn64 = 0x03f79d71b4ca8b09
+
+var deBruijn64tab = [64]byte{
+ 0, 1, 56, 2, 57, 49, 28, 3, 61, 58, 42, 50, 38, 29, 17, 4,
+ 62, 47, 59, 36, 45, 43, 51, 22, 53, 39, 33, 30, 24, 18, 12, 5,
+ 63, 55, 48, 27, 60, 41, 37, 16, 46, 35, 44, 21, 52, 32, 23, 11,
+ 54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6,
+}
+
+// TrailingZeros returns the number of trailing zero bits in x; the result is UintSize for x == 0.
+func TrailingZeros(x uint) int {
+ if UintSize == 32 {
+ return TrailingZeros32(uint32(x))
+ }
+ return TrailingZeros64(uint64(x))
+}
+
+// TrailingZeros8 returns the number of trailing zero bits in x; the result is 8 for x == 0.
+func TrailingZeros8(x uint8) int {
+ return int(ntz8tab[x])
+}
+
+// TrailingZeros16 returns the number of trailing zero bits in x; the result is 16 for x == 0.
+func TrailingZeros16(x uint16) int {
+ if x == 0 {
+ return 16
+ }
+ // see comment in TrailingZeros64
+ return int(deBruijn32tab[uint32(x&-x)*deBruijn32>>(32-5)])
+}
+
+// TrailingZeros32 returns the number of trailing zero bits in x; the result is 32 for x == 0.
+func TrailingZeros32(x uint32) int {
+ if x == 0 {
+ return 32
+ }
+ // see comment in TrailingZeros64
+ return int(deBruijn32tab[(x&-x)*deBruijn32>>(32-5)])
+}
+
+// TrailingZeros64 returns the number of trailing zero bits in x; the result is 64 for x == 0.
+func TrailingZeros64(x uint64) int {
+ if x == 0 {
+ return 64
+ }
+ // If popcount is fast, replace code below with return popcount(^x & (x - 1)).
+ //
+ // x & -x leaves only the right-most bit set in the word. Let k be the
+ // index of that bit. Since only a single bit is set, the value is two
+ // to the power of k. Multiplying by a power of two is equivalent to
+ // left shifting, in this case by k bits. The de Bruijn (64 bit) constant
+ // is such that all six bit, consecutive substrings are distinct.
+ // Therefore, if we have a left shifted version of this constant we can
+ // find by how many bits it was shifted by looking at which six bit
+ // substring ended up at the top of the word.
+ // (Knuth, volume 4, section 7.3.1)
+ return int(deBruijn64tab[(x&-x)*deBruijn64>>(64-6)])
+}
+
+// --- OnesCount ---
+
+const m0 = 0x5555555555555555 // 01010101 ...
+const m1 = 0x3333333333333333 // 00110011 ...
+const m2 = 0x0f0f0f0f0f0f0f0f // 00001111 ...
+const m3 = 0x00ff00ff00ff00ff // etc.
+const m4 = 0x0000ffff0000ffff
+
+// OnesCount returns the number of one bits ("population count") in x.
+func OnesCount(x uint) int {
+ if UintSize == 32 {
+ return OnesCount32(uint32(x))
+ }
+ return OnesCount64(uint64(x))
+}
+
+// OnesCount8 returns the number of one bits ("population count") in x.
+func OnesCount8(x uint8) int {
+ return int(pop8tab[x])
+}
+
+// OnesCount16 returns the number of one bits ("population count") in x.
+func OnesCount16(x uint16) int {
+ return int(pop8tab[x>>8] + pop8tab[x&0xff])
+}
+
+// OnesCount32 returns the number of one bits ("population count") in x.
+func OnesCount32(x uint32) int {
+ return int(pop8tab[x>>24] + pop8tab[x>>16&0xff] + pop8tab[x>>8&0xff] + pop8tab[x&0xff])
+}
+
+// OnesCount64 returns the number of one bits ("population count") in x.
+func OnesCount64(x uint64) int {
+ // Implementation: Parallel summing of adjacent bits.
+ // See "Hacker's Delight", Chap. 5: Counting Bits.
+ // The following pattern shows the general approach:
+ //
+ // x = x>>1&(m0&m) + x&(m0&m)
+ // x = x>>2&(m1&m) + x&(m1&m)
+ // x = x>>4&(m2&m) + x&(m2&m)
+ // x = x>>8&(m3&m) + x&(m3&m)
+ // x = x>>16&(m4&m) + x&(m4&m)
+ // x = x>>32&(m5&m) + x&(m5&m)
+ // return int(x)
+ //
+ // Masking (& operations) can be left away when there's no
+ // danger that a field's sum will carry over into the next
+ // field: Since the result cannot be > 64, 8 bits is enough
+ // and we can ignore the masks for the shifts by 8 and up.
+ // Per "Hacker's Delight", the first line can be simplified
+ // more, but it saves at best one instruction, so we leave
+ // it alone for clarity.
+ const m = 1<<64 - 1
+ x = x>>1&(m0&m) + x&(m0&m)
+ x = x>>2&(m1&m) + x&(m1&m)
+ x = (x>>4 + x) & (m2 & m)
+ x += x >> 8
+ x += x >> 16
+ x += x >> 32
+ return int(x) & (1<<7 - 1)
+}
+
+// --- RotateLeft ---
+
+// RotateLeft returns the value of x rotated left by (k mod UintSize) bits.
+// To rotate x right by k bits, call RotateLeft(x, -k).
+//
+// This function's execution time does not depend on the inputs.
+func RotateLeft(x uint, k int) uint {
+ if UintSize == 32 {
+ return uint(RotateLeft32(uint32(x), k))
+ }
+ return uint(RotateLeft64(uint64(x), k))
+}
+
+// RotateLeft8 returns the value of x rotated left by (k mod 8) bits.
+// To rotate x right by k bits, call RotateLeft8(x, -k).
+//
+// This function's execution time does not depend on the inputs.
+func RotateLeft8(x uint8, k int) uint8 {
+ const n = 8
+ s := uint(k) & (n - 1)
+ return x<<s | x>>(n-s)
+}
+
+// RotateLeft16 returns the value of x rotated left by (k mod 16) bits.
+// To rotate x right by k bits, call RotateLeft16(x, -k).
+//
+// This function's execution time does not depend on the inputs.
+func RotateLeft16(x uint16, k int) uint16 {
+ const n = 16
+ s := uint(k) & (n - 1)
+ return x<<s | x>>(n-s)
+}
+
+// RotateLeft32 returns the value of x rotated left by (k mod 32) bits.
+// To rotate x right by k bits, call RotateLeft32(x, -k).
+//
+// This function's execution time does not depend on the inputs.
+func RotateLeft32(x uint32, k int) uint32 {
+ const n = 32
+ s := uint(k) & (n - 1)
+ return x<<s | x>>(n-s)
+}
+
+// RotateLeft64 returns the value of x rotated left by (k mod 64) bits.
+// To rotate x right by k bits, call RotateLeft64(x, -k).
+//
+// This function's execution time does not depend on the inputs.
+func RotateLeft64(x uint64, k int) uint64 {
+ const n = 64
+ s := uint(k) & (n - 1)
+ return x<<s | x>>(n-s)
+}
+
+// --- Reverse ---
+
+// Reverse returns the value of x with its bits in reversed order.
+func Reverse(x uint) uint {
+ if UintSize == 32 {
+ return uint(Reverse32(uint32(x)))
+ }
+ return uint(Reverse64(uint64(x)))
+}
+
+// Reverse8 returns the value of x with its bits in reversed order.
+func Reverse8(x uint8) uint8 {
+ return rev8tab[x]
+}
+
+// Reverse16 returns the value of x with its bits in reversed order.
+func Reverse16(x uint16) uint16 {
+ return uint16(rev8tab[x>>8]) | uint16(rev8tab[x&0xff])<<8
+}
+
+// Reverse32 returns the value of x with its bits in reversed order.
+func Reverse32(x uint32) uint32 {
+ const m = 1<<32 - 1
+ x = x>>1&(m0&m) | x&(m0&m)<<1
+ x = x>>2&(m1&m) | x&(m1&m)<<2
+ x = x>>4&(m2&m) | x&(m2&m)<<4
+ return ReverseBytes32(x)
+}
+
+// Reverse64 returns the value of x with its bits in reversed order.
+func Reverse64(x uint64) uint64 {
+ const m = 1<<64 - 1
+ x = x>>1&(m0&m) | x&(m0&m)<<1
+ x = x>>2&(m1&m) | x&(m1&m)<<2
+ x = x>>4&(m2&m) | x&(m2&m)<<4
+ return ReverseBytes64(x)
+}
+
+// --- ReverseBytes ---
+
+// ReverseBytes returns the value of x with its bytes in reversed order.
+//
+// This function's execution time does not depend on the inputs.
+func ReverseBytes(x uint) uint {
+ if UintSize == 32 {
+ return uint(ReverseBytes32(uint32(x)))
+ }
+ return uint(ReverseBytes64(uint64(x)))
+}
+
+// ReverseBytes16 returns the value of x with its bytes in reversed order.
+//
+// This function's execution time does not depend on the inputs.
+func ReverseBytes16(x uint16) uint16 {
+ return x>>8 | x<<8
+}
+
+// ReverseBytes32 returns the value of x with its bytes in reversed order.
+//
+// This function's execution time does not depend on the inputs.
+func ReverseBytes32(x uint32) uint32 {
+ const m = 1<<32 - 1
+ x = x>>8&(m3&m) | x&(m3&m)<<8
+ return x>>16 | x<<16
+}
+
+// ReverseBytes64 returns the value of x with its bytes in reversed order.
+//
+// This function's execution time does not depend on the inputs.
+func ReverseBytes64(x uint64) uint64 {
+ const m = 1<<64 - 1
+ x = x>>8&(m3&m) | x&(m3&m)<<8
+ x = x>>16&(m4&m) | x&(m4&m)<<16
+ return x>>32 | x<<32
+}
+
+// --- Len ---
+
+// Len returns the minimum number of bits required to represent x; the result is 0 for x == 0.
+func Len(x uint) int {
+ if UintSize == 32 {
+ return Len32(uint32(x))
+ }
+ return Len64(uint64(x))
+}
+
+// Len8 returns the minimum number of bits required to represent x; the result is 0 for x == 0.
+func Len8(x uint8) int {
+ return int(len8tab[x])
+}
+
+// Len16 returns the minimum number of bits required to represent x; the result is 0 for x == 0.
+func Len16(x uint16) (n int) {
+ if x >= 1<<8 {
+ x >>= 8
+ n = 8
+ }
+ return n + int(len8tab[x])
+}
+
+// Len32 returns the minimum number of bits required to represent x; the result is 0 for x == 0.
+func Len32(x uint32) (n int) {
+ if x >= 1<<16 {
+ x >>= 16
+ n = 16
+ }
+ if x >= 1<<8 {
+ x >>= 8
+ n += 8
+ }
+ return n + int(len8tab[x])
+}
+
+// Len64 returns the minimum number of bits required to represent x; the result is 0 for x == 0.
+func Len64(x uint64) (n int) {
+ if x >= 1<<32 {
+ x >>= 32
+ n = 32
+ }
+ if x >= 1<<16 {
+ x >>= 16
+ n += 16
+ }
+ if x >= 1<<8 {
+ x >>= 8
+ n += 8
+ }
+ return n + int(len8tab[x])
+}
+
+// --- Add with carry ---
+
+// Add returns the sum with carry of x, y and carry: sum = x + y + carry.
+// The carry input must be 0 or 1; otherwise the behavior is undefined.
+// The carryOut output is guaranteed to be 0 or 1.
+//
+// This function's execution time does not depend on the inputs.
+func Add(x, y, carry uint) (sum, carryOut uint) {
+ if UintSize == 32 {
+ s32, c32 := Add32(uint32(x), uint32(y), uint32(carry))
+ return uint(s32), uint(c32)
+ }
+ s64, c64 := Add64(uint64(x), uint64(y), uint64(carry))
+ return uint(s64), uint(c64)
+}
+
+// Add32 returns the sum with carry of x, y and carry: sum = x + y + carry.
+// The carry input must be 0 or 1; otherwise the behavior is undefined.
+// The carryOut output is guaranteed to be 0 or 1.
+//
+// This function's execution time does not depend on the inputs.
+func Add32(x, y, carry uint32) (sum, carryOut uint32) {
+ sum64 := uint64(x) + uint64(y) + uint64(carry)
+ sum = uint32(sum64)
+ carryOut = uint32(sum64 >> 32)
+ return
+}
+
+// Add64 returns the sum with carry of x, y and carry: sum = x + y + carry.
+// The carry input must be 0 or 1; otherwise the behavior is undefined.
+// The carryOut output is guaranteed to be 0 or 1.
+//
+// This function's execution time does not depend on the inputs.
+func Add64(x, y, carry uint64) (sum, carryOut uint64) {
+ sum = x + y + carry
+ // The sum will overflow if both top bits are set (x & y) or if one of them
+ // is (x | y), and a carry from the lower place happened. If such a carry
+ // happens, the top bit will be 1 + 0 + 1 = 0 (&^ sum).
+ carryOut = ((x & y) | ((x | y) &^ sum)) >> 63
+ return
+}
+
+// --- Subtract with borrow ---
+
+// Sub returns the difference of x, y and borrow: diff = x - y - borrow.
+// The borrow input must be 0 or 1; otherwise the behavior is undefined.
+// The borrowOut output is guaranteed to be 0 or 1.
+//
+// This function's execution time does not depend on the inputs.
+func Sub(x, y, borrow uint) (diff, borrowOut uint) {
+ if UintSize == 32 {
+ d32, b32 := Sub32(uint32(x), uint32(y), uint32(borrow))
+ return uint(d32), uint(b32)
+ }
+ d64, b64 := Sub64(uint64(x), uint64(y), uint64(borrow))
+ return uint(d64), uint(b64)
+}
+
+// Sub32 returns the difference of x, y and borrow, diff = x - y - borrow.
+// The borrow input must be 0 or 1; otherwise the behavior is undefined.
+// The borrowOut output is guaranteed to be 0 or 1.
+//
+// This function's execution time does not depend on the inputs.
+func Sub32(x, y, borrow uint32) (diff, borrowOut uint32) {
+ diff = x - y - borrow
+ // The difference will underflow if the top bit of x is not set and the top
+ // bit of y is set (^x & y) or if they are the same (^(x ^ y)) and a borrow
+ // from the lower place happens. If that borrow happens, the result will be
+ // 1 - 1 - 1 = 0 - 0 - 1 = 1 (& diff).
+ borrowOut = ((^x & y) | (^(x ^ y) & diff)) >> 31
+ return
+}
+
+// Sub64 returns the difference of x, y and borrow: diff = x - y - borrow.
+// The borrow input must be 0 or 1; otherwise the behavior is undefined.
+// The borrowOut output is guaranteed to be 0 or 1.
+//
+// This function's execution time does not depend on the inputs.
+func Sub64(x, y, borrow uint64) (diff, borrowOut uint64) {
+ diff = x - y - borrow
+ // See Sub32 for the bit logic.
+ borrowOut = ((^x & y) | (^(x ^ y) & diff)) >> 63
+ return
+}
+
+// --- Full-width multiply ---
+
+// Mul returns the full-width product of x and y: (hi, lo) = x * y
+// with the product bits' upper half returned in hi and the lower
+// half returned in lo.
+//
+// This function's execution time does not depend on the inputs.
+func Mul(x, y uint) (hi, lo uint) {
+ if UintSize == 32 {
+ h, l := Mul32(uint32(x), uint32(y))
+ return uint(h), uint(l)
+ }
+ h, l := Mul64(uint64(x), uint64(y))
+ return uint(h), uint(l)
+}
+
+// Mul32 returns the 64-bit product of x and y: (hi, lo) = x * y
+// with the product bits' upper half returned in hi and the lower
+// half returned in lo.
+//
+// This function's execution time does not depend on the inputs.
+func Mul32(x, y uint32) (hi, lo uint32) {
+ tmp := uint64(x) * uint64(y)
+ hi, lo = uint32(tmp>>32), uint32(tmp)
+ return
+}
+
+// Mul64 returns the 128-bit product of x and y: (hi, lo) = x * y
+// with the product bits' upper half returned in hi and the lower
+// half returned in lo.
+//
+// This function's execution time does not depend on the inputs.
+func Mul64(x, y uint64) (hi, lo uint64) {
+ const mask32 = 1<<32 - 1
+ x0 := x & mask32
+ x1 := x >> 32
+ y0 := y & mask32
+ y1 := y >> 32
+ w0 := x0 * y0
+ t := x1*y0 + w0>>32
+ w1 := t & mask32
+ w2 := t >> 32
+ w1 += x0 * y1
+ hi = x1*y1 + w2 + w1>>32
+ lo = x * y
+ return
+}
+
+// --- Full-width divide ---
+
+// Div returns the quotient and remainder of (hi, lo) divided by y:
+// quo = (hi, lo)/y, rem = (hi, lo)%y with the dividend bits' upper
+// half in parameter hi and the lower half in parameter lo.
+// Div panics for y == 0 (division by zero) or y <= hi (quotient overflow).
+func Div(hi, lo, y uint) (quo, rem uint) {
+ if UintSize == 32 {
+ q, r := Div32(uint32(hi), uint32(lo), uint32(y))
+ return uint(q), uint(r)
+ }
+ q, r := Div64(uint64(hi), uint64(lo), uint64(y))
+ return uint(q), uint(r)
+}
+
+// Div32 returns the quotient and remainder of (hi, lo) divided by y:
+// quo = (hi, lo)/y, rem = (hi, lo)%y with the dividend bits' upper
+// half in parameter hi and the lower half in parameter lo.
+// Div32 panics for y == 0 (division by zero) or y <= hi (quotient overflow).
+func Div32(hi, lo, y uint32) (quo, rem uint32) {
+ if y != 0 && y <= hi {
+ panic(overflowError)
+ }
+ z := uint64(hi)<<32 | uint64(lo)
+ quo, rem = uint32(z/uint64(y)), uint32(z%uint64(y))
+ return
+}
+
+// Div64 returns the quotient and remainder of (hi, lo) divided by y:
+// quo = (hi, lo)/y, rem = (hi, lo)%y with the dividend bits' upper
+// half in parameter hi and the lower half in parameter lo.
+// Div64 panics for y == 0 (division by zero) or y <= hi (quotient overflow).
+func Div64(hi, lo, y uint64) (quo, rem uint64) {
+ const (
+ two32 = 1 << 32
+ mask32 = two32 - 1
+ )
+ if y == 0 {
+ panic(divideError)
+ }
+ if y <= hi {
+ panic(overflowError)
+ }
+
+ s := uint(LeadingZeros64(y))
+ y <<= s
+
+ yn1 := y >> 32
+ yn0 := y & mask32
+ un32 := hi<<s | lo>>(64-s)
+ un10 := lo << s
+ un1 := un10 >> 32
+ un0 := un10 & mask32
+ q1 := un32 / yn1
+ rhat := un32 - q1*yn1
+
+ for q1 >= two32 || q1*yn0 > two32*rhat+un1 {
+ q1--
+ rhat += yn1
+ if rhat >= two32 {
+ break
+ }
+ }
+
+ un21 := un32*two32 + un1 - q1*y
+ q0 := un21 / yn1
+ rhat = un21 - q0*yn1
+
+ for q0 >= two32 || q0*yn0 > two32*rhat+un0 {
+ q0--
+ rhat += yn1
+ if rhat >= two32 {
+ break
+ }
+ }
+
+ return q1*two32 + q0, (un21*two32 + un0 - q0*y) >> s
+}
+
+// Rem returns the remainder of (hi, lo) divided by y. Rem panics for
+// y == 0 (division by zero) but, unlike Div, it doesn't panic on a
+// quotient overflow.
+func Rem(hi, lo, y uint) uint {
+ if UintSize == 32 {
+ return uint(Rem32(uint32(hi), uint32(lo), uint32(y)))
+ }
+ return uint(Rem64(uint64(hi), uint64(lo), uint64(y)))
+}
+
+// Rem32 returns the remainder of (hi, lo) divided by y. Rem32 panics
+// for y == 0 (division by zero) but, unlike Div32, it doesn't panic
+// on a quotient overflow.
+func Rem32(hi, lo, y uint32) uint32 {
+ return uint32((uint64(hi)<<32 | uint64(lo)) % uint64(y))
+}
+
+// Rem64 returns the remainder of (hi, lo) divided by y. Rem64 panics
+// for y == 0 (division by zero) but, unlike Div64, it doesn't panic
+// on a quotient overflow.
+func Rem64(hi, lo, y uint64) uint64 {
+ // We scale down hi so that hi < y, then use Div64 to compute the
+ // rem with the guarantee that it won't panic on quotient overflow.
+ // Given that
+ // hi ≡ hi%y (mod y)
+ // we have
+ // hi<<64 + lo ≡ (hi%y)<<64 + lo (mod y)
+ _, rem := Div64(hi%y, lo, y)
+ return rem
+}
diff --git a/src/math/bits/bits_errors.go b/src/math/bits/bits_errors.go
new file mode 100644
index 0000000..192b4be
--- /dev/null
+++ b/src/math/bits/bits_errors.go
@@ -0,0 +1,15 @@
+// Copyright 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build !compiler_bootstrap
+
+package bits
+
+import _ "unsafe"
+
+//go:linkname overflowError runtime.overflowError
+var overflowError error
+
+//go:linkname divideError runtime.divideError
+var divideError error
diff --git a/src/math/bits/bits_errors_bootstrap.go b/src/math/bits/bits_errors_bootstrap.go
new file mode 100644
index 0000000..5df5738
--- /dev/null
+++ b/src/math/bits/bits_errors_bootstrap.go
@@ -0,0 +1,22 @@
+// Copyright 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build compiler_bootstrap
+
+// This version used only for bootstrap (on this path we want
+// to avoid use of go:linkname as applied to variables).
+
+package bits
+
+type errorString string
+
+func (e errorString) RuntimeError() {}
+
+func (e errorString) Error() string {
+ return "runtime error: " + string(e)
+}
+
+var overflowError = error(errorString("integer overflow"))
+
+var divideError = error(errorString("integer divide by zero"))
diff --git a/src/math/bits/bits_tables.go b/src/math/bits/bits_tables.go
new file mode 100644
index 0000000..f1e15a0
--- /dev/null
+++ b/src/math/bits/bits_tables.go
@@ -0,0 +1,83 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Code generated by go run make_tables.go. DO NOT EDIT.
+
+package bits
+
+var ntz8tab = [256]uint8{
+ 0x08, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x05, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x06, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x05, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x07, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x05, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x06, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x05, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+ 0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
+}
+
+var pop8tab = [256]uint8{
+ 0x00, 0x01, 0x01, 0x02, 0x01, 0x02, 0x02, 0x03, 0x01, 0x02, 0x02, 0x03, 0x02, 0x03, 0x03, 0x04,
+ 0x01, 0x02, 0x02, 0x03, 0x02, 0x03, 0x03, 0x04, 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05,
+ 0x01, 0x02, 0x02, 0x03, 0x02, 0x03, 0x03, 0x04, 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05,
+ 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
+ 0x01, 0x02, 0x02, 0x03, 0x02, 0x03, 0x03, 0x04, 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05,
+ 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
+ 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
+ 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06, 0x04, 0x05, 0x05, 0x06, 0x05, 0x06, 0x06, 0x07,
+ 0x01, 0x02, 0x02, 0x03, 0x02, 0x03, 0x03, 0x04, 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05,
+ 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
+ 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
+ 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06, 0x04, 0x05, 0x05, 0x06, 0x05, 0x06, 0x06, 0x07,
+ 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
+ 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06, 0x04, 0x05, 0x05, 0x06, 0x05, 0x06, 0x06, 0x07,
+ 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06, 0x04, 0x05, 0x05, 0x06, 0x05, 0x06, 0x06, 0x07,
+ 0x04, 0x05, 0x05, 0x06, 0x05, 0x06, 0x06, 0x07, 0x05, 0x06, 0x06, 0x07, 0x06, 0x07, 0x07, 0x08,
+}
+
+var rev8tab = [256]uint8{
+ 0x00, 0x80, 0x40, 0xc0, 0x20, 0xa0, 0x60, 0xe0, 0x10, 0x90, 0x50, 0xd0, 0x30, 0xb0, 0x70, 0xf0,
+ 0x08, 0x88, 0x48, 0xc8, 0x28, 0xa8, 0x68, 0xe8, 0x18, 0x98, 0x58, 0xd8, 0x38, 0xb8, 0x78, 0xf8,
+ 0x04, 0x84, 0x44, 0xc4, 0x24, 0xa4, 0x64, 0xe4, 0x14, 0x94, 0x54, 0xd4, 0x34, 0xb4, 0x74, 0xf4,
+ 0x0c, 0x8c, 0x4c, 0xcc, 0x2c, 0xac, 0x6c, 0xec, 0x1c, 0x9c, 0x5c, 0xdc, 0x3c, 0xbc, 0x7c, 0xfc,
+ 0x02, 0x82, 0x42, 0xc2, 0x22, 0xa2, 0x62, 0xe2, 0x12, 0x92, 0x52, 0xd2, 0x32, 0xb2, 0x72, 0xf2,
+ 0x0a, 0x8a, 0x4a, 0xca, 0x2a, 0xaa, 0x6a, 0xea, 0x1a, 0x9a, 0x5a, 0xda, 0x3a, 0xba, 0x7a, 0xfa,
+ 0x06, 0x86, 0x46, 0xc6, 0x26, 0xa6, 0x66, 0xe6, 0x16, 0x96, 0x56, 0xd6, 0x36, 0xb6, 0x76, 0xf6,
+ 0x0e, 0x8e, 0x4e, 0xce, 0x2e, 0xae, 0x6e, 0xee, 0x1e, 0x9e, 0x5e, 0xde, 0x3e, 0xbe, 0x7e, 0xfe,
+ 0x01, 0x81, 0x41, 0xc1, 0x21, 0xa1, 0x61, 0xe1, 0x11, 0x91, 0x51, 0xd1, 0x31, 0xb1, 0x71, 0xf1,
+ 0x09, 0x89, 0x49, 0xc9, 0x29, 0xa9, 0x69, 0xe9, 0x19, 0x99, 0x59, 0xd9, 0x39, 0xb9, 0x79, 0xf9,
+ 0x05, 0x85, 0x45, 0xc5, 0x25, 0xa5, 0x65, 0xe5, 0x15, 0x95, 0x55, 0xd5, 0x35, 0xb5, 0x75, 0xf5,
+ 0x0d, 0x8d, 0x4d, 0xcd, 0x2d, 0xad, 0x6d, 0xed, 0x1d, 0x9d, 0x5d, 0xdd, 0x3d, 0xbd, 0x7d, 0xfd,
+ 0x03, 0x83, 0x43, 0xc3, 0x23, 0xa3, 0x63, 0xe3, 0x13, 0x93, 0x53, 0xd3, 0x33, 0xb3, 0x73, 0xf3,
+ 0x0b, 0x8b, 0x4b, 0xcb, 0x2b, 0xab, 0x6b, 0xeb, 0x1b, 0x9b, 0x5b, 0xdb, 0x3b, 0xbb, 0x7b, 0xfb,
+ 0x07, 0x87, 0x47, 0xc7, 0x27, 0xa7, 0x67, 0xe7, 0x17, 0x97, 0x57, 0xd7, 0x37, 0xb7, 0x77, 0xf7,
+ 0x0f, 0x8f, 0x4f, 0xcf, 0x2f, 0xaf, 0x6f, 0xef, 0x1f, 0x9f, 0x5f, 0xdf, 0x3f, 0xbf, 0x7f, 0xff,
+}
+
+var len8tab = [256]uint8{
+ 0x00, 0x01, 0x02, 0x02, 0x03, 0x03, 0x03, 0x03, 0x04, 0x04, 0x04, 0x04, 0x04, 0x04, 0x04, 0x04,
+ 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05,
+ 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06,
+ 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06,
+ 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07,
+ 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07,
+ 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07,
+ 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07,
+ 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
+ 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
+ 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
+ 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
+ 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
+ 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
+ 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
+ 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
+}
diff --git a/src/math/bits/bits_test.go b/src/math/bits/bits_test.go
new file mode 100644
index 0000000..23b4539
--- /dev/null
+++ b/src/math/bits/bits_test.go
@@ -0,0 +1,1347 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package bits_test
+
+import (
+ . "math/bits"
+ "runtime"
+ "testing"
+ "unsafe"
+)
+
+func TestUintSize(t *testing.T) {
+ var x uint
+ if want := unsafe.Sizeof(x) * 8; UintSize != want {
+ t.Fatalf("UintSize = %d; want %d", UintSize, want)
+ }
+}
+
+func TestLeadingZeros(t *testing.T) {
+ for i := 0; i < 256; i++ {
+ nlz := tab[i].nlz
+ for k := 0; k < 64-8; k++ {
+ x := uint64(i) << uint(k)
+ if x <= 1<<8-1 {
+ got := LeadingZeros8(uint8(x))
+ want := nlz - k + (8 - 8)
+ if x == 0 {
+ want = 8
+ }
+ if got != want {
+ t.Fatalf("LeadingZeros8(%#02x) == %d; want %d", x, got, want)
+ }
+ }
+
+ if x <= 1<<16-1 {
+ got := LeadingZeros16(uint16(x))
+ want := nlz - k + (16 - 8)
+ if x == 0 {
+ want = 16
+ }
+ if got != want {
+ t.Fatalf("LeadingZeros16(%#04x) == %d; want %d", x, got, want)
+ }
+ }
+
+ if x <= 1<<32-1 {
+ got := LeadingZeros32(uint32(x))
+ want := nlz - k + (32 - 8)
+ if x == 0 {
+ want = 32
+ }
+ if got != want {
+ t.Fatalf("LeadingZeros32(%#08x) == %d; want %d", x, got, want)
+ }
+ if UintSize == 32 {
+ got = LeadingZeros(uint(x))
+ if got != want {
+ t.Fatalf("LeadingZeros(%#08x) == %d; want %d", x, got, want)
+ }
+ }
+ }
+
+ if x <= 1<<64-1 {
+ got := LeadingZeros64(uint64(x))
+ want := nlz - k + (64 - 8)
+ if x == 0 {
+ want = 64
+ }
+ if got != want {
+ t.Fatalf("LeadingZeros64(%#016x) == %d; want %d", x, got, want)
+ }
+ if UintSize == 64 {
+ got = LeadingZeros(uint(x))
+ if got != want {
+ t.Fatalf("LeadingZeros(%#016x) == %d; want %d", x, got, want)
+ }
+ }
+ }
+ }
+ }
+}
+
+// Exported (global) variable serving as input for some
+// of the benchmarks to ensure side-effect free calls
+// are not optimized away.
+var Input uint64 = DeBruijn64
+
+// Exported (global) variable to store function results
+// during benchmarking to ensure side-effect free calls
+// are not optimized away.
+var Output int
+
+func BenchmarkLeadingZeros(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += LeadingZeros(uint(Input) >> (uint(i) % UintSize))
+ }
+ Output = s
+}
+
+func BenchmarkLeadingZeros8(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += LeadingZeros8(uint8(Input) >> (uint(i) % 8))
+ }
+ Output = s
+}
+
+func BenchmarkLeadingZeros16(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += LeadingZeros16(uint16(Input) >> (uint(i) % 16))
+ }
+ Output = s
+}
+
+func BenchmarkLeadingZeros32(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += LeadingZeros32(uint32(Input) >> (uint(i) % 32))
+ }
+ Output = s
+}
+
+func BenchmarkLeadingZeros64(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += LeadingZeros64(uint64(Input) >> (uint(i) % 64))
+ }
+ Output = s
+}
+
+func TestTrailingZeros(t *testing.T) {
+ for i := 0; i < 256; i++ {
+ ntz := tab[i].ntz
+ for k := 0; k < 64-8; k++ {
+ x := uint64(i) << uint(k)
+ want := ntz + k
+ if x <= 1<<8-1 {
+ got := TrailingZeros8(uint8(x))
+ if x == 0 {
+ want = 8
+ }
+ if got != want {
+ t.Fatalf("TrailingZeros8(%#02x) == %d; want %d", x, got, want)
+ }
+ }
+
+ if x <= 1<<16-1 {
+ got := TrailingZeros16(uint16(x))
+ if x == 0 {
+ want = 16
+ }
+ if got != want {
+ t.Fatalf("TrailingZeros16(%#04x) == %d; want %d", x, got, want)
+ }
+ }
+
+ if x <= 1<<32-1 {
+ got := TrailingZeros32(uint32(x))
+ if x == 0 {
+ want = 32
+ }
+ if got != want {
+ t.Fatalf("TrailingZeros32(%#08x) == %d; want %d", x, got, want)
+ }
+ if UintSize == 32 {
+ got = TrailingZeros(uint(x))
+ if got != want {
+ t.Fatalf("TrailingZeros(%#08x) == %d; want %d", x, got, want)
+ }
+ }
+ }
+
+ if x <= 1<<64-1 {
+ got := TrailingZeros64(uint64(x))
+ if x == 0 {
+ want = 64
+ }
+ if got != want {
+ t.Fatalf("TrailingZeros64(%#016x) == %d; want %d", x, got, want)
+ }
+ if UintSize == 64 {
+ got = TrailingZeros(uint(x))
+ if got != want {
+ t.Fatalf("TrailingZeros(%#016x) == %d; want %d", x, got, want)
+ }
+ }
+ }
+ }
+ }
+}
+
+func BenchmarkTrailingZeros(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += TrailingZeros(uint(Input) << (uint(i) % UintSize))
+ }
+ Output = s
+}
+
+func BenchmarkTrailingZeros8(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += TrailingZeros8(uint8(Input) << (uint(i) % 8))
+ }
+ Output = s
+}
+
+func BenchmarkTrailingZeros16(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += TrailingZeros16(uint16(Input) << (uint(i) % 16))
+ }
+ Output = s
+}
+
+func BenchmarkTrailingZeros32(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += TrailingZeros32(uint32(Input) << (uint(i) % 32))
+ }
+ Output = s
+}
+
+func BenchmarkTrailingZeros64(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += TrailingZeros64(uint64(Input) << (uint(i) % 64))
+ }
+ Output = s
+}
+
+func TestOnesCount(t *testing.T) {
+ var x uint64
+ for i := 0; i <= 64; i++ {
+ testOnesCount(t, x, i)
+ x = x<<1 | 1
+ }
+
+ for i := 64; i >= 0; i-- {
+ testOnesCount(t, x, i)
+ x = x << 1
+ }
+
+ for i := 0; i < 256; i++ {
+ for k := 0; k < 64-8; k++ {
+ testOnesCount(t, uint64(i)<<uint(k), tab[i].pop)
+ }
+ }
+}
+
+func testOnesCount(t *testing.T, x uint64, want int) {
+ if x <= 1<<8-1 {
+ got := OnesCount8(uint8(x))
+ if got != want {
+ t.Fatalf("OnesCount8(%#02x) == %d; want %d", uint8(x), got, want)
+ }
+ }
+
+ if x <= 1<<16-1 {
+ got := OnesCount16(uint16(x))
+ if got != want {
+ t.Fatalf("OnesCount16(%#04x) == %d; want %d", uint16(x), got, want)
+ }
+ }
+
+ if x <= 1<<32-1 {
+ got := OnesCount32(uint32(x))
+ if got != want {
+ t.Fatalf("OnesCount32(%#08x) == %d; want %d", uint32(x), got, want)
+ }
+ if UintSize == 32 {
+ got = OnesCount(uint(x))
+ if got != want {
+ t.Fatalf("OnesCount(%#08x) == %d; want %d", uint32(x), got, want)
+ }
+ }
+ }
+
+ if x <= 1<<64-1 {
+ got := OnesCount64(uint64(x))
+ if got != want {
+ t.Fatalf("OnesCount64(%#016x) == %d; want %d", x, got, want)
+ }
+ if UintSize == 64 {
+ got = OnesCount(uint(x))
+ if got != want {
+ t.Fatalf("OnesCount(%#016x) == %d; want %d", x, got, want)
+ }
+ }
+ }
+}
+
+func BenchmarkOnesCount(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += OnesCount(uint(Input))
+ }
+ Output = s
+}
+
+func BenchmarkOnesCount8(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += OnesCount8(uint8(Input))
+ }
+ Output = s
+}
+
+func BenchmarkOnesCount16(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += OnesCount16(uint16(Input))
+ }
+ Output = s
+}
+
+func BenchmarkOnesCount32(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += OnesCount32(uint32(Input))
+ }
+ Output = s
+}
+
+func BenchmarkOnesCount64(b *testing.B) {
+ var s int
+ for i := 0; i < b.N; i++ {
+ s += OnesCount64(uint64(Input))
+ }
+ Output = s
+}
+
+func TestRotateLeft(t *testing.T) {
+ var m uint64 = DeBruijn64
+
+ for k := uint(0); k < 128; k++ {
+ x8 := uint8(m)
+ got8 := RotateLeft8(x8, int(k))
+ want8 := x8<<(k&0x7) | x8>>(8-k&0x7)
+ if got8 != want8 {
+ t.Fatalf("RotateLeft8(%#02x, %d) == %#02x; want %#02x", x8, k, got8, want8)
+ }
+ got8 = RotateLeft8(want8, -int(k))
+ if got8 != x8 {
+ t.Fatalf("RotateLeft8(%#02x, -%d) == %#02x; want %#02x", want8, k, got8, x8)
+ }
+
+ x16 := uint16(m)
+ got16 := RotateLeft16(x16, int(k))
+ want16 := x16<<(k&0xf) | x16>>(16-k&0xf)
+ if got16 != want16 {
+ t.Fatalf("RotateLeft16(%#04x, %d) == %#04x; want %#04x", x16, k, got16, want16)
+ }
+ got16 = RotateLeft16(want16, -int(k))
+ if got16 != x16 {
+ t.Fatalf("RotateLeft16(%#04x, -%d) == %#04x; want %#04x", want16, k, got16, x16)
+ }
+
+ x32 := uint32(m)
+ got32 := RotateLeft32(x32, int(k))
+ want32 := x32<<(k&0x1f) | x32>>(32-k&0x1f)
+ if got32 != want32 {
+ t.Fatalf("RotateLeft32(%#08x, %d) == %#08x; want %#08x", x32, k, got32, want32)
+ }
+ got32 = RotateLeft32(want32, -int(k))
+ if got32 != x32 {
+ t.Fatalf("RotateLeft32(%#08x, -%d) == %#08x; want %#08x", want32, k, got32, x32)
+ }
+ if UintSize == 32 {
+ x := uint(m)
+ got := RotateLeft(x, int(k))
+ want := x<<(k&0x1f) | x>>(32-k&0x1f)
+ if got != want {
+ t.Fatalf("RotateLeft(%#08x, %d) == %#08x; want %#08x", x, k, got, want)
+ }
+ got = RotateLeft(want, -int(k))
+ if got != x {
+ t.Fatalf("RotateLeft(%#08x, -%d) == %#08x; want %#08x", want, k, got, x)
+ }
+ }
+
+ x64 := uint64(m)
+ got64 := RotateLeft64(x64, int(k))
+ want64 := x64<<(k&0x3f) | x64>>(64-k&0x3f)
+ if got64 != want64 {
+ t.Fatalf("RotateLeft64(%#016x, %d) == %#016x; want %#016x", x64, k, got64, want64)
+ }
+ got64 = RotateLeft64(want64, -int(k))
+ if got64 != x64 {
+ t.Fatalf("RotateLeft64(%#016x, -%d) == %#016x; want %#016x", want64, k, got64, x64)
+ }
+ if UintSize == 64 {
+ x := uint(m)
+ got := RotateLeft(x, int(k))
+ want := x<<(k&0x3f) | x>>(64-k&0x3f)
+ if got != want {
+ t.Fatalf("RotateLeft(%#016x, %d) == %#016x; want %#016x", x, k, got, want)
+ }
+ got = RotateLeft(want, -int(k))
+ if got != x {
+ t.Fatalf("RotateLeft(%#08x, -%d) == %#08x; want %#08x", want, k, got, x)
+ }
+ }
+ }
+}
+
+func BenchmarkRotateLeft(b *testing.B) {
+ var s uint
+ for i := 0; i < b.N; i++ {
+ s += RotateLeft(uint(Input), i)
+ }
+ Output = int(s)
+}
+
+func BenchmarkRotateLeft8(b *testing.B) {
+ var s uint8
+ for i := 0; i < b.N; i++ {
+ s += RotateLeft8(uint8(Input), i)
+ }
+ Output = int(s)
+}
+
+func BenchmarkRotateLeft16(b *testing.B) {
+ var s uint16
+ for i := 0; i < b.N; i++ {
+ s += RotateLeft16(uint16(Input), i)
+ }
+ Output = int(s)
+}
+
+func BenchmarkRotateLeft32(b *testing.B) {
+ var s uint32
+ for i := 0; i < b.N; i++ {
+ s += RotateLeft32(uint32(Input), i)
+ }
+ Output = int(s)
+}
+
+func BenchmarkRotateLeft64(b *testing.B) {
+ var s uint64
+ for i := 0; i < b.N; i++ {
+ s += RotateLeft64(uint64(Input), i)
+ }
+ Output = int(s)
+}
+
+func TestReverse(t *testing.T) {
+ // test each bit
+ for i := uint(0); i < 64; i++ {
+ testReverse(t, uint64(1)<<i, uint64(1)<<(63-i))
+ }
+
+ // test a few patterns
+ for _, test := range []struct {
+ x, r uint64
+ }{
+ {0, 0},
+ {0x1, 0x8 << 60},
+ {0x2, 0x4 << 60},
+ {0x3, 0xc << 60},
+ {0x4, 0x2 << 60},
+ {0x5, 0xa << 60},
+ {0x6, 0x6 << 60},
+ {0x7, 0xe << 60},
+ {0x8, 0x1 << 60},
+ {0x9, 0x9 << 60},
+ {0xa, 0x5 << 60},
+ {0xb, 0xd << 60},
+ {0xc, 0x3 << 60},
+ {0xd, 0xb << 60},
+ {0xe, 0x7 << 60},
+ {0xf, 0xf << 60},
+ {0x5686487, 0xe12616a000000000},
+ {0x0123456789abcdef, 0xf7b3d591e6a2c480},
+ } {
+ testReverse(t, test.x, test.r)
+ testReverse(t, test.r, test.x)
+ }
+}
+
+func testReverse(t *testing.T, x64, want64 uint64) {
+ x8 := uint8(x64)
+ got8 := Reverse8(x8)
+ want8 := uint8(want64 >> (64 - 8))
+ if got8 != want8 {
+ t.Fatalf("Reverse8(%#02x) == %#02x; want %#02x", x8, got8, want8)
+ }
+
+ x16 := uint16(x64)
+ got16 := Reverse16(x16)
+ want16 := uint16(want64 >> (64 - 16))
+ if got16 != want16 {
+ t.Fatalf("Reverse16(%#04x) == %#04x; want %#04x", x16, got16, want16)
+ }
+
+ x32 := uint32(x64)
+ got32 := Reverse32(x32)
+ want32 := uint32(want64 >> (64 - 32))
+ if got32 != want32 {
+ t.Fatalf("Reverse32(%#08x) == %#08x; want %#08x", x32, got32, want32)
+ }
+ if UintSize == 32 {
+ x := uint(x32)
+ got := Reverse(x)
+ want := uint(want32)
+ if got != want {
+ t.Fatalf("Reverse(%#08x) == %#08x; want %#08x", x, got, want)
+ }
+ }
+
+ got64 := Reverse64(x64)
+ if got64 != want64 {
+ t.Fatalf("Reverse64(%#016x) == %#016x; want %#016x", x64, got64, want64)
+ }
+ if UintSize == 64 {
+ x := uint(x64)
+ got := Reverse(x)
+ want := uint(want64)
+ if got != want {
+ t.Fatalf("Reverse(%#08x) == %#016x; want %#016x", x, got, want)
+ }
+ }
+}
+
+func BenchmarkReverse(b *testing.B) {
+ var s uint
+ for i := 0; i < b.N; i++ {
+ s += Reverse(uint(i))
+ }
+ Output = int(s)
+}
+
+func BenchmarkReverse8(b *testing.B) {
+ var s uint8
+ for i := 0; i < b.N; i++ {
+ s += Reverse8(uint8(i))
+ }
+ Output = int(s)
+}
+
+func BenchmarkReverse16(b *testing.B) {
+ var s uint16
+ for i := 0; i < b.N; i++ {
+ s += Reverse16(uint16(i))
+ }
+ Output = int(s)
+}
+
+func BenchmarkReverse32(b *testing.B) {
+ var s uint32
+ for i := 0; i < b.N; i++ {
+ s += Reverse32(uint32(i))
+ }
+ Output = int(s)
+}
+
+func BenchmarkReverse64(b *testing.B) {
+ var s uint64
+ for i := 0; i < b.N; i++ {
+ s += Reverse64(uint64(i))
+ }
+ Output = int(s)
+}
+
+func TestReverseBytes(t *testing.T) {
+ for _, test := range []struct {
+ x, r uint64
+ }{
+ {0, 0},
+ {0x01, 0x01 << 56},
+ {0x0123, 0x2301 << 48},
+ {0x012345, 0x452301 << 40},
+ {0x01234567, 0x67452301 << 32},
+ {0x0123456789, 0x8967452301 << 24},
+ {0x0123456789ab, 0xab8967452301 << 16},
+ {0x0123456789abcd, 0xcdab8967452301 << 8},
+ {0x0123456789abcdef, 0xefcdab8967452301 << 0},
+ } {
+ testReverseBytes(t, test.x, test.r)
+ testReverseBytes(t, test.r, test.x)
+ }
+}
+
+func testReverseBytes(t *testing.T, x64, want64 uint64) {
+ x16 := uint16(x64)
+ got16 := ReverseBytes16(x16)
+ want16 := uint16(want64 >> (64 - 16))
+ if got16 != want16 {
+ t.Fatalf("ReverseBytes16(%#04x) == %#04x; want %#04x", x16, got16, want16)
+ }
+
+ x32 := uint32(x64)
+ got32 := ReverseBytes32(x32)
+ want32 := uint32(want64 >> (64 - 32))
+ if got32 != want32 {
+ t.Fatalf("ReverseBytes32(%#08x) == %#08x; want %#08x", x32, got32, want32)
+ }
+ if UintSize == 32 {
+ x := uint(x32)
+ got := ReverseBytes(x)
+ want := uint(want32)
+ if got != want {
+ t.Fatalf("ReverseBytes(%#08x) == %#08x; want %#08x", x, got, want)
+ }
+ }
+
+ got64 := ReverseBytes64(x64)
+ if got64 != want64 {
+ t.Fatalf("ReverseBytes64(%#016x) == %#016x; want %#016x", x64, got64, want64)
+ }
+ if UintSize == 64 {
+ x := uint(x64)
+ got := ReverseBytes(x)
+ want := uint(want64)
+ if got != want {
+ t.Fatalf("ReverseBytes(%#016x) == %#016x; want %#016x", x, got, want)
+ }
+ }
+}
+
+func BenchmarkReverseBytes(b *testing.B) {
+ var s uint
+ for i := 0; i < b.N; i++ {
+ s += ReverseBytes(uint(i))
+ }
+ Output = int(s)
+}
+
+func BenchmarkReverseBytes16(b *testing.B) {
+ var s uint16
+ for i := 0; i < b.N; i++ {
+ s += ReverseBytes16(uint16(i))
+ }
+ Output = int(s)
+}
+
+func BenchmarkReverseBytes32(b *testing.B) {
+ var s uint32
+ for i := 0; i < b.N; i++ {
+ s += ReverseBytes32(uint32(i))
+ }
+ Output = int(s)
+}
+
+func BenchmarkReverseBytes64(b *testing.B) {
+ var s uint64
+ for i := 0; i < b.N; i++ {
+ s += ReverseBytes64(uint64(i))
+ }
+ Output = int(s)
+}
+
+func TestLen(t *testing.T) {
+ for i := 0; i < 256; i++ {
+ len := 8 - tab[i].nlz
+ for k := 0; k < 64-8; k++ {
+ x := uint64(i) << uint(k)
+ want := 0
+ if x != 0 {
+ want = len + k
+ }
+ if x <= 1<<8-1 {
+ got := Len8(uint8(x))
+ if got != want {
+ t.Fatalf("Len8(%#02x) == %d; want %d", x, got, want)
+ }
+ }
+
+ if x <= 1<<16-1 {
+ got := Len16(uint16(x))
+ if got != want {
+ t.Fatalf("Len16(%#04x) == %d; want %d", x, got, want)
+ }
+ }
+
+ if x <= 1<<32-1 {
+ got := Len32(uint32(x))
+ if got != want {
+ t.Fatalf("Len32(%#08x) == %d; want %d", x, got, want)
+ }
+ if UintSize == 32 {
+ got := Len(uint(x))
+ if got != want {
+ t.Fatalf("Len(%#08x) == %d; want %d", x, got, want)
+ }
+ }
+ }
+
+ if x <= 1<<64-1 {
+ got := Len64(uint64(x))
+ if got != want {
+ t.Fatalf("Len64(%#016x) == %d; want %d", x, got, want)
+ }
+ if UintSize == 64 {
+ got := Len(uint(x))
+ if got != want {
+ t.Fatalf("Len(%#016x) == %d; want %d", x, got, want)
+ }
+ }
+ }
+ }
+ }
+}
+
+const (
+ _M = 1<<UintSize - 1
+ _M32 = 1<<32 - 1
+ _M64 = 1<<64 - 1
+)
+
+func TestAddSubUint(t *testing.T) {
+ test := func(msg string, f func(x, y, c uint) (z, cout uint), x, y, c, z, cout uint) {
+ z1, cout1 := f(x, y, c)
+ if z1 != z || cout1 != cout {
+ t.Errorf("%s: got z:cout = %#x:%#x; want %#x:%#x", msg, z1, cout1, z, cout)
+ }
+ }
+ for _, a := range []struct{ x, y, c, z, cout uint }{
+ {0, 0, 0, 0, 0},
+ {0, 1, 0, 1, 0},
+ {0, 0, 1, 1, 0},
+ {0, 1, 1, 2, 0},
+ {12345, 67890, 0, 80235, 0},
+ {12345, 67890, 1, 80236, 0},
+ {_M, 1, 0, 0, 1},
+ {_M, 0, 1, 0, 1},
+ {_M, 1, 1, 1, 1},
+ {_M, _M, 0, _M - 1, 1},
+ {_M, _M, 1, _M, 1},
+ } {
+ test("Add", Add, a.x, a.y, a.c, a.z, a.cout)
+ test("Add symmetric", Add, a.y, a.x, a.c, a.z, a.cout)
+ test("Sub", Sub, a.z, a.x, a.c, a.y, a.cout)
+ test("Sub symmetric", Sub, a.z, a.y, a.c, a.x, a.cout)
+ // The above code can't test intrinsic implementation, because the passed function is not called directly.
+ // The following code uses a closure to test the intrinsic version in case the function is intrinsified.
+ test("Add intrinsic", func(x, y, c uint) (uint, uint) { return Add(x, y, c) }, a.x, a.y, a.c, a.z, a.cout)
+ test("Add intrinsic symmetric", func(x, y, c uint) (uint, uint) { return Add(x, y, c) }, a.y, a.x, a.c, a.z, a.cout)
+ test("Sub intrinsic", func(x, y, c uint) (uint, uint) { return Sub(x, y, c) }, a.z, a.x, a.c, a.y, a.cout)
+ test("Sub intrinsic symmetric", func(x, y, c uint) (uint, uint) { return Sub(x, y, c) }, a.z, a.y, a.c, a.x, a.cout)
+
+ }
+}
+
+func TestAddSubUint32(t *testing.T) {
+ test := func(msg string, f func(x, y, c uint32) (z, cout uint32), x, y, c, z, cout uint32) {
+ z1, cout1 := f(x, y, c)
+ if z1 != z || cout1 != cout {
+ t.Errorf("%s: got z:cout = %#x:%#x; want %#x:%#x", msg, z1, cout1, z, cout)
+ }
+ }
+ for _, a := range []struct{ x, y, c, z, cout uint32 }{
+ {0, 0, 0, 0, 0},
+ {0, 1, 0, 1, 0},
+ {0, 0, 1, 1, 0},
+ {0, 1, 1, 2, 0},
+ {12345, 67890, 0, 80235, 0},
+ {12345, 67890, 1, 80236, 0},
+ {_M32, 1, 0, 0, 1},
+ {_M32, 0, 1, 0, 1},
+ {_M32, 1, 1, 1, 1},
+ {_M32, _M32, 0, _M32 - 1, 1},
+ {_M32, _M32, 1, _M32, 1},
+ } {
+ test("Add32", Add32, a.x, a.y, a.c, a.z, a.cout)
+ test("Add32 symmetric", Add32, a.y, a.x, a.c, a.z, a.cout)
+ test("Sub32", Sub32, a.z, a.x, a.c, a.y, a.cout)
+ test("Sub32 symmetric", Sub32, a.z, a.y, a.c, a.x, a.cout)
+ }
+}
+
+func TestAddSubUint64(t *testing.T) {
+ test := func(msg string, f func(x, y, c uint64) (z, cout uint64), x, y, c, z, cout uint64) {
+ z1, cout1 := f(x, y, c)
+ if z1 != z || cout1 != cout {
+ t.Errorf("%s: got z:cout = %#x:%#x; want %#x:%#x", msg, z1, cout1, z, cout)
+ }
+ }
+ for _, a := range []struct{ x, y, c, z, cout uint64 }{
+ {0, 0, 0, 0, 0},
+ {0, 1, 0, 1, 0},
+ {0, 0, 1, 1, 0},
+ {0, 1, 1, 2, 0},
+ {12345, 67890, 0, 80235, 0},
+ {12345, 67890, 1, 80236, 0},
+ {_M64, 1, 0, 0, 1},
+ {_M64, 0, 1, 0, 1},
+ {_M64, 1, 1, 1, 1},
+ {_M64, _M64, 0, _M64 - 1, 1},
+ {_M64, _M64, 1, _M64, 1},
+ } {
+ test("Add64", Add64, a.x, a.y, a.c, a.z, a.cout)
+ test("Add64 symmetric", Add64, a.y, a.x, a.c, a.z, a.cout)
+ test("Sub64", Sub64, a.z, a.x, a.c, a.y, a.cout)
+ test("Sub64 symmetric", Sub64, a.z, a.y, a.c, a.x, a.cout)
+ // The above code can't test intrinsic implementation, because the passed function is not called directly.
+ // The following code uses a closure to test the intrinsic version in case the function is intrinsified.
+ test("Add64 intrinsic", func(x, y, c uint64) (uint64, uint64) { return Add64(x, y, c) }, a.x, a.y, a.c, a.z, a.cout)
+ test("Add64 intrinsic symmetric", func(x, y, c uint64) (uint64, uint64) { return Add64(x, y, c) }, a.y, a.x, a.c, a.z, a.cout)
+ test("Sub64 intrinsic", func(x, y, c uint64) (uint64, uint64) { return Sub64(x, y, c) }, a.z, a.x, a.c, a.y, a.cout)
+ test("Sub64 intrinsic symmetric", func(x, y, c uint64) (uint64, uint64) { return Sub64(x, y, c) }, a.z, a.y, a.c, a.x, a.cout)
+ }
+}
+
+func TestAdd64OverflowPanic(t *testing.T) {
+ // Test that 64-bit overflow panics fire correctly.
+ // These are designed to improve coverage of compiler intrinsics.
+ tests := []func(uint64, uint64) uint64{
+ func(a, b uint64) uint64 {
+ x, c := Add64(a, b, 0)
+ if c > 0 {
+ panic("overflow")
+ }
+ return x
+ },
+ func(a, b uint64) uint64 {
+ x, c := Add64(a, b, 0)
+ if c != 0 {
+ panic("overflow")
+ }
+ return x
+ },
+ func(a, b uint64) uint64 {
+ x, c := Add64(a, b, 0)
+ if c == 1 {
+ panic("overflow")
+ }
+ return x
+ },
+ func(a, b uint64) uint64 {
+ x, c := Add64(a, b, 0)
+ if c != 1 {
+ return x
+ }
+ panic("overflow")
+ },
+ func(a, b uint64) uint64 {
+ x, c := Add64(a, b, 0)
+ if c == 0 {
+ return x
+ }
+ panic("overflow")
+ },
+ }
+ for _, test := range tests {
+ shouldPanic := func(f func()) {
+ defer func() {
+ if err := recover(); err == nil {
+ t.Fatalf("expected panic")
+ }
+ }()
+ f()
+ }
+
+ // overflow
+ shouldPanic(func() { test(_M64, 1) })
+ shouldPanic(func() { test(1, _M64) })
+ shouldPanic(func() { test(_M64, _M64) })
+
+ // no overflow
+ test(_M64, 0)
+ test(0, 0)
+ test(1, 1)
+ }
+}
+
+func TestSub64OverflowPanic(t *testing.T) {
+ // Test that 64-bit overflow panics fire correctly.
+ // These are designed to improve coverage of compiler intrinsics.
+ tests := []func(uint64, uint64) uint64{
+ func(a, b uint64) uint64 {
+ x, c := Sub64(a, b, 0)
+ if c > 0 {
+ panic("overflow")
+ }
+ return x
+ },
+ func(a, b uint64) uint64 {
+ x, c := Sub64(a, b, 0)
+ if c != 0 {
+ panic("overflow")
+ }
+ return x
+ },
+ func(a, b uint64) uint64 {
+ x, c := Sub64(a, b, 0)
+ if c == 1 {
+ panic("overflow")
+ }
+ return x
+ },
+ func(a, b uint64) uint64 {
+ x, c := Sub64(a, b, 0)
+ if c != 1 {
+ return x
+ }
+ panic("overflow")
+ },
+ func(a, b uint64) uint64 {
+ x, c := Sub64(a, b, 0)
+ if c == 0 {
+ return x
+ }
+ panic("overflow")
+ },
+ }
+ for _, test := range tests {
+ shouldPanic := func(f func()) {
+ defer func() {
+ if err := recover(); err == nil {
+ t.Fatalf("expected panic")
+ }
+ }()
+ f()
+ }
+
+ // overflow
+ shouldPanic(func() { test(0, 1) })
+ shouldPanic(func() { test(1, _M64) })
+ shouldPanic(func() { test(_M64-1, _M64) })
+
+ // no overflow
+ test(_M64, 0)
+ test(0, 0)
+ test(1, 1)
+ }
+}
+
+func TestMulDiv(t *testing.T) {
+ testMul := func(msg string, f func(x, y uint) (hi, lo uint), x, y, hi, lo uint) {
+ hi1, lo1 := f(x, y)
+ if hi1 != hi || lo1 != lo {
+ t.Errorf("%s: got hi:lo = %#x:%#x; want %#x:%#x", msg, hi1, lo1, hi, lo)
+ }
+ }
+ testDiv := func(msg string, f func(hi, lo, y uint) (q, r uint), hi, lo, y, q, r uint) {
+ q1, r1 := f(hi, lo, y)
+ if q1 != q || r1 != r {
+ t.Errorf("%s: got q:r = %#x:%#x; want %#x:%#x", msg, q1, r1, q, r)
+ }
+ }
+ for _, a := range []struct {
+ x, y uint
+ hi, lo, r uint
+ }{
+ {1 << (UintSize - 1), 2, 1, 0, 1},
+ {_M, _M, _M - 1, 1, 42},
+ } {
+ testMul("Mul", Mul, a.x, a.y, a.hi, a.lo)
+ testMul("Mul symmetric", Mul, a.y, a.x, a.hi, a.lo)
+ testDiv("Div", Div, a.hi, a.lo+a.r, a.y, a.x, a.r)
+ testDiv("Div symmetric", Div, a.hi, a.lo+a.r, a.x, a.y, a.r)
+ // The above code can't test intrinsic implementation, because the passed function is not called directly.
+ // The following code uses a closure to test the intrinsic version in case the function is intrinsified.
+ testMul("Mul intrinsic", func(x, y uint) (uint, uint) { return Mul(x, y) }, a.x, a.y, a.hi, a.lo)
+ testMul("Mul intrinsic symmetric", func(x, y uint) (uint, uint) { return Mul(x, y) }, a.y, a.x, a.hi, a.lo)
+ testDiv("Div intrinsic", func(hi, lo, y uint) (uint, uint) { return Div(hi, lo, y) }, a.hi, a.lo+a.r, a.y, a.x, a.r)
+ testDiv("Div intrinsic symmetric", func(hi, lo, y uint) (uint, uint) { return Div(hi, lo, y) }, a.hi, a.lo+a.r, a.x, a.y, a.r)
+ }
+}
+
+func TestMulDiv32(t *testing.T) {
+ testMul := func(msg string, f func(x, y uint32) (hi, lo uint32), x, y, hi, lo uint32) {
+ hi1, lo1 := f(x, y)
+ if hi1 != hi || lo1 != lo {
+ t.Errorf("%s: got hi:lo = %#x:%#x; want %#x:%#x", msg, hi1, lo1, hi, lo)
+ }
+ }
+ testDiv := func(msg string, f func(hi, lo, y uint32) (q, r uint32), hi, lo, y, q, r uint32) {
+ q1, r1 := f(hi, lo, y)
+ if q1 != q || r1 != r {
+ t.Errorf("%s: got q:r = %#x:%#x; want %#x:%#x", msg, q1, r1, q, r)
+ }
+ }
+ for _, a := range []struct {
+ x, y uint32
+ hi, lo, r uint32
+ }{
+ {1 << 31, 2, 1, 0, 1},
+ {0xc47dfa8c, 50911, 0x98a4, 0x998587f4, 13},
+ {_M32, _M32, _M32 - 1, 1, 42},
+ } {
+ testMul("Mul32", Mul32, a.x, a.y, a.hi, a.lo)
+ testMul("Mul32 symmetric", Mul32, a.y, a.x, a.hi, a.lo)
+ testDiv("Div32", Div32, a.hi, a.lo+a.r, a.y, a.x, a.r)
+ testDiv("Div32 symmetric", Div32, a.hi, a.lo+a.r, a.x, a.y, a.r)
+ }
+}
+
+func TestMulDiv64(t *testing.T) {
+ testMul := func(msg string, f func(x, y uint64) (hi, lo uint64), x, y, hi, lo uint64) {
+ hi1, lo1 := f(x, y)
+ if hi1 != hi || lo1 != lo {
+ t.Errorf("%s: got hi:lo = %#x:%#x; want %#x:%#x", msg, hi1, lo1, hi, lo)
+ }
+ }
+ testDiv := func(msg string, f func(hi, lo, y uint64) (q, r uint64), hi, lo, y, q, r uint64) {
+ q1, r1 := f(hi, lo, y)
+ if q1 != q || r1 != r {
+ t.Errorf("%s: got q:r = %#x:%#x; want %#x:%#x", msg, q1, r1, q, r)
+ }
+ }
+ for _, a := range []struct {
+ x, y uint64
+ hi, lo, r uint64
+ }{
+ {1 << 63, 2, 1, 0, 1},
+ {0x3626229738a3b9, 0xd8988a9f1cc4a61, 0x2dd0712657fe8, 0x9dd6a3364c358319, 13},
+ {_M64, _M64, _M64 - 1, 1, 42},
+ } {
+ testMul("Mul64", Mul64, a.x, a.y, a.hi, a.lo)
+ testMul("Mul64 symmetric", Mul64, a.y, a.x, a.hi, a.lo)
+ testDiv("Div64", Div64, a.hi, a.lo+a.r, a.y, a.x, a.r)
+ testDiv("Div64 symmetric", Div64, a.hi, a.lo+a.r, a.x, a.y, a.r)
+ // The above code can't test intrinsic implementation, because the passed function is not called directly.
+ // The following code uses a closure to test the intrinsic version in case the function is intrinsified.
+ testMul("Mul64 intrinsic", func(x, y uint64) (uint64, uint64) { return Mul64(x, y) }, a.x, a.y, a.hi, a.lo)
+ testMul("Mul64 intrinsic symmetric", func(x, y uint64) (uint64, uint64) { return Mul64(x, y) }, a.y, a.x, a.hi, a.lo)
+ testDiv("Div64 intrinsic", func(hi, lo, y uint64) (uint64, uint64) { return Div64(hi, lo, y) }, a.hi, a.lo+a.r, a.y, a.x, a.r)
+ testDiv("Div64 intrinsic symmetric", func(hi, lo, y uint64) (uint64, uint64) { return Div64(hi, lo, y) }, a.hi, a.lo+a.r, a.x, a.y, a.r)
+ }
+}
+
+const (
+ divZeroError = "runtime error: integer divide by zero"
+ overflowError = "runtime error: integer overflow"
+)
+
+func TestDivPanicOverflow(t *testing.T) {
+ // Expect a panic
+ defer func() {
+ if err := recover(); err == nil {
+ t.Error("Div should have panicked when y<=hi")
+ } else if e, ok := err.(runtime.Error); !ok || e.Error() != overflowError {
+ t.Errorf("Div expected panic: %q, got: %q ", overflowError, e.Error())
+ }
+ }()
+ q, r := Div(1, 0, 1)
+ t.Errorf("undefined q, r = %v, %v calculated when Div should have panicked", q, r)
+}
+
+func TestDiv32PanicOverflow(t *testing.T) {
+ // Expect a panic
+ defer func() {
+ if err := recover(); err == nil {
+ t.Error("Div32 should have panicked when y<=hi")
+ } else if e, ok := err.(runtime.Error); !ok || e.Error() != overflowError {
+ t.Errorf("Div32 expected panic: %q, got: %q ", overflowError, e.Error())
+ }
+ }()
+ q, r := Div32(1, 0, 1)
+ t.Errorf("undefined q, r = %v, %v calculated when Div32 should have panicked", q, r)
+}
+
+func TestDiv64PanicOverflow(t *testing.T) {
+ // Expect a panic
+ defer func() {
+ if err := recover(); err == nil {
+ t.Error("Div64 should have panicked when y<=hi")
+ } else if e, ok := err.(runtime.Error); !ok || e.Error() != overflowError {
+ t.Errorf("Div64 expected panic: %q, got: %q ", overflowError, e.Error())
+ }
+ }()
+ q, r := Div64(1, 0, 1)
+ t.Errorf("undefined q, r = %v, %v calculated when Div64 should have panicked", q, r)
+}
+
+func TestDivPanicZero(t *testing.T) {
+ // Expect a panic
+ defer func() {
+ if err := recover(); err == nil {
+ t.Error("Div should have panicked when y==0")
+ } else if e, ok := err.(runtime.Error); !ok || e.Error() != divZeroError {
+ t.Errorf("Div expected panic: %q, got: %q ", divZeroError, e.Error())
+ }
+ }()
+ q, r := Div(1, 1, 0)
+ t.Errorf("undefined q, r = %v, %v calculated when Div should have panicked", q, r)
+}
+
+func TestDiv32PanicZero(t *testing.T) {
+ // Expect a panic
+ defer func() {
+ if err := recover(); err == nil {
+ t.Error("Div32 should have panicked when y==0")
+ } else if e, ok := err.(runtime.Error); !ok || e.Error() != divZeroError {
+ t.Errorf("Div32 expected panic: %q, got: %q ", divZeroError, e.Error())
+ }
+ }()
+ q, r := Div32(1, 1, 0)
+ t.Errorf("undefined q, r = %v, %v calculated when Div32 should have panicked", q, r)
+}
+
+func TestDiv64PanicZero(t *testing.T) {
+ // Expect a panic
+ defer func() {
+ if err := recover(); err == nil {
+ t.Error("Div64 should have panicked when y==0")
+ } else if e, ok := err.(runtime.Error); !ok || e.Error() != divZeroError {
+ t.Errorf("Div64 expected panic: %q, got: %q ", divZeroError, e.Error())
+ }
+ }()
+ q, r := Div64(1, 1, 0)
+ t.Errorf("undefined q, r = %v, %v calculated when Div64 should have panicked", q, r)
+}
+
+func TestRem32(t *testing.T) {
+ // Sanity check: for non-oveflowing dividends, the result is the
+ // same as the rem returned by Div32
+ hi, lo, y := uint32(510510), uint32(9699690), uint32(510510+1) // ensure hi < y
+ for i := 0; i < 1000; i++ {
+ r := Rem32(hi, lo, y)
+ _, r2 := Div32(hi, lo, y)
+ if r != r2 {
+ t.Errorf("Rem32(%v, %v, %v) returned %v, but Div32 returned rem %v", hi, lo, y, r, r2)
+ }
+ y += 13
+ }
+}
+
+func TestRem32Overflow(t *testing.T) {
+ // To trigger a quotient overflow, we need y <= hi
+ hi, lo, y := uint32(510510), uint32(9699690), uint32(7)
+ for i := 0; i < 1000; i++ {
+ r := Rem32(hi, lo, y)
+ _, r2 := Div64(0, uint64(hi)<<32|uint64(lo), uint64(y))
+ if r != uint32(r2) {
+ t.Errorf("Rem32(%v, %v, %v) returned %v, but Div64 returned rem %v", hi, lo, y, r, r2)
+ }
+ y += 13
+ }
+}
+
+func TestRem64(t *testing.T) {
+ // Sanity check: for non-oveflowing dividends, the result is the
+ // same as the rem returned by Div64
+ hi, lo, y := uint64(510510), uint64(9699690), uint64(510510+1) // ensure hi < y
+ for i := 0; i < 1000; i++ {
+ r := Rem64(hi, lo, y)
+ _, r2 := Div64(hi, lo, y)
+ if r != r2 {
+ t.Errorf("Rem64(%v, %v, %v) returned %v, but Div64 returned rem %v", hi, lo, y, r, r2)
+ }
+ y += 13
+ }
+}
+
+func TestRem64Overflow(t *testing.T) {
+ Rem64Tests := []struct {
+ hi, lo, y uint64
+ rem uint64
+ }{
+ // Testcases computed using Python 3, as:
+ // >>> hi = 42; lo = 1119; y = 42
+ // >>> ((hi<<64)+lo) % y
+ {42, 1119, 42, 27},
+ {42, 1119, 38, 9},
+ {42, 1119, 26, 23},
+ {469, 0, 467, 271},
+ {469, 0, 113, 58},
+ {111111, 111111, 1171, 803},
+ {3968194946088682615, 3192705705065114702, 1000037, 56067},
+ }
+
+ for _, rt := range Rem64Tests {
+ if rt.hi < rt.y {
+ t.Fatalf("Rem64(%v, %v, %v) is not a test with quo overflow", rt.hi, rt.lo, rt.y)
+ }
+ rem := Rem64(rt.hi, rt.lo, rt.y)
+ if rem != rt.rem {
+ t.Errorf("Rem64(%v, %v, %v) returned %v, wanted %v",
+ rt.hi, rt.lo, rt.y, rem, rt.rem)
+ }
+ }
+}
+
+func BenchmarkAdd(b *testing.B) {
+ var z, c uint
+ for i := 0; i < b.N; i++ {
+ z, c = Add(uint(Input), uint(i), c)
+ }
+ Output = int(z + c)
+}
+
+func BenchmarkAdd32(b *testing.B) {
+ var z, c uint32
+ for i := 0; i < b.N; i++ {
+ z, c = Add32(uint32(Input), uint32(i), c)
+ }
+ Output = int(z + c)
+}
+
+func BenchmarkAdd64(b *testing.B) {
+ var z, c uint64
+ for i := 0; i < b.N; i++ {
+ z, c = Add64(uint64(Input), uint64(i), c)
+ }
+ Output = int(z + c)
+}
+
+func BenchmarkAdd64multiple(b *testing.B) {
+ var z0 = uint64(Input)
+ var z1 = uint64(Input)
+ var z2 = uint64(Input)
+ var z3 = uint64(Input)
+ for i := 0; i < b.N; i++ {
+ var c uint64
+ z0, c = Add64(z0, uint64(i), c)
+ z1, c = Add64(z1, uint64(i), c)
+ z2, c = Add64(z2, uint64(i), c)
+ z3, _ = Add64(z3, uint64(i), c)
+ }
+ Output = int(z0 + z1 + z2 + z3)
+}
+
+func BenchmarkSub(b *testing.B) {
+ var z, c uint
+ for i := 0; i < b.N; i++ {
+ z, c = Sub(uint(Input), uint(i), c)
+ }
+ Output = int(z + c)
+}
+
+func BenchmarkSub32(b *testing.B) {
+ var z, c uint32
+ for i := 0; i < b.N; i++ {
+ z, c = Sub32(uint32(Input), uint32(i), c)
+ }
+ Output = int(z + c)
+}
+
+func BenchmarkSub64(b *testing.B) {
+ var z, c uint64
+ for i := 0; i < b.N; i++ {
+ z, c = Sub64(uint64(Input), uint64(i), c)
+ }
+ Output = int(z + c)
+}
+
+func BenchmarkSub64multiple(b *testing.B) {
+ var z0 = uint64(Input)
+ var z1 = uint64(Input)
+ var z2 = uint64(Input)
+ var z3 = uint64(Input)
+ for i := 0; i < b.N; i++ {
+ var c uint64
+ z0, c = Sub64(z0, uint64(i), c)
+ z1, c = Sub64(z1, uint64(i), c)
+ z2, c = Sub64(z2, uint64(i), c)
+ z3, _ = Sub64(z3, uint64(i), c)
+ }
+ Output = int(z0 + z1 + z2 + z3)
+}
+
+func BenchmarkMul(b *testing.B) {
+ var hi, lo uint
+ for i := 0; i < b.N; i++ {
+ hi, lo = Mul(uint(Input), uint(i))
+ }
+ Output = int(hi + lo)
+}
+
+func BenchmarkMul32(b *testing.B) {
+ var hi, lo uint32
+ for i := 0; i < b.N; i++ {
+ hi, lo = Mul32(uint32(Input), uint32(i))
+ }
+ Output = int(hi + lo)
+}
+
+func BenchmarkMul64(b *testing.B) {
+ var hi, lo uint64
+ for i := 0; i < b.N; i++ {
+ hi, lo = Mul64(uint64(Input), uint64(i))
+ }
+ Output = int(hi + lo)
+}
+
+func BenchmarkDiv(b *testing.B) {
+ var q, r uint
+ for i := 0; i < b.N; i++ {
+ q, r = Div(1, uint(i), uint(Input))
+ }
+ Output = int(q + r)
+}
+
+func BenchmarkDiv32(b *testing.B) {
+ var q, r uint32
+ for i := 0; i < b.N; i++ {
+ q, r = Div32(1, uint32(i), uint32(Input))
+ }
+ Output = int(q + r)
+}
+
+func BenchmarkDiv64(b *testing.B) {
+ var q, r uint64
+ for i := 0; i < b.N; i++ {
+ q, r = Div64(1, uint64(i), uint64(Input))
+ }
+ Output = int(q + r)
+}
+
+// ----------------------------------------------------------------------------
+// Testing support
+
+type entry = struct {
+ nlz, ntz, pop int
+}
+
+// tab contains results for all uint8 values
+var tab [256]entry
+
+func init() {
+ tab[0] = entry{8, 8, 0}
+ for i := 1; i < len(tab); i++ {
+ // nlz
+ x := i // x != 0
+ n := 0
+ for x&0x80 == 0 {
+ n++
+ x <<= 1
+ }
+ tab[i].nlz = n
+
+ // ntz
+ x = i // x != 0
+ n = 0
+ for x&1 == 0 {
+ n++
+ x >>= 1
+ }
+ tab[i].ntz = n
+
+ // pop
+ x = i // x != 0
+ n = 0
+ for x != 0 {
+ n += int(x & 1)
+ x >>= 1
+ }
+ tab[i].pop = n
+ }
+}
diff --git a/src/math/bits/example_test.go b/src/math/bits/example_test.go
new file mode 100644
index 0000000..b2ed2cb
--- /dev/null
+++ b/src/math/bits/example_test.go
@@ -0,0 +1,210 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Code generated by go run make_examples.go. DO NOT EDIT.
+
+package bits_test
+
+import (
+ "fmt"
+ "math/bits"
+)
+
+func ExampleLeadingZeros8() {
+ fmt.Printf("LeadingZeros8(%08b) = %d\n", 1, bits.LeadingZeros8(1))
+ // Output:
+ // LeadingZeros8(00000001) = 7
+}
+
+func ExampleLeadingZeros16() {
+ fmt.Printf("LeadingZeros16(%016b) = %d\n", 1, bits.LeadingZeros16(1))
+ // Output:
+ // LeadingZeros16(0000000000000001) = 15
+}
+
+func ExampleLeadingZeros32() {
+ fmt.Printf("LeadingZeros32(%032b) = %d\n", 1, bits.LeadingZeros32(1))
+ // Output:
+ // LeadingZeros32(00000000000000000000000000000001) = 31
+}
+
+func ExampleLeadingZeros64() {
+ fmt.Printf("LeadingZeros64(%064b) = %d\n", 1, bits.LeadingZeros64(1))
+ // Output:
+ // LeadingZeros64(0000000000000000000000000000000000000000000000000000000000000001) = 63
+}
+
+func ExampleTrailingZeros8() {
+ fmt.Printf("TrailingZeros8(%08b) = %d\n", 14, bits.TrailingZeros8(14))
+ // Output:
+ // TrailingZeros8(00001110) = 1
+}
+
+func ExampleTrailingZeros16() {
+ fmt.Printf("TrailingZeros16(%016b) = %d\n", 14, bits.TrailingZeros16(14))
+ // Output:
+ // TrailingZeros16(0000000000001110) = 1
+}
+
+func ExampleTrailingZeros32() {
+ fmt.Printf("TrailingZeros32(%032b) = %d\n", 14, bits.TrailingZeros32(14))
+ // Output:
+ // TrailingZeros32(00000000000000000000000000001110) = 1
+}
+
+func ExampleTrailingZeros64() {
+ fmt.Printf("TrailingZeros64(%064b) = %d\n", 14, bits.TrailingZeros64(14))
+ // Output:
+ // TrailingZeros64(0000000000000000000000000000000000000000000000000000000000001110) = 1
+}
+
+func ExampleOnesCount() {
+ fmt.Printf("OnesCount(%b) = %d\n", 14, bits.OnesCount(14))
+ // Output:
+ // OnesCount(1110) = 3
+}
+
+func ExampleOnesCount8() {
+ fmt.Printf("OnesCount8(%08b) = %d\n", 14, bits.OnesCount8(14))
+ // Output:
+ // OnesCount8(00001110) = 3
+}
+
+func ExampleOnesCount16() {
+ fmt.Printf("OnesCount16(%016b) = %d\n", 14, bits.OnesCount16(14))
+ // Output:
+ // OnesCount16(0000000000001110) = 3
+}
+
+func ExampleOnesCount32() {
+ fmt.Printf("OnesCount32(%032b) = %d\n", 14, bits.OnesCount32(14))
+ // Output:
+ // OnesCount32(00000000000000000000000000001110) = 3
+}
+
+func ExampleOnesCount64() {
+ fmt.Printf("OnesCount64(%064b) = %d\n", 14, bits.OnesCount64(14))
+ // Output:
+ // OnesCount64(0000000000000000000000000000000000000000000000000000000000001110) = 3
+}
+
+func ExampleRotateLeft8() {
+ fmt.Printf("%08b\n", 15)
+ fmt.Printf("%08b\n", bits.RotateLeft8(15, 2))
+ fmt.Printf("%08b\n", bits.RotateLeft8(15, -2))
+ // Output:
+ // 00001111
+ // 00111100
+ // 11000011
+}
+
+func ExampleRotateLeft16() {
+ fmt.Printf("%016b\n", 15)
+ fmt.Printf("%016b\n", bits.RotateLeft16(15, 2))
+ fmt.Printf("%016b\n", bits.RotateLeft16(15, -2))
+ // Output:
+ // 0000000000001111
+ // 0000000000111100
+ // 1100000000000011
+}
+
+func ExampleRotateLeft32() {
+ fmt.Printf("%032b\n", 15)
+ fmt.Printf("%032b\n", bits.RotateLeft32(15, 2))
+ fmt.Printf("%032b\n", bits.RotateLeft32(15, -2))
+ // Output:
+ // 00000000000000000000000000001111
+ // 00000000000000000000000000111100
+ // 11000000000000000000000000000011
+}
+
+func ExampleRotateLeft64() {
+ fmt.Printf("%064b\n", 15)
+ fmt.Printf("%064b\n", bits.RotateLeft64(15, 2))
+ fmt.Printf("%064b\n", bits.RotateLeft64(15, -2))
+ // Output:
+ // 0000000000000000000000000000000000000000000000000000000000001111
+ // 0000000000000000000000000000000000000000000000000000000000111100
+ // 1100000000000000000000000000000000000000000000000000000000000011
+}
+
+func ExampleReverse8() {
+ fmt.Printf("%08b\n", 19)
+ fmt.Printf("%08b\n", bits.Reverse8(19))
+ // Output:
+ // 00010011
+ // 11001000
+}
+
+func ExampleReverse16() {
+ fmt.Printf("%016b\n", 19)
+ fmt.Printf("%016b\n", bits.Reverse16(19))
+ // Output:
+ // 0000000000010011
+ // 1100100000000000
+}
+
+func ExampleReverse32() {
+ fmt.Printf("%032b\n", 19)
+ fmt.Printf("%032b\n", bits.Reverse32(19))
+ // Output:
+ // 00000000000000000000000000010011
+ // 11001000000000000000000000000000
+}
+
+func ExampleReverse64() {
+ fmt.Printf("%064b\n", 19)
+ fmt.Printf("%064b\n", bits.Reverse64(19))
+ // Output:
+ // 0000000000000000000000000000000000000000000000000000000000010011
+ // 1100100000000000000000000000000000000000000000000000000000000000
+}
+
+func ExampleReverseBytes16() {
+ fmt.Printf("%016b\n", 15)
+ fmt.Printf("%016b\n", bits.ReverseBytes16(15))
+ // Output:
+ // 0000000000001111
+ // 0000111100000000
+}
+
+func ExampleReverseBytes32() {
+ fmt.Printf("%032b\n", 15)
+ fmt.Printf("%032b\n", bits.ReverseBytes32(15))
+ // Output:
+ // 00000000000000000000000000001111
+ // 00001111000000000000000000000000
+}
+
+func ExampleReverseBytes64() {
+ fmt.Printf("%064b\n", 15)
+ fmt.Printf("%064b\n", bits.ReverseBytes64(15))
+ // Output:
+ // 0000000000000000000000000000000000000000000000000000000000001111
+ // 0000111100000000000000000000000000000000000000000000000000000000
+}
+
+func ExampleLen8() {
+ fmt.Printf("Len8(%08b) = %d\n", 8, bits.Len8(8))
+ // Output:
+ // Len8(00001000) = 4
+}
+
+func ExampleLen16() {
+ fmt.Printf("Len16(%016b) = %d\n", 8, bits.Len16(8))
+ // Output:
+ // Len16(0000000000001000) = 4
+}
+
+func ExampleLen32() {
+ fmt.Printf("Len32(%032b) = %d\n", 8, bits.Len32(8))
+ // Output:
+ // Len32(00000000000000000000000000001000) = 4
+}
+
+func ExampleLen64() {
+ fmt.Printf("Len64(%064b) = %d\n", 8, bits.Len64(8))
+ // Output:
+ // Len64(0000000000000000000000000000000000000000000000000000000000001000) = 4
+}
diff --git a/src/math/bits/export_test.go b/src/math/bits/export_test.go
new file mode 100644
index 0000000..8c6f933
--- /dev/null
+++ b/src/math/bits/export_test.go
@@ -0,0 +1,7 @@
+// Copyright 2018 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package bits
+
+const DeBruijn64 = deBruijn64
diff --git a/src/math/bits/make_examples.go b/src/math/bits/make_examples.go
new file mode 100644
index 0000000..1d3ad53
--- /dev/null
+++ b/src/math/bits/make_examples.go
@@ -0,0 +1,112 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build ignore
+
+// This program generates example_test.go.
+
+package main
+
+import (
+ "bytes"
+ "fmt"
+ "log"
+ "math/bits"
+ "os"
+)
+
+const header = `// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Code generated by go run make_examples.go. DO NOT EDIT.
+
+package bits_test
+
+import (
+ "fmt"
+ "math/bits"
+)
+`
+
+func main() {
+ w := bytes.NewBuffer([]byte(header))
+
+ for _, e := range []struct {
+ name string
+ in int
+ out [4]interface{}
+ out2 [4]interface{}
+ }{
+ {
+ name: "LeadingZeros",
+ in: 1,
+ out: [4]interface{}{bits.LeadingZeros8(1), bits.LeadingZeros16(1), bits.LeadingZeros32(1), bits.LeadingZeros64(1)},
+ },
+ {
+ name: "TrailingZeros",
+ in: 14,
+ out: [4]interface{}{bits.TrailingZeros8(14), bits.TrailingZeros16(14), bits.TrailingZeros32(14), bits.TrailingZeros64(14)},
+ },
+ {
+ name: "OnesCount",
+ in: 14,
+ out: [4]interface{}{bits.OnesCount8(14), bits.OnesCount16(14), bits.OnesCount32(14), bits.OnesCount64(14)},
+ },
+ {
+ name: "RotateLeft",
+ in: 15,
+ out: [4]interface{}{bits.RotateLeft8(15, 2), bits.RotateLeft16(15, 2), bits.RotateLeft32(15, 2), bits.RotateLeft64(15, 2)},
+ out2: [4]interface{}{bits.RotateLeft8(15, -2), bits.RotateLeft16(15, -2), bits.RotateLeft32(15, -2), bits.RotateLeft64(15, -2)},
+ },
+ {
+ name: "Reverse",
+ in: 19,
+ out: [4]interface{}{bits.Reverse8(19), bits.Reverse16(19), bits.Reverse32(19), bits.Reverse64(19)},
+ },
+ {
+ name: "ReverseBytes",
+ in: 15,
+ out: [4]interface{}{nil, bits.ReverseBytes16(15), bits.ReverseBytes32(15), bits.ReverseBytes64(15)},
+ },
+ {
+ name: "Len",
+ in: 8,
+ out: [4]interface{}{bits.Len8(8), bits.Len16(8), bits.Len32(8), bits.Len64(8)},
+ },
+ } {
+ for i, size := range []int{8, 16, 32, 64} {
+ if e.out[i] == nil {
+ continue // function doesn't exist
+ }
+ f := fmt.Sprintf("%s%d", e.name, size)
+ fmt.Fprintf(w, "\nfunc Example%s() {\n", f)
+ switch e.name {
+ case "RotateLeft", "Reverse", "ReverseBytes":
+ fmt.Fprintf(w, "\tfmt.Printf(\"%%0%db\\n\", %d)\n", size, e.in)
+ if e.name == "RotateLeft" {
+ fmt.Fprintf(w, "\tfmt.Printf(\"%%0%db\\n\", bits.%s(%d, 2))\n", size, f, e.in)
+ fmt.Fprintf(w, "\tfmt.Printf(\"%%0%db\\n\", bits.%s(%d, -2))\n", size, f, e.in)
+ } else {
+ fmt.Fprintf(w, "\tfmt.Printf(\"%%0%db\\n\", bits.%s(%d))\n", size, f, e.in)
+ }
+ fmt.Fprintf(w, "\t// Output:\n")
+ fmt.Fprintf(w, "\t// %0*b\n", size, e.in)
+ fmt.Fprintf(w, "\t// %0*b\n", size, e.out[i])
+ if e.name == "RotateLeft" && e.out2[i] != nil {
+ fmt.Fprintf(w, "\t// %0*b\n", size, e.out2[i])
+ }
+ default:
+ fmt.Fprintf(w, "\tfmt.Printf(\"%s(%%0%db) = %%d\\n\", %d, bits.%s(%d))\n", f, size, e.in, f, e.in)
+ fmt.Fprintf(w, "\t// Output:\n")
+ fmt.Fprintf(w, "\t// %s(%0*b) = %d\n", f, size, e.in, e.out[i])
+ }
+ fmt.Fprintf(w, "}\n")
+ }
+ }
+
+ if err := os.WriteFile("example_test.go", w.Bytes(), 0666); err != nil {
+ log.Fatal(err)
+ }
+}
diff --git a/src/math/bits/make_tables.go b/src/math/bits/make_tables.go
new file mode 100644
index 0000000..b068d5e
--- /dev/null
+++ b/src/math/bits/make_tables.go
@@ -0,0 +1,92 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build ignore
+
+// This program generates bits_tables.go.
+
+package main
+
+import (
+ "bytes"
+ "fmt"
+ "go/format"
+ "io"
+ "log"
+ "os"
+)
+
+var header = []byte(`// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Code generated by go run make_tables.go. DO NOT EDIT.
+
+package bits
+
+`)
+
+func main() {
+ buf := bytes.NewBuffer(header)
+
+ gen(buf, "ntz8tab", ntz8)
+ gen(buf, "pop8tab", pop8)
+ gen(buf, "rev8tab", rev8)
+ gen(buf, "len8tab", len8)
+
+ out, err := format.Source(buf.Bytes())
+ if err != nil {
+ log.Fatal(err)
+ }
+
+ err = os.WriteFile("bits_tables.go", out, 0666)
+ if err != nil {
+ log.Fatal(err)
+ }
+}
+
+func gen(w io.Writer, name string, f func(uint8) uint8) {
+ fmt.Fprintf(w, "var %s = [256]uint8{", name)
+ for i := 0; i < 256; i++ {
+ if i%16 == 0 {
+ fmt.Fprint(w, "\n\t")
+ } else {
+ fmt.Fprint(w, " ")
+ }
+ fmt.Fprintf(w, "%#02x,", f(uint8(i)))
+ }
+ fmt.Fprint(w, "\n}\n\n")
+}
+
+func ntz8(x uint8) (n uint8) {
+ for x&1 == 0 && n < 8 {
+ x >>= 1
+ n++
+ }
+ return
+}
+
+func pop8(x uint8) (n uint8) {
+ for x != 0 {
+ x &= x - 1
+ n++
+ }
+ return
+}
+
+func rev8(x uint8) (r uint8) {
+ for i := 8; i > 0; i-- {
+ r = r<<1 | x&1
+ x >>= 1
+ }
+ return
+}
+
+func len8(x uint8) (n uint8) {
+ for x != 0 {
+ x >>= 1
+ n++
+ }
+ return
+}
diff --git a/src/math/cbrt.go b/src/math/cbrt.go
new file mode 100644
index 0000000..4afd4a8
--- /dev/null
+++ b/src/math/cbrt.go
@@ -0,0 +1,79 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The go code is a modified version of the original C code from
+// http://www.netlib.org/fdlibm/s_cbrt.c and came with this notice.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunSoft, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+
+// Cbrt returns the cube root of x.
+//
+// Special cases are:
+// Cbrt(±0) = ±0
+// Cbrt(±Inf) = ±Inf
+// Cbrt(NaN) = NaN
+func Cbrt(x float64) float64
+
+func cbrt(x float64) float64 {
+ const (
+ B1 = 715094163 // (682-0.03306235651)*2**20
+ B2 = 696219795 // (664-0.03306235651)*2**20
+ C = 5.42857142857142815906e-01 // 19/35 = 0x3FE15F15F15F15F1
+ D = -7.05306122448979611050e-01 // -864/1225 = 0xBFE691DE2532C834
+ E = 1.41428571428571436819e+00 // 99/70 = 0x3FF6A0EA0EA0EA0F
+ F = 1.60714285714285720630e+00 // 45/28 = 0x3FF9B6DB6DB6DB6E
+ G = 3.57142857142857150787e-01 // 5/14 = 0x3FD6DB6DB6DB6DB7
+ SmallestNormal = 2.22507385850720138309e-308 // 2**-1022 = 0x0010000000000000
+ )
+ // special cases
+ switch {
+ case x == 0 || IsNaN(x) || IsInf(x, 0):
+ return x
+ }
+
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+
+ // rough cbrt to 5 bits
+ t := Float64frombits(Float64bits(x)/3 + B1<<32)
+ if x < SmallestNormal {
+ // subnormal number
+ t = float64(1 << 54) // set t= 2**54
+ t *= x
+ t = Float64frombits(Float64bits(t)/3 + B2<<32)
+ }
+
+ // new cbrt to 23 bits
+ r := t * t / x
+ s := C + r*t
+ t *= G + F/(s+E+D/s)
+
+ // chop to 22 bits, make larger than cbrt(x)
+ t = Float64frombits(Float64bits(t)&(0xFFFFFFFFC<<28) + 1<<30)
+
+ // one step newton iteration to 53 bits with error less than 0.667ulps
+ s = t * t // t*t is exact
+ r = x / s
+ w := t + t
+ r = (r - t) / (w + r) // r-s is exact
+ t = t + t*r
+
+ // restore the sign bit
+ if sign {
+ t = -t
+ }
+ return t
+}
diff --git a/src/math/cbrt_s390x.s b/src/math/cbrt_s390x.s
new file mode 100644
index 0000000..87bba53
--- /dev/null
+++ b/src/math/cbrt_s390x.s
@@ -0,0 +1,156 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial coefficients and other constants
+DATA ·cbrtrodataL9<> + 0(SB)/8, $-.00016272731015974436E+00
+DATA ·cbrtrodataL9<> + 8(SB)/8, $0.66639548758285293179E+00
+DATA ·cbrtrodataL9<> + 16(SB)/8, $0.55519402697349815993E+00
+DATA ·cbrtrodataL9<> + 24(SB)/8, $0.49338566048766782004E+00
+DATA ·cbrtrodataL9<> + 32(SB)/8, $0.45208160036325611486E+00
+DATA ·cbrtrodataL9<> + 40(SB)/8, $0.43099892837778637816E+00
+DATA ·cbrtrodataL9<> + 48(SB)/8, $1.000244140625
+DATA ·cbrtrodataL9<> + 56(SB)/8, $0.33333333333333333333E+00
+DATA ·cbrtrodataL9<> + 64(SB)/8, $79228162514264337593543950336.
+GLOBL ·cbrtrodataL9<> + 0(SB), RODATA, $72
+
+// Index tables
+DATA ·cbrttab32069<> + 0(SB)/8, $0x404030303020202
+DATA ·cbrttab32069<> + 8(SB)/8, $0x101010101000000
+DATA ·cbrttab32069<> + 16(SB)/8, $0x808070706060605
+DATA ·cbrttab32069<> + 24(SB)/8, $0x505040404040303
+DATA ·cbrttab32069<> + 32(SB)/8, $0xe0d0c0c0b0b0b0a
+DATA ·cbrttab32069<> + 40(SB)/8, $0xa09090908080808
+DATA ·cbrttab32069<> + 48(SB)/8, $0x11111010100f0f0f
+DATA ·cbrttab32069<> + 56(SB)/8, $0xe0e0e0e0e0d0d0d
+DATA ·cbrttab32069<> + 64(SB)/8, $0x1515141413131312
+DATA ·cbrttab32069<> + 72(SB)/8, $0x1212111111111010
+GLOBL ·cbrttab32069<> + 0(SB), RODATA, $80
+
+DATA ·cbrttab22068<> + 0(SB)/8, $0x151015001420141
+DATA ·cbrttab22068<> + 8(SB)/8, $0x140013201310130
+DATA ·cbrttab22068<> + 16(SB)/8, $0x122012101200112
+DATA ·cbrttab22068<> + 24(SB)/8, $0x111011001020101
+DATA ·cbrttab22068<> + 32(SB)/8, $0x10000f200f100f0
+DATA ·cbrttab22068<> + 40(SB)/8, $0xe200e100e000d2
+DATA ·cbrttab22068<> + 48(SB)/8, $0xd100d000c200c1
+DATA ·cbrttab22068<> + 56(SB)/8, $0xc000b200b100b0
+DATA ·cbrttab22068<> + 64(SB)/8, $0xa200a100a00092
+DATA ·cbrttab22068<> + 72(SB)/8, $0x91009000820081
+DATA ·cbrttab22068<> + 80(SB)/8, $0x80007200710070
+DATA ·cbrttab22068<> + 88(SB)/8, $0x62006100600052
+DATA ·cbrttab22068<> + 96(SB)/8, $0x51005000420041
+DATA ·cbrttab22068<> + 104(SB)/8, $0x40003200310030
+DATA ·cbrttab22068<> + 112(SB)/8, $0x22002100200012
+DATA ·cbrttab22068<> + 120(SB)/8, $0x11001000020001
+GLOBL ·cbrttab22068<> + 0(SB), RODATA, $128
+
+DATA ·cbrttab12067<> + 0(SB)/8, $0x53e1529051324fe1
+DATA ·cbrttab12067<> + 8(SB)/8, $0x4e904d324be14a90
+DATA ·cbrttab12067<> + 16(SB)/8, $0x493247e146904532
+DATA ·cbrttab12067<> + 24(SB)/8, $0x43e1429041323fe1
+DATA ·cbrttab12067<> + 32(SB)/8, $0x3e903d323be13a90
+DATA ·cbrttab12067<> + 40(SB)/8, $0x393237e136903532
+DATA ·cbrttab12067<> + 48(SB)/8, $0x33e1329031322fe1
+DATA ·cbrttab12067<> + 56(SB)/8, $0x2e902d322be12a90
+DATA ·cbrttab12067<> + 64(SB)/8, $0xd3e1d290d132cfe1
+DATA ·cbrttab12067<> + 72(SB)/8, $0xce90cd32cbe1ca90
+DATA ·cbrttab12067<> + 80(SB)/8, $0xc932c7e1c690c532
+DATA ·cbrttab12067<> + 88(SB)/8, $0xc3e1c290c132bfe1
+DATA ·cbrttab12067<> + 96(SB)/8, $0xbe90bd32bbe1ba90
+DATA ·cbrttab12067<> + 104(SB)/8, $0xb932b7e1b690b532
+DATA ·cbrttab12067<> + 112(SB)/8, $0xb3e1b290b132afe1
+DATA ·cbrttab12067<> + 120(SB)/8, $0xae90ad32abe1aa90
+GLOBL ·cbrttab12067<> + 0(SB), RODATA, $128
+
+// Cbrt returns the cube root of the argument.
+//
+// Special cases are:
+// Cbrt(±0) = ±0
+// Cbrt(±Inf) = ±Inf
+// Cbrt(NaN) = NaN
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·cbrtAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·cbrtrodataL9<>+0(SB), R9
+ LGDR F0, R2
+ WORD $0xC039000F //iilf %r3,1048575
+ BYTE $0xFF
+ BYTE $0xFF
+ SRAD $32, R2
+ WORD $0xB9170012 //llgtr %r1,%r2
+ MOVW R1, R6
+ MOVW R3, R7
+ CMPBLE R6, R7, L2
+ WORD $0xC0397FEF //iilf %r3,2146435071
+ BYTE $0xFF
+ BYTE $0xFF
+ MOVW R3, R7
+ CMPBLE R6, R7, L8
+L1:
+ FMOVD F0, ret+8(FP)
+ RET
+L3:
+L2:
+ LTDBR F0, F0
+ BEQ L1
+ FMOVD F0, F2
+ WORD $0xED209040 //mdb %f2,.L10-.L9(%r9)
+ BYTE $0x00
+ BYTE $0x1C
+ MOVH $0x200, R4
+ LGDR F2, R2
+ SRAD $32, R2
+L4:
+ RISBGZ $57, $62, $39, R2, R3
+ MOVD $·cbrttab12067<>+0(SB), R1
+ WORD $0x48131000 //lh %r1,0(%r3,%r1)
+ RISBGZ $57, $62, $45, R2, R3
+ MOVD $·cbrttab22068<>+0(SB), R5
+ RISBGNZ $60, $63, $48, R2, R2
+ WORD $0x4A135000 //ah %r1,0(%r3,%r5)
+ BYTE $0x18 //lr %r3,%r1
+ BYTE $0x31
+ MOVD $·cbrttab32069<>+0(SB), R1
+ FMOVD 56(R9), F1
+ FMOVD 48(R9), F5
+ WORD $0xEC23393B //rosbg %r2,%r3,57,59,4
+ BYTE $0x04
+ BYTE $0x56
+ WORD $0xE3121000 //llc %r1,0(%r2,%r1)
+ BYTE $0x00
+ BYTE $0x94
+ ADDW R3, R1
+ ADDW R4, R1
+ SLW $16, R1, R1
+ SLD $32, R1, R1
+ LDGR R1, F2
+ WFMDB V2, V2, V4
+ WFMDB V4, V0, V6
+ WFMSDB V4, V6, V2, V4
+ FMOVD 40(R9), F6
+ FMSUB F1, F4, F2
+ FMOVD 32(R9), F4
+ WFMDB V2, V2, V3
+ FMOVD 24(R9), F1
+ FMUL F3, F0
+ FMOVD 16(R9), F3
+ WFMADB V2, V0, V5, V2
+ FMOVD 8(R9), F5
+ FMADD F6, F2, F4
+ WFMADB V2, V1, V3, V1
+ WFMDB V2, V2, V6
+ FMOVD 0(R9), F3
+ WFMADB V4, V6, V1, V4
+ WFMADB V2, V5, V3, V2
+ FMADD F4, F6, F2
+ FMADD F2, F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+L8:
+ MOVH $0x0, R4
+ BR L4
diff --git a/src/math/cmplx/abs.go b/src/math/cmplx/abs.go
new file mode 100644
index 0000000..2f89d1b
--- /dev/null
+++ b/src/math/cmplx/abs.go
@@ -0,0 +1,13 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package cmplx provides basic constants and mathematical functions for
+// complex numbers. Special case handling conforms to the C99 standard
+// Annex G IEC 60559-compatible complex arithmetic.
+package cmplx
+
+import "math"
+
+// Abs returns the absolute value (also called the modulus) of x.
+func Abs(x complex128) float64 { return math.Hypot(real(x), imag(x)) }
diff --git a/src/math/cmplx/asin.go b/src/math/cmplx/asin.go
new file mode 100644
index 0000000..30d019e
--- /dev/null
+++ b/src/math/cmplx/asin.go
@@ -0,0 +1,221 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex circular arc sine
+//
+// DESCRIPTION:
+//
+// Inverse complex sine:
+// 2
+// w = -i clog( iz + csqrt( 1 - z ) ).
+//
+// casin(z) = -i casinh(iz)
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 10100 2.1e-15 3.4e-16
+// IEEE -10,+10 30000 2.2e-14 2.7e-15
+// Larger relative error can be observed for z near zero.
+// Also tested by csin(casin(z)) = z.
+
+// Asin returns the inverse sine of x.
+func Asin(x complex128) complex128 {
+ switch re, im := real(x), imag(x); {
+ case im == 0 && math.Abs(re) <= 1:
+ return complex(math.Asin(re), im)
+ case re == 0 && math.Abs(im) <= 1:
+ return complex(re, math.Asinh(im))
+ case math.IsNaN(im):
+ switch {
+ case re == 0:
+ return complex(re, math.NaN())
+ case math.IsInf(re, 0):
+ return complex(math.NaN(), re)
+ default:
+ return NaN()
+ }
+ case math.IsInf(im, 0):
+ switch {
+ case math.IsNaN(re):
+ return x
+ case math.IsInf(re, 0):
+ return complex(math.Copysign(math.Pi/4, re), im)
+ default:
+ return complex(math.Copysign(0, re), im)
+ }
+ case math.IsInf(re, 0):
+ return complex(math.Copysign(math.Pi/2, re), math.Copysign(re, im))
+ }
+ ct := complex(-imag(x), real(x)) // i * x
+ xx := x * x
+ x1 := complex(1-real(xx), -imag(xx)) // 1 - x*x
+ x2 := Sqrt(x1) // x2 = sqrt(1 - x*x)
+ w := Log(ct + x2)
+ return complex(imag(w), -real(w)) // -i * w
+}
+
+// Asinh returns the inverse hyperbolic sine of x.
+func Asinh(x complex128) complex128 {
+ switch re, im := real(x), imag(x); {
+ case im == 0 && math.Abs(re) <= 1:
+ return complex(math.Asinh(re), im)
+ case re == 0 && math.Abs(im) <= 1:
+ return complex(re, math.Asin(im))
+ case math.IsInf(re, 0):
+ switch {
+ case math.IsInf(im, 0):
+ return complex(re, math.Copysign(math.Pi/4, im))
+ case math.IsNaN(im):
+ return x
+ default:
+ return complex(re, math.Copysign(0.0, im))
+ }
+ case math.IsNaN(re):
+ switch {
+ case im == 0:
+ return x
+ case math.IsInf(im, 0):
+ return complex(im, re)
+ default:
+ return NaN()
+ }
+ case math.IsInf(im, 0):
+ return complex(math.Copysign(im, re), math.Copysign(math.Pi/2, im))
+ }
+ xx := x * x
+ x1 := complex(1+real(xx), imag(xx)) // 1 + x*x
+ return Log(x + Sqrt(x1)) // log(x + sqrt(1 + x*x))
+}
+
+// Complex circular arc cosine
+//
+// DESCRIPTION:
+//
+// w = arccos z = PI/2 - arcsin z.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 5200 1.6e-15 2.8e-16
+// IEEE -10,+10 30000 1.8e-14 2.2e-15
+
+// Acos returns the inverse cosine of x.
+func Acos(x complex128) complex128 {
+ w := Asin(x)
+ return complex(math.Pi/2-real(w), -imag(w))
+}
+
+// Acosh returns the inverse hyperbolic cosine of x.
+func Acosh(x complex128) complex128 {
+ if x == 0 {
+ return complex(0, math.Copysign(math.Pi/2, imag(x)))
+ }
+ w := Acos(x)
+ if imag(w) <= 0 {
+ return complex(-imag(w), real(w)) // i * w
+ }
+ return complex(imag(w), -real(w)) // -i * w
+}
+
+// Complex circular arc tangent
+//
+// DESCRIPTION:
+//
+// If
+// z = x + iy,
+//
+// then
+// 1 ( 2x )
+// Re w = - arctan(-----------) + k PI
+// 2 ( 2 2)
+// (1 - x - y )
+//
+// ( 2 2)
+// 1 (x + (y+1) )
+// Im w = - log(------------)
+// 4 ( 2 2)
+// (x + (y-1) )
+//
+// Where k is an arbitrary integer.
+//
+// catan(z) = -i catanh(iz).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 5900 1.3e-16 7.8e-18
+// IEEE -10,+10 30000 2.3e-15 8.5e-17
+// The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
+// had peak relative error 1.5e-16, rms relative error
+// 2.9e-17. See also clog().
+
+// Atan returns the inverse tangent of x.
+func Atan(x complex128) complex128 {
+ switch re, im := real(x), imag(x); {
+ case im == 0:
+ return complex(math.Atan(re), im)
+ case re == 0 && math.Abs(im) <= 1:
+ return complex(re, math.Atanh(im))
+ case math.IsInf(im, 0) || math.IsInf(re, 0):
+ if math.IsNaN(re) {
+ return complex(math.NaN(), math.Copysign(0, im))
+ }
+ return complex(math.Copysign(math.Pi/2, re), math.Copysign(0, im))
+ case math.IsNaN(re) || math.IsNaN(im):
+ return NaN()
+ }
+ x2 := real(x) * real(x)
+ a := 1 - x2 - imag(x)*imag(x)
+ if a == 0 {
+ return NaN()
+ }
+ t := 0.5 * math.Atan2(2*real(x), a)
+ w := reducePi(t)
+
+ t = imag(x) - 1
+ b := x2 + t*t
+ if b == 0 {
+ return NaN()
+ }
+ t = imag(x) + 1
+ c := (x2 + t*t) / b
+ return complex(w, 0.25*math.Log(c))
+}
+
+// Atanh returns the inverse hyperbolic tangent of x.
+func Atanh(x complex128) complex128 {
+ z := complex(-imag(x), real(x)) // z = i * x
+ z = Atan(z)
+ return complex(imag(z), -real(z)) // z = -i * z
+}
diff --git a/src/math/cmplx/cmath_test.go b/src/math/cmplx/cmath_test.go
new file mode 100644
index 0000000..3011e83
--- /dev/null
+++ b/src/math/cmplx/cmath_test.go
@@ -0,0 +1,1589 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import (
+ "math"
+ "testing"
+)
+
+// The higher-precision values in vc26 were used to derive the
+// input arguments vc (see also comment below). For reference
+// only (do not delete).
+var vc26 = []complex128{
+ (4.97901192488367350108546816 + 7.73887247457810456552351752i),
+ (7.73887247457810456552351752 - 0.27688005719200159404635997i),
+ (-0.27688005719200159404635997 - 5.01060361827107492160848778i),
+ (-5.01060361827107492160848778 + 9.63629370719841737980004837i),
+ (9.63629370719841737980004837 + 2.92637723924396464525443662i),
+ (2.92637723924396464525443662 + 5.22908343145930665230025625i),
+ (5.22908343145930665230025625 + 2.72793991043601025126008608i),
+ (2.72793991043601025126008608 + 1.82530809168085506044576505i),
+ (1.82530809168085506044576505 - 8.68592476857560136238589621i),
+ (-8.68592476857560136238589621 + 4.97901192488367350108546816i),
+}
+
+var vc = []complex128{
+ (4.9790119248836735e+00 + 7.7388724745781045e+00i),
+ (7.7388724745781045e+00 - 2.7688005719200159e-01i),
+ (-2.7688005719200159e-01 - 5.0106036182710749e+00i),
+ (-5.0106036182710749e+00 + 9.6362937071984173e+00i),
+ (9.6362937071984173e+00 + 2.9263772392439646e+00i),
+ (2.9263772392439646e+00 + 5.2290834314593066e+00i),
+ (5.2290834314593066e+00 + 2.7279399104360102e+00i),
+ (2.7279399104360102e+00 + 1.8253080916808550e+00i),
+ (1.8253080916808550e+00 - 8.6859247685756013e+00i),
+ (-8.6859247685756013e+00 + 4.9790119248836735e+00i),
+}
+
+// The expected results below were computed by the high precision calculators
+// at https://keisan.casio.com/. More exact input values (array vc[], above)
+// were obtained by printing them with "%.26f". The answers were calculated
+// to 26 digits (by using the "Digit number" drop-down control of each
+// calculator).
+
+var abs = []float64{
+ 9.2022120669932650313380972e+00,
+ 7.7438239742296106616261394e+00,
+ 5.0182478202557746902556648e+00,
+ 1.0861137372799545160704002e+01,
+ 1.0070841084922199607011905e+01,
+ 5.9922447613166942183705192e+00,
+ 5.8978784056736762299945176e+00,
+ 3.2822866700678709020367184e+00,
+ 8.8756430028990417290744307e+00,
+ 1.0011785496777731986390856e+01,
+}
+
+var acos = []complex128{
+ (1.0017679804707456328694569 - 2.9138232718554953784519807i),
+ (0.03606427612041407369636057 + 2.7358584434576260925091256i),
+ (1.6249365462333796703711823 + 2.3159537454335901187730929i),
+ (2.0485650849650740120660391 - 3.0795576791204117911123886i),
+ (0.29621132089073067282488147 - 3.0007392508200622519398814i),
+ (1.0664555914934156601503632 - 2.4872865024796011364747111i),
+ (0.48681307452231387690013905 - 2.463655912283054555225301i),
+ (0.6116977071277574248407752 - 1.8734458851737055262693056i),
+ (1.3649311280370181331184214 + 2.8793528632328795424123832i),
+ (2.6189310485682988308904501 - 2.9956543302898767795858704i),
+}
+var acosh = []complex128{
+ (2.9138232718554953784519807 + 1.0017679804707456328694569i),
+ (2.7358584434576260925091256 - 0.03606427612041407369636057i),
+ (2.3159537454335901187730929 - 1.6249365462333796703711823i),
+ (3.0795576791204117911123886 + 2.0485650849650740120660391i),
+ (3.0007392508200622519398814 + 0.29621132089073067282488147i),
+ (2.4872865024796011364747111 + 1.0664555914934156601503632i),
+ (2.463655912283054555225301 + 0.48681307452231387690013905i),
+ (1.8734458851737055262693056 + 0.6116977071277574248407752i),
+ (2.8793528632328795424123832 - 1.3649311280370181331184214i),
+ (2.9956543302898767795858704 + 2.6189310485682988308904501i),
+}
+var asin = []complex128{
+ (0.56902834632415098636186476 + 2.9138232718554953784519807i),
+ (1.5347320506744825455349611 - 2.7358584434576260925091256i),
+ (-0.054140219438483051139860579 - 2.3159537454335901187730929i),
+ (-0.47776875817017739283471738 + 3.0795576791204117911123886i),
+ (1.2745850059041659464064402 + 3.0007392508200622519398814i),
+ (0.50434073530148095908095852 + 2.4872865024796011364747111i),
+ (1.0839832522725827423311826 + 2.463655912283054555225301i),
+ (0.9590986196671391943905465 + 1.8734458851737055262693056i),
+ (0.20586519875787848611290031 - 2.8793528632328795424123832i),
+ (-1.0481347217734022116591284 + 2.9956543302898767795858704i),
+}
+var asinh = []complex128{
+ (2.9113760469415295679342185 + 0.99639459545704326759805893i),
+ (2.7441755423994259061579029 - 0.035468308789000500601119392i),
+ (-2.2962136462520690506126678 - 1.5144663565690151885726707i),
+ (-3.0771233459295725965402455 + 1.0895577967194013849422294i),
+ (3.0048366100923647417557027 + 0.29346979169819220036454168i),
+ (2.4800059370795363157364643 + 1.0545868606049165710424232i),
+ (2.4718773838309585611141821 + 0.47502344364250803363708842i),
+ (1.8910743588080159144378396 + 0.56882925572563602341139174i),
+ (2.8735426423367341878069406 - 1.362376149648891420997548i),
+ (-2.9981750586172477217567878 + 0.5183571985225367505624207i),
+}
+var atan = []complex128{
+ (1.5115747079332741358607654 + 0.091324403603954494382276776i),
+ (1.4424504323482602560806727 - 0.0045416132642803911503770933i),
+ (-1.5593488703630532674484026 - 0.20163295409248362456446431i),
+ (-1.5280619472445889867794105 + 0.081721556230672003746956324i),
+ (1.4759909163240799678221039 + 0.028602969320691644358773586i),
+ (1.4877353772046548932715555 + 0.14566877153207281663773599i),
+ (1.4206983927779191889826 + 0.076830486127880702249439993i),
+ (1.3162236060498933364869556 + 0.16031313000467530644933363i),
+ (1.5473450684303703578810093 - 0.11064907507939082484935782i),
+ (-1.4841462340185253987375812 + 0.049341850305024399493142411i),
+}
+var atanh = []complex128{
+ (0.058375027938968509064640438 + 1.4793488495105334458167782i),
+ (0.12977343497790381229915667 - 1.5661009410463561327262499i),
+ (-0.010576456067347252072200088 - 1.3743698658402284549750563i),
+ (-0.042218595678688358882784918 + 1.4891433968166405606692604i),
+ (0.095218997991316722061828397 + 1.5416884098777110330499698i),
+ (0.079965459366890323857556487 + 1.4252510353873192700350435i),
+ (0.15051245471980726221708301 + 1.4907432533016303804884461i),
+ (0.25082072933993987714470373 + 1.392057665392187516442986i),
+ (0.022896108815797135846276662 - 1.4609224989282864208963021i),
+ (-0.08665624101841876130537396 + 1.5207902036935093480142159i),
+}
+var conj = []complex128{
+ (4.9790119248836735e+00 - 7.7388724745781045e+00i),
+ (7.7388724745781045e+00 + 2.7688005719200159e-01i),
+ (-2.7688005719200159e-01 + 5.0106036182710749e+00i),
+ (-5.0106036182710749e+00 - 9.6362937071984173e+00i),
+ (9.6362937071984173e+00 - 2.9263772392439646e+00i),
+ (2.9263772392439646e+00 - 5.2290834314593066e+00i),
+ (5.2290834314593066e+00 - 2.7279399104360102e+00i),
+ (2.7279399104360102e+00 - 1.8253080916808550e+00i),
+ (1.8253080916808550e+00 + 8.6859247685756013e+00i),
+ (-8.6859247685756013e+00 - 4.9790119248836735e+00i),
+}
+var cos = []complex128{
+ (3.024540920601483938336569e+02 + 1.1073797572517071650045357e+03i),
+ (1.192858682649064973252758e-01 + 2.7857554122333065540970207e-01i),
+ (7.2144394304528306603857962e+01 - 2.0500129667076044169954205e+01i),
+ (2.24921952538403984190541e+03 - 7.317363745602773587049329e+03i),
+ (-9.148222970032421760015498e+00 + 1.953124661113563541862227e+00i),
+ (-9.116081175857732248227078e+01 - 1.992669213569952232487371e+01i),
+ (3.795639179042704640002918e+00 + 6.623513350981458399309662e+00i),
+ (-2.9144840732498869560679084e+00 - 1.214620271628002917638748e+00i),
+ (-7.45123482501299743872481e+02 + 2.8641692314488080814066734e+03i),
+ (-5.371977967039319076416747e+01 + 4.893348341339375830564624e+01i),
+}
+var cosh = []complex128{
+ (8.34638383523018249366948e+00 + 7.2181057886425846415112064e+01i),
+ (1.10421967379919366952251e+03 - 3.1379638689277575379469861e+02i),
+ (3.051485206773701584738512e-01 - 2.6805384730105297848044485e-01i),
+ (-7.33294728684187933370938e+01 + 1.574445942284918251038144e+01i),
+ (-7.478643293945957535757355e+03 + 1.6348382209913353929473321e+03i),
+ (4.622316522966235701630926e+00 - 8.088695185566375256093098e+00i),
+ (-8.544333183278877406197712e+01 + 3.7505836120128166455231717e+01i),
+ (-1.934457815021493925115198e+00 + 7.3725859611767228178358673e+00i),
+ (-2.352958770061749348353548e+00 - 2.034982010440878358915409e+00i),
+ (7.79756457532134748165069e+02 + 2.8549350716819176560377717e+03i),
+}
+var exp = []complex128{
+ (1.669197736864670815125146e+01 + 1.4436895109507663689174096e+02i),
+ (2.2084389286252583447276212e+03 - 6.2759289284909211238261917e+02i),
+ (2.227538273122775173434327e-01 + 7.2468284028334191250470034e-01i),
+ (-6.5182985958153548997881627e-03 - 1.39965837915193860879044e-03i),
+ (-1.4957286524084015746110777e+04 + 3.269676455931135688988042e+03i),
+ (9.218158701983105935659273e+00 - 1.6223985291084956009304582e+01i),
+ (-1.7088175716853040841444505e+02 + 7.501382609870410713795546e+01i),
+ (-3.852461315830959613132505e+00 + 1.4808420423156073221970892e+01i),
+ (-4.586775503301407379786695e+00 - 4.178501081246873415144744e+00i),
+ (4.451337963005453491095747e-05 - 1.62977574205442915935263e-04i),
+}
+var log = []complex128{
+ (2.2194438972179194425697051e+00 + 9.9909115046919291062461269e-01i),
+ (2.0468956191154167256337289e+00 - 3.5762575021856971295156489e-02i),
+ (1.6130808329853860438751244e+00 - 1.6259990074019058442232221e+00i),
+ (2.3851910394823008710032651e+00 + 2.0502936359659111755031062e+00i),
+ (2.3096442270679923004800651e+00 + 2.9483213155446756211881774e-01i),
+ (1.7904660933974656106951860e+00 + 1.0605860367252556281902109e+00i),
+ (1.7745926939841751666177512e+00 + 4.8084556083358307819310911e-01i),
+ (1.1885403350045342425648780e+00 + 5.8969634164776659423195222e-01i),
+ (2.1833107837679082586772505e+00 - 1.3636647724582455028314573e+00i),
+ (2.3037629487273259170991671e+00 + 2.6210913895386013290915234e+00i),
+}
+var log10 = []complex128{
+ (9.6389223745559042474184943e-01 + 4.338997735671419492599631e-01i),
+ (8.8895547241376579493490892e-01 - 1.5531488990643548254864806e-02i),
+ (7.0055210462945412305244578e-01 - 7.0616239649481243222248404e-01i),
+ (1.0358753067322445311676952e+00 + 8.9043121238134980156490909e-01i),
+ (1.003065742975330237172029e+00 + 1.2804396782187887479857811e-01i),
+ (7.7758954439739162532085157e-01 + 4.6060666333341810869055108e-01i),
+ (7.7069581462315327037689152e-01 + 2.0882857371769952195512475e-01i),
+ (5.1617650901191156135137239e-01 + 2.5610186717615977620363299e-01i),
+ (9.4819982567026639742663212e-01 - 5.9223208584446952284914289e-01i),
+ (1.0005115362454417135973429e+00 + 1.1383255270407412817250921e+00i),
+}
+
+type ff struct {
+ r, theta float64
+}
+
+var polar = []ff{
+ {9.2022120669932650313380972e+00, 9.9909115046919291062461269e-01},
+ {7.7438239742296106616261394e+00, -3.5762575021856971295156489e-02},
+ {5.0182478202557746902556648e+00, -1.6259990074019058442232221e+00},
+ {1.0861137372799545160704002e+01, 2.0502936359659111755031062e+00},
+ {1.0070841084922199607011905e+01, 2.9483213155446756211881774e-01},
+ {5.9922447613166942183705192e+00, 1.0605860367252556281902109e+00},
+ {5.8978784056736762299945176e+00, 4.8084556083358307819310911e-01},
+ {3.2822866700678709020367184e+00, 5.8969634164776659423195222e-01},
+ {8.8756430028990417290744307e+00, -1.3636647724582455028314573e+00},
+ {1.0011785496777731986390856e+01, 2.6210913895386013290915234e+00},
+}
+var pow = []complex128{
+ (-2.499956739197529585028819e+00 + 1.759751724335650228957144e+00i),
+ (7.357094338218116311191939e+04 - 5.089973412479151648145882e+04i),
+ (1.320777296067768517259592e+01 - 3.165621914333901498921986e+01i),
+ (-3.123287828297300934072149e-07 - 1.9849567521490553032502223e-7i),
+ (8.0622651468477229614813e+04 - 7.80028727944573092944363e+04i),
+ (-1.0268824572103165858577141e+00 - 4.716844738244989776610672e-01i),
+ (-4.35953819012244175753187e+01 + 2.2036445974645306917648585e+02i),
+ (8.3556092283250594950239e-01 - 1.2261571947167240272593282e+01i),
+ (1.582292972120769306069625e+03 + 1.273564263524278244782512e+04i),
+ (6.592208301642122149025369e-08 + 2.584887236651661903526389e-08i),
+}
+var sin = []complex128{
+ (-1.1073801774240233539648544e+03 + 3.024539773002502192425231e+02i),
+ (1.0317037521400759359744682e+00 - 3.2208979799929570242818e-02i),
+ (-2.0501952097271429804261058e+01 - 7.2137981348240798841800967e+01i),
+ (7.3173638080346338642193078e+03 + 2.249219506193664342566248e+03i),
+ (-1.964375633631808177565226e+00 - 9.0958264713870404464159683e+00i),
+ (1.992783647158514838337674e+01 - 9.11555769410191350416942e+01i),
+ (-6.680335650741921444300349e+00 + 3.763353833142432513086117e+00i),
+ (1.2794028166657459148245993e+00 - 2.7669092099795781155109602e+00i),
+ (2.8641693949535259594188879e+03 + 7.451234399649871202841615e+02i),
+ (-4.893811726244659135553033e+01 - 5.371469305562194635957655e+01i),
+}
+var sinh = []complex128{
+ (8.34559353341652565758198e+00 + 7.2187893208650790476628899e+01i),
+ (1.1042192548260646752051112e+03 - 3.1379650595631635858792056e+02i),
+ (-8.239469336509264113041849e-02 + 9.9273668758439489098514519e-01i),
+ (7.332295456982297798219401e+01 - 1.574585908122833444899023e+01i),
+ (-7.4786432301380582103534216e+03 + 1.63483823493980029604071e+03i),
+ (4.595842179016870234028347e+00 - 8.135290105518580753211484e+00i),
+ (-8.543842533574163435246793e+01 + 3.750798997857594068272375e+01i),
+ (-1.918003500809465688017307e+00 + 7.4358344619793504041350251e+00i),
+ (-2.233816733239658031433147e+00 - 2.143519070805995056229335e+00i),
+ (-7.797564130187551181105341e+02 - 2.8549352346594918614806877e+03i),
+}
+var sqrt = []complex128{
+ (2.6628203086086130543813948e+00 + 1.4531345674282185229796902e+00i),
+ (2.7823278427251986247149295e+00 - 4.9756907317005224529115567e-02i),
+ (1.5397025302089642757361015e+00 - 1.6271336573016637535695727e+00i),
+ (1.7103411581506875260277898e+00 + 2.8170677122737589676157029e+00i),
+ (3.1390392472953103383607947e+00 + 4.6612625849858653248980849e-01i),
+ (2.1117080764822417640789287e+00 + 1.2381170223514273234967850e+00i),
+ (2.3587032281672256703926939e+00 + 5.7827111903257349935720172e-01i),
+ (1.7335262588873410476661577e+00 + 5.2647258220721269141550382e-01i),
+ (2.3131094974708716531499282e+00 - 1.8775429304303785570775490e+00i),
+ (8.1420535745048086240947359e-01 + 3.0575897587277248522656113e+00i),
+}
+var tan = []complex128{
+ (-1.928757919086441129134525e-07 + 1.0000003267499169073251826e+00i),
+ (1.242412685364183792138948e+00 - 3.17149693883133370106696e+00i),
+ (-4.6745126251587795225571826e-05 - 9.9992439225263959286114298e-01i),
+ (4.792363401193648192887116e-09 + 1.0000000070589333451557723e+00i),
+ (2.345740824080089140287315e-03 + 9.947733046570988661022763e-01i),
+ (-2.396030789494815566088809e-05 + 9.9994781345418591429826779e-01i),
+ (-7.370204836644931340905303e-03 + 1.0043553413417138987717748e+00i),
+ (-3.691803847992048527007457e-02 + 9.6475071993469548066328894e-01i),
+ (-2.781955256713729368401878e-08 - 1.000000049848910609006646e+00i),
+ (9.4281590064030478879791249e-05 + 9.9999119340863718183758545e-01i),
+}
+var tanh = []complex128{
+ (1.0000921981225144748819918e+00 + 2.160986245871518020231507e-05i),
+ (9.9999967727531993209562591e-01 - 1.9953763222959658873657676e-07i),
+ (-1.765485739548037260789686e+00 + 1.7024216325552852445168471e+00i),
+ (-9.999189442732736452807108e-01 + 3.64906070494473701938098e-05i),
+ (9.9999999224622333738729767e-01 - 3.560088949517914774813046e-09i),
+ (1.0029324933367326862499343e+00 - 4.948790309797102353137528e-03i),
+ (9.9996113064788012488693567e-01 - 4.226995742097032481451259e-05i),
+ (1.0074784189316340029873945e+00 - 4.194050814891697808029407e-03i),
+ (9.9385534229718327109131502e-01 + 5.144217985914355502713437e-02i),
+ (-1.0000000491604982429364892e+00 - 2.901873195374433112227349e-08i),
+}
+
+// huge values along the real axis for testing reducePi in Tan
+var hugeIn = []complex128{
+ 1 << 28,
+ 1 << 29,
+ 1 << 30,
+ 1 << 35,
+ -1 << 120,
+ 1 << 240,
+ 1 << 300,
+ -1 << 480,
+ 1234567891234567 << 180,
+ -1234567891234567 << 300,
+}
+
+// Results for tanHuge[i] calculated with https://github.com/robpike/ivy
+// using 4096 bits of working precision.
+var tanHuge = []complex128{
+ 5.95641897939639421,
+ -0.34551069233430392,
+ -0.78469661331920043,
+ 0.84276385870875983,
+ 0.40806638884180424,
+ -0.37603456702698076,
+ 4.60901287677810962,
+ 3.39135965054779932,
+ -6.76813854009065030,
+ -0.76417695016604922,
+}
+
+// special cases conform to C99 standard appendix G.6 Complex arithmetic
+var inf, nan = math.Inf(1), math.NaN()
+
+var vcAbsSC = []complex128{
+ NaN(),
+}
+var absSC = []float64{
+ math.NaN(),
+}
+var acosSC = []struct {
+ in,
+ want complex128
+}{
+ // G.6.1.1
+ {complex(zero, zero),
+ complex(math.Pi/2, -zero)},
+ {complex(-zero, zero),
+ complex(math.Pi/2, -zero)},
+ {complex(zero, nan),
+ complex(math.Pi/2, nan)},
+ {complex(-zero, nan),
+ complex(math.Pi/2, nan)},
+ {complex(1.0, inf),
+ complex(math.Pi/2, -inf)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(-inf, 1.0),
+ complex(math.Pi, -inf)},
+ {complex(inf, 1.0),
+ complex(0.0, -inf)},
+ {complex(-inf, inf),
+ complex(3*math.Pi/4, -inf)},
+ {complex(inf, inf),
+ complex(math.Pi/4, -inf)},
+ {complex(inf, nan),
+ complex(nan, -inf)}, // imaginary sign unspecified
+ {complex(-inf, nan),
+ complex(nan, inf)}, // imaginary sign unspecified
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ complex(nan, -inf)},
+ {NaN(),
+ NaN()},
+}
+var acoshSC = []struct {
+ in,
+ want complex128
+}{
+ // G.6.2.1
+ {complex(zero, zero),
+ complex(zero, math.Pi/2)},
+ {complex(-zero, zero),
+ complex(zero, math.Pi/2)},
+ {complex(1.0, inf),
+ complex(inf, math.Pi/2)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(-inf, 1.0),
+ complex(inf, math.Pi)},
+ {complex(inf, 1.0),
+ complex(inf, zero)},
+ {complex(-inf, inf),
+ complex(inf, 3*math.Pi/4)},
+ {complex(inf, inf),
+ complex(inf, math.Pi/4)},
+ {complex(inf, nan),
+ complex(inf, nan)},
+ {complex(-inf, nan),
+ complex(inf, nan)},
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ complex(inf, nan)},
+ {NaN(),
+ NaN()},
+}
+var asinSC = []struct {
+ in,
+ want complex128
+}{
+ // Derived from Asin(z) = -i * Asinh(i * z), G.6 #7
+ {complex(zero, zero),
+ complex(zero, zero)},
+ {complex(1.0, inf),
+ complex(0, inf)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, 1),
+ complex(math.Pi/2, inf)},
+ {complex(inf, inf),
+ complex(math.Pi/4, inf)},
+ {complex(inf, nan),
+ complex(nan, inf)}, // imaginary sign unspecified
+ {complex(nan, zero),
+ NaN()},
+ {complex(nan, 1),
+ NaN()},
+ {complex(nan, inf),
+ complex(nan, inf)},
+ {NaN(),
+ NaN()},
+}
+var asinhSC = []struct {
+ in,
+ want complex128
+}{
+ // G.6.2.2
+ {complex(zero, zero),
+ complex(zero, zero)},
+ {complex(1.0, inf),
+ complex(inf, math.Pi/2)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, 1.0),
+ complex(inf, zero)},
+ {complex(inf, inf),
+ complex(inf, math.Pi/4)},
+ {complex(inf, nan),
+ complex(inf, nan)},
+ {complex(nan, zero),
+ complex(nan, zero)},
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ complex(inf, nan)}, // sign of real part unspecified
+ {NaN(),
+ NaN()},
+}
+var atanSC = []struct {
+ in,
+ want complex128
+}{
+ // Derived from Atan(z) = -i * Atanh(i * z), G.6 #7
+ {complex(0, zero),
+ complex(0, zero)},
+ {complex(0, nan),
+ NaN()},
+ {complex(1.0, zero),
+ complex(math.Pi/4, zero)},
+ {complex(1.0, inf),
+ complex(math.Pi/2, zero)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, 1),
+ complex(math.Pi/2, zero)},
+ {complex(inf, inf),
+ complex(math.Pi/2, zero)},
+ {complex(inf, nan),
+ complex(math.Pi/2, zero)},
+ {complex(nan, 1),
+ NaN()},
+ {complex(nan, inf),
+ complex(nan, zero)},
+ {NaN(),
+ NaN()},
+}
+var atanhSC = []struct {
+ in,
+ want complex128
+}{
+ // G.6.2.3
+ {complex(zero, zero),
+ complex(zero, zero)},
+ {complex(zero, nan),
+ complex(zero, nan)},
+ {complex(1.0, zero),
+ complex(inf, zero)},
+ {complex(1.0, inf),
+ complex(0, math.Pi/2)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, 1.0),
+ complex(zero, math.Pi/2)},
+ {complex(inf, inf),
+ complex(zero, math.Pi/2)},
+ {complex(inf, nan),
+ complex(0, nan)},
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ complex(zero, math.Pi/2)}, // sign of real part not specified.
+ {NaN(),
+ NaN()},
+}
+var vcConjSC = []complex128{
+ NaN(),
+}
+var conjSC = []complex128{
+ NaN(),
+}
+var cosSC = []struct {
+ in,
+ want complex128
+}{
+ // Derived from Cos(z) = Cosh(i * z), G.6 #7
+ {complex(zero, zero),
+ complex(1.0, -zero)},
+ {complex(zero, inf),
+ complex(inf, -zero)},
+ {complex(zero, nan),
+ complex(nan, zero)}, // imaginary sign unspecified
+ {complex(1.0, inf),
+ complex(inf, -inf)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, zero),
+ complex(nan, -zero)},
+ {complex(inf, 1.0),
+ NaN()},
+ {complex(inf, inf),
+ complex(inf, nan)}, // real sign unspecified
+ {complex(inf, nan),
+ NaN()},
+ {complex(nan, zero),
+ complex(nan, -zero)}, // imaginary sign unspecified
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ complex(inf, nan)},
+ {NaN(),
+ NaN()},
+}
+var coshSC = []struct {
+ in,
+ want complex128
+}{
+ // G.6.2.4
+ {complex(zero, zero),
+ complex(1.0, zero)},
+ {complex(zero, inf),
+ complex(nan, zero)}, // imaginary sign unspecified
+ {complex(zero, nan),
+ complex(nan, zero)}, // imaginary sign unspecified
+ {complex(1.0, inf),
+ NaN()},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, zero),
+ complex(inf, zero)},
+ {complex(inf, 1.0),
+ complex(inf*math.Cos(1.0), inf*math.Sin(1.0))}, // +inf cis(y)
+ {complex(inf, inf),
+ complex(inf, nan)}, // real sign unspecified
+ {complex(inf, nan),
+ complex(inf, nan)},
+ {complex(nan, zero),
+ complex(nan, zero)}, // imaginary sign unspecified
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ NaN()},
+ {NaN(),
+ NaN()},
+}
+var expSC = []struct {
+ in,
+ want complex128
+}{
+ // G.6.3.1
+ {complex(zero, zero),
+ complex(1.0, zero)},
+ {complex(-zero, zero),
+ complex(1.0, zero)},
+ {complex(1.0, inf),
+ NaN()},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, zero),
+ complex(inf, zero)},
+ {complex(-inf, 1.0),
+ complex(math.Copysign(0.0, math.Cos(1.0)), math.Copysign(0.0, math.Sin(1.0)))}, // +0 cis(y)
+ {complex(inf, 1.0),
+ complex(inf*math.Cos(1.0), inf*math.Sin(1.0))}, // +inf cis(y)
+ {complex(-inf, inf),
+ complex(zero, zero)}, // real and imaginary sign unspecified
+ {complex(inf, inf),
+ complex(inf, nan)}, // real sign unspecified
+ {complex(-inf, nan),
+ complex(zero, zero)}, // real and imaginary sign unspecified
+ {complex(inf, nan),
+ complex(inf, nan)}, // real sign unspecified
+ {complex(nan, zero),
+ complex(nan, zero)},
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ NaN()},
+ {NaN(),
+ NaN()},
+}
+var vcIsNaNSC = []complex128{
+ complex(math.Inf(-1), math.Inf(-1)),
+ complex(math.Inf(-1), math.NaN()),
+ complex(math.NaN(), math.Inf(-1)),
+ complex(0, math.NaN()),
+ complex(math.NaN(), 0),
+ complex(math.Inf(1), math.Inf(1)),
+ complex(math.Inf(1), math.NaN()),
+ complex(math.NaN(), math.Inf(1)),
+ complex(math.NaN(), math.NaN()),
+}
+var isNaNSC = []bool{
+ false,
+ false,
+ false,
+ true,
+ true,
+ false,
+ false,
+ false,
+ true,
+}
+
+var logSC = []struct {
+ in,
+ want complex128
+}{
+ // G.6.3.2
+ {complex(zero, zero),
+ complex(-inf, zero)},
+ {complex(-zero, zero),
+ complex(-inf, math.Pi)},
+ {complex(1.0, inf),
+ complex(inf, math.Pi/2)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(-inf, 1.0),
+ complex(inf, math.Pi)},
+ {complex(inf, 1.0),
+ complex(inf, 0.0)},
+ {complex(-inf, inf),
+ complex(inf, 3*math.Pi/4)},
+ {complex(inf, inf),
+ complex(inf, math.Pi/4)},
+ {complex(-inf, nan),
+ complex(inf, nan)},
+ {complex(inf, nan),
+ complex(inf, nan)},
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ complex(inf, nan)},
+ {NaN(),
+ NaN()},
+}
+var log10SC = []struct {
+ in,
+ want complex128
+}{
+ // derived from Log special cases via Log10(x) = math.Log10E*Log(x)
+ {complex(zero, zero),
+ complex(-inf, zero)},
+ {complex(-zero, zero),
+ complex(-inf, float64(math.Log10E)*float64(math.Pi))},
+ {complex(1.0, inf),
+ complex(inf, float64(math.Log10E)*float64(math.Pi/2))},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(-inf, 1.0),
+ complex(inf, float64(math.Log10E)*float64(math.Pi))},
+ {complex(inf, 1.0),
+ complex(inf, 0.0)},
+ {complex(-inf, inf),
+ complex(inf, float64(math.Log10E)*float64(3*math.Pi/4))},
+ {complex(inf, inf),
+ complex(inf, float64(math.Log10E)*float64(math.Pi/4))},
+ {complex(-inf, nan),
+ complex(inf, nan)},
+ {complex(inf, nan),
+ complex(inf, nan)},
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ complex(inf, nan)},
+ {NaN(),
+ NaN()},
+}
+var vcPolarSC = []complex128{
+ NaN(),
+}
+var polarSC = []ff{
+ {math.NaN(), math.NaN()},
+}
+var vcPowSC = [][2]complex128{
+ {NaN(), NaN()},
+ {0, NaN()},
+}
+var powSC = []complex128{
+ NaN(),
+ NaN(),
+}
+var sinSC = []struct {
+ in,
+ want complex128
+}{
+ // Derived from Sin(z) = -i * Sinh(i * z), G.6 #7
+ {complex(zero, zero),
+ complex(zero, zero)},
+ {complex(zero, inf),
+ complex(zero, inf)},
+ {complex(zero, nan),
+ complex(zero, nan)},
+ {complex(1.0, inf),
+ complex(inf, inf)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, zero),
+ complex(nan, zero)},
+ {complex(inf, 1.0),
+ NaN()},
+ {complex(inf, inf),
+ complex(nan, inf)},
+ {complex(inf, nan),
+ NaN()},
+ {complex(nan, zero),
+ complex(nan, zero)},
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ complex(nan, inf)},
+ {NaN(),
+ NaN()},
+}
+
+var sinhSC = []struct {
+ in,
+ want complex128
+}{
+ // G.6.2.5
+ {complex(zero, zero),
+ complex(zero, zero)},
+ {complex(zero, inf),
+ complex(zero, nan)}, // real sign unspecified
+ {complex(zero, nan),
+ complex(zero, nan)}, // real sign unspecified
+ {complex(1.0, inf),
+ NaN()},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, zero),
+ complex(inf, zero)},
+ {complex(inf, 1.0),
+ complex(inf*math.Cos(1.0), inf*math.Sin(1.0))}, // +inf cis(y)
+ {complex(inf, inf),
+ complex(inf, nan)}, // real sign unspecified
+ {complex(inf, nan),
+ complex(inf, nan)}, // real sign unspecified
+ {complex(nan, zero),
+ complex(nan, zero)},
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ NaN()},
+ {NaN(),
+ NaN()},
+}
+
+var sqrtSC = []struct {
+ in,
+ want complex128
+}{
+ // G.6.4.2
+ {complex(zero, zero),
+ complex(zero, zero)},
+ {complex(-zero, zero),
+ complex(zero, zero)},
+ {complex(1.0, inf),
+ complex(inf, inf)},
+ {complex(nan, inf),
+ complex(inf, inf)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(-inf, 1.0),
+ complex(zero, inf)},
+ {complex(inf, 1.0),
+ complex(inf, zero)},
+ {complex(-inf, nan),
+ complex(nan, inf)}, // imaginary sign unspecified
+ {complex(inf, nan),
+ complex(inf, nan)},
+ {complex(nan, 1.0),
+ NaN()},
+ {NaN(),
+ NaN()},
+}
+var tanSC = []struct {
+ in,
+ want complex128
+}{
+ // Derived from Tan(z) = -i * Tanh(i * z), G.6 #7
+ {complex(zero, zero),
+ complex(zero, zero)},
+ {complex(zero, nan),
+ complex(zero, nan)},
+ {complex(1.0, inf),
+ complex(zero, 1.0)},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, 1.0),
+ NaN()},
+ {complex(inf, inf),
+ complex(zero, 1.0)},
+ {complex(inf, nan),
+ NaN()},
+ {complex(nan, zero),
+ NaN()},
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ complex(zero, 1.0)},
+ {NaN(),
+ NaN()},
+}
+var tanhSC = []struct {
+ in,
+ want complex128
+}{
+ // G.6.2.6
+ {complex(zero, zero),
+ complex(zero, zero)},
+ {complex(1.0, inf),
+ NaN()},
+ {complex(1.0, nan),
+ NaN()},
+ {complex(inf, 1.0),
+ complex(1.0, math.Copysign(0.0, math.Sin(2*1.0)))}, // 1 + i 0 sin(2y)
+ {complex(inf, inf),
+ complex(1.0, zero)}, // imaginary sign unspecified
+ {complex(inf, nan),
+ complex(1.0, zero)}, // imaginary sign unspecified
+ {complex(nan, zero),
+ complex(nan, zero)},
+ {complex(nan, 1.0),
+ NaN()},
+ {complex(nan, inf),
+ NaN()},
+ {NaN(),
+ NaN()},
+}
+
+// branch cut continuity checks
+// points on each axis at |z| > 1 are checked for one-sided continuity from both the positive and negative side
+// all possible branch cuts for the elementary functions are at one of these points
+
+var zero = 0.0
+var eps = 1.0 / (1 << 53)
+
+var branchPoints = [][2]complex128{
+ {complex(2.0, zero), complex(2.0, eps)},
+ {complex(2.0, -zero), complex(2.0, -eps)},
+ {complex(-2.0, zero), complex(-2.0, eps)},
+ {complex(-2.0, -zero), complex(-2.0, -eps)},
+ {complex(zero, 2.0), complex(eps, 2.0)},
+ {complex(-zero, 2.0), complex(-eps, 2.0)},
+ {complex(zero, -2.0), complex(eps, -2.0)},
+ {complex(-zero, -2.0), complex(-eps, -2.0)},
+}
+
+// functions borrowed from pkg/math/all_test.go
+func tolerance(a, b, e float64) bool {
+ d := a - b
+ if d < 0 {
+ d = -d
+ }
+
+ // note: b is correct (expected) value, a is actual value.
+ // make error tolerance a fraction of b, not a.
+ if b != 0 {
+ e = e * b
+ if e < 0 {
+ e = -e
+ }
+ }
+ return d < e
+}
+func veryclose(a, b float64) bool { return tolerance(a, b, 4e-16) }
+func alike(a, b float64) bool {
+ switch {
+ case a != a && b != b: // math.IsNaN(a) && math.IsNaN(b):
+ return true
+ case a == b:
+ return math.Signbit(a) == math.Signbit(b)
+ }
+ return false
+}
+
+func cTolerance(a, b complex128, e float64) bool {
+ d := Abs(a - b)
+ if b != 0 {
+ e = e * Abs(b)
+ if e < 0 {
+ e = -e
+ }
+ }
+ return d < e
+}
+func cSoclose(a, b complex128, e float64) bool { return cTolerance(a, b, e) }
+func cVeryclose(a, b complex128) bool { return cTolerance(a, b, 4e-16) }
+func cAlike(a, b complex128) bool {
+ var realAlike, imagAlike bool
+ if isExact(real(b)) {
+ realAlike = alike(real(a), real(b))
+ } else {
+ // Allow non-exact special cases to have errors in ULP.
+ realAlike = veryclose(real(a), real(b))
+ }
+ if isExact(imag(b)) {
+ imagAlike = alike(imag(a), imag(b))
+ } else {
+ // Allow non-exact special cases to have errors in ULP.
+ imagAlike = veryclose(imag(a), imag(b))
+ }
+ return realAlike && imagAlike
+}
+func isExact(x float64) bool {
+ // Special cases that should match exactly. Other cases are multiples
+ // of Pi that may not be last bit identical on all platforms.
+ return math.IsNaN(x) || math.IsInf(x, 0) || x == 0 || x == 1 || x == -1
+}
+
+func TestAbs(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Abs(vc[i]); !veryclose(abs[i], f) {
+ t.Errorf("Abs(%g) = %g, want %g", vc[i], f, abs[i])
+ }
+ }
+ for i := 0; i < len(vcAbsSC); i++ {
+ if f := Abs(vcAbsSC[i]); !alike(absSC[i], f) {
+ t.Errorf("Abs(%g) = %g, want %g", vcAbsSC[i], f, absSC[i])
+ }
+ }
+}
+func TestAcos(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Acos(vc[i]); !cSoclose(acos[i], f, 1e-14) {
+ t.Errorf("Acos(%g) = %g, want %g", vc[i], f, acos[i])
+ }
+ }
+ for _, v := range acosSC {
+ if f := Acos(v.in); !cAlike(v.want, f) {
+ t.Errorf("Acos(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Acos(Conj(z)) == Conj(Acos(z))
+ if f := Acos(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Acos(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ }
+ for _, pt := range branchPoints {
+ if f0, f1 := Acos(pt[0]), Acos(pt[1]); !cVeryclose(f0, f1) {
+ t.Errorf("Acos(%g) not continuous, got %g want %g", pt[0], f0, f1)
+ }
+ }
+}
+func TestAcosh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Acosh(vc[i]); !cSoclose(acosh[i], f, 1e-14) {
+ t.Errorf("Acosh(%g) = %g, want %g", vc[i], f, acosh[i])
+ }
+ }
+ for _, v := range acoshSC {
+ if f := Acosh(v.in); !cAlike(v.want, f) {
+ t.Errorf("Acosh(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Acosh(Conj(z)) == Conj(Acosh(z))
+ if f := Acosh(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Acosh(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+
+ }
+ for _, pt := range branchPoints {
+ if f0, f1 := Acosh(pt[0]), Acosh(pt[1]); !cVeryclose(f0, f1) {
+ t.Errorf("Acosh(%g) not continuous, got %g want %g", pt[0], f0, f1)
+ }
+ }
+}
+func TestAsin(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Asin(vc[i]); !cSoclose(asin[i], f, 1e-14) {
+ t.Errorf("Asin(%g) = %g, want %g", vc[i], f, asin[i])
+ }
+ }
+ for _, v := range asinSC {
+ if f := Asin(v.in); !cAlike(v.want, f) {
+ t.Errorf("Asin(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Asin(Conj(z)) == Asin(Sinh(z))
+ if f := Asin(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Asin(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ if math.IsNaN(real(v.in)) || math.IsNaN(real(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Asin(-z) == -Asin(z)
+ if f := Asin(-v.in); !cAlike(-v.want, f) && !cAlike(v.in, -v.in) {
+ t.Errorf("Asin(%g) = %g, want %g", -v.in, f, -v.want)
+ }
+ }
+ for _, pt := range branchPoints {
+ if f0, f1 := Asin(pt[0]), Asin(pt[1]); !cVeryclose(f0, f1) {
+ t.Errorf("Asin(%g) not continuous, got %g want %g", pt[0], f0, f1)
+ }
+ }
+}
+func TestAsinh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Asinh(vc[i]); !cSoclose(asinh[i], f, 4e-15) {
+ t.Errorf("Asinh(%g) = %g, want %g", vc[i], f, asinh[i])
+ }
+ }
+ for _, v := range asinhSC {
+ if f := Asinh(v.in); !cAlike(v.want, f) {
+ t.Errorf("Asinh(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Asinh(Conj(z)) == Asinh(Sinh(z))
+ if f := Asinh(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Asinh(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ if math.IsNaN(real(v.in)) || math.IsNaN(real(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Asinh(-z) == -Asinh(z)
+ if f := Asinh(-v.in); !cAlike(-v.want, f) && !cAlike(v.in, -v.in) {
+ t.Errorf("Asinh(%g) = %g, want %g", -v.in, f, -v.want)
+ }
+ }
+ for _, pt := range branchPoints {
+ if f0, f1 := Asinh(pt[0]), Asinh(pt[1]); !cVeryclose(f0, f1) {
+ t.Errorf("Asinh(%g) not continuous, got %g want %g", pt[0], f0, f1)
+ }
+ }
+}
+func TestAtan(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Atan(vc[i]); !cVeryclose(atan[i], f) {
+ t.Errorf("Atan(%g) = %g, want %g", vc[i], f, atan[i])
+ }
+ }
+ for _, v := range atanSC {
+ if f := Atan(v.in); !cAlike(v.want, f) {
+ t.Errorf("Atan(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Atan(Conj(z)) == Conj(Atan(z))
+ if f := Atan(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Atan(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ if math.IsNaN(real(v.in)) || math.IsNaN(real(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Atan(-z) == -Atan(z)
+ if f := Atan(-v.in); !cAlike(-v.want, f) && !cAlike(v.in, -v.in) {
+ t.Errorf("Atan(%g) = %g, want %g", -v.in, f, -v.want)
+ }
+ }
+ for _, pt := range branchPoints {
+ if f0, f1 := Atan(pt[0]), Atan(pt[1]); !cVeryclose(f0, f1) {
+ t.Errorf("Atan(%g) not continuous, got %g want %g", pt[0], f0, f1)
+ }
+ }
+}
+func TestAtanh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Atanh(vc[i]); !cVeryclose(atanh[i], f) {
+ t.Errorf("Atanh(%g) = %g, want %g", vc[i], f, atanh[i])
+ }
+ }
+ for _, v := range atanhSC {
+ if f := Atanh(v.in); !cAlike(v.want, f) {
+ t.Errorf("Atanh(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Atanh(Conj(z)) == Conj(Atanh(z))
+ if f := Atanh(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Atanh(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ if math.IsNaN(real(v.in)) || math.IsNaN(real(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Atanh(-z) == -Atanh(z)
+ if f := Atanh(-v.in); !cAlike(-v.want, f) && !cAlike(v.in, -v.in) {
+ t.Errorf("Atanh(%g) = %g, want %g", -v.in, f, -v.want)
+ }
+ }
+ for _, pt := range branchPoints {
+ if f0, f1 := Atanh(pt[0]), Atanh(pt[1]); !cVeryclose(f0, f1) {
+ t.Errorf("Atanh(%g) not continuous, got %g want %g", pt[0], f0, f1)
+ }
+ }
+}
+func TestConj(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Conj(vc[i]); !cVeryclose(conj[i], f) {
+ t.Errorf("Conj(%g) = %g, want %g", vc[i], f, conj[i])
+ }
+ }
+ for i := 0; i < len(vcConjSC); i++ {
+ if f := Conj(vcConjSC[i]); !cAlike(conjSC[i], f) {
+ t.Errorf("Conj(%g) = %g, want %g", vcConjSC[i], f, conjSC[i])
+ }
+ }
+}
+func TestCos(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Cos(vc[i]); !cSoclose(cos[i], f, 3e-15) {
+ t.Errorf("Cos(%g) = %g, want %g", vc[i], f, cos[i])
+ }
+ }
+ for _, v := range cosSC {
+ if f := Cos(v.in); !cAlike(v.want, f) {
+ t.Errorf("Cos(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Cos(Conj(z)) == Cos(Cosh(z))
+ if f := Cos(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Cos(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ if math.IsNaN(real(v.in)) || math.IsNaN(real(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Cos(-z) == Cos(z)
+ if f := Cos(-v.in); !cAlike(v.want, f) && !cAlike(v.in, -v.in) {
+ t.Errorf("Cos(%g) = %g, want %g", -v.in, f, v.want)
+ }
+ }
+}
+func TestCosh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Cosh(vc[i]); !cSoclose(cosh[i], f, 2e-15) {
+ t.Errorf("Cosh(%g) = %g, want %g", vc[i], f, cosh[i])
+ }
+ }
+ for _, v := range coshSC {
+ if f := Cosh(v.in); !cAlike(v.want, f) {
+ t.Errorf("Cosh(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Cosh(Conj(z)) == Conj(Cosh(z))
+ if f := Cosh(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Cosh(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ if math.IsNaN(real(v.in)) || math.IsNaN(real(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Cosh(-z) == Cosh(z)
+ if f := Cosh(-v.in); !cAlike(v.want, f) && !cAlike(v.in, -v.in) {
+ t.Errorf("Cosh(%g) = %g, want %g", -v.in, f, v.want)
+ }
+ }
+}
+func TestExp(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Exp(vc[i]); !cSoclose(exp[i], f, 1e-15) {
+ t.Errorf("Exp(%g) = %g, want %g", vc[i], f, exp[i])
+ }
+ }
+ for _, v := range expSC {
+ if f := Exp(v.in); !cAlike(v.want, f) {
+ t.Errorf("Exp(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Exp(Conj(z)) == Exp(Cosh(z))
+ if f := Exp(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Exp(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ }
+}
+func TestIsNaN(t *testing.T) {
+ for i := 0; i < len(vcIsNaNSC); i++ {
+ if f := IsNaN(vcIsNaNSC[i]); isNaNSC[i] != f {
+ t.Errorf("IsNaN(%v) = %v, want %v", vcIsNaNSC[i], f, isNaNSC[i])
+ }
+ }
+}
+func TestLog(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Log(vc[i]); !cVeryclose(log[i], f) {
+ t.Errorf("Log(%g) = %g, want %g", vc[i], f, log[i])
+ }
+ }
+ for _, v := range logSC {
+ if f := Log(v.in); !cAlike(v.want, f) {
+ t.Errorf("Log(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Log(Conj(z)) == Conj(Log(z))
+ if f := Log(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Log(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ }
+ for _, pt := range branchPoints {
+ if f0, f1 := Log(pt[0]), Log(pt[1]); !cVeryclose(f0, f1) {
+ t.Errorf("Log(%g) not continuous, got %g want %g", pt[0], f0, f1)
+ }
+ }
+}
+func TestLog10(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Log10(vc[i]); !cVeryclose(log10[i], f) {
+ t.Errorf("Log10(%g) = %g, want %g", vc[i], f, log10[i])
+ }
+ }
+ for _, v := range log10SC {
+ if f := Log10(v.in); !cAlike(v.want, f) {
+ t.Errorf("Log10(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Log10(Conj(z)) == Conj(Log10(z))
+ if f := Log10(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Log10(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ }
+}
+func TestPolar(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if r, theta := Polar(vc[i]); !veryclose(polar[i].r, r) && !veryclose(polar[i].theta, theta) {
+ t.Errorf("Polar(%g) = %g, %g want %g, %g", vc[i], r, theta, polar[i].r, polar[i].theta)
+ }
+ }
+ for i := 0; i < len(vcPolarSC); i++ {
+ if r, theta := Polar(vcPolarSC[i]); !alike(polarSC[i].r, r) && !alike(polarSC[i].theta, theta) {
+ t.Errorf("Polar(%g) = %g, %g, want %g, %g", vcPolarSC[i], r, theta, polarSC[i].r, polarSC[i].theta)
+ }
+ }
+}
+func TestPow(t *testing.T) {
+ // Special cases for Pow(0, c).
+ var zero = complex(0, 0)
+ zeroPowers := [][2]complex128{
+ {0, 1 + 0i},
+ {1.5, 0 + 0i},
+ {-1.5, complex(math.Inf(0), 0)},
+ {-1.5 + 1.5i, Inf()},
+ }
+ for _, zp := range zeroPowers {
+ if f := Pow(zero, zp[0]); f != zp[1] {
+ t.Errorf("Pow(%g, %g) = %g, want %g", zero, zp[0], f, zp[1])
+ }
+ }
+ var a = complex(3.0, 3.0)
+ for i := 0; i < len(vc); i++ {
+ if f := Pow(a, vc[i]); !cSoclose(pow[i], f, 4e-15) {
+ t.Errorf("Pow(%g, %g) = %g, want %g", a, vc[i], f, pow[i])
+ }
+ }
+ for i := 0; i < len(vcPowSC); i++ {
+ if f := Pow(vcPowSC[i][0], vcPowSC[i][1]); !cAlike(powSC[i], f) {
+ t.Errorf("Pow(%g, %g) = %g, want %g", vcPowSC[i][0], vcPowSC[i][1], f, powSC[i])
+ }
+ }
+ for _, pt := range branchPoints {
+ if f0, f1 := Pow(pt[0], 0.1), Pow(pt[1], 0.1); !cVeryclose(f0, f1) {
+ t.Errorf("Pow(%g, 0.1) not continuous, got %g want %g", pt[0], f0, f1)
+ }
+ }
+}
+func TestRect(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Rect(polar[i].r, polar[i].theta); !cVeryclose(vc[i], f) {
+ t.Errorf("Rect(%g, %g) = %g want %g", polar[i].r, polar[i].theta, f, vc[i])
+ }
+ }
+ for i := 0; i < len(vcPolarSC); i++ {
+ if f := Rect(polarSC[i].r, polarSC[i].theta); !cAlike(vcPolarSC[i], f) {
+ t.Errorf("Rect(%g, %g) = %g, want %g", polarSC[i].r, polarSC[i].theta, f, vcPolarSC[i])
+ }
+ }
+}
+func TestSin(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Sin(vc[i]); !cSoclose(sin[i], f, 2e-15) {
+ t.Errorf("Sin(%g) = %g, want %g", vc[i], f, sin[i])
+ }
+ }
+ for _, v := range sinSC {
+ if f := Sin(v.in); !cAlike(v.want, f) {
+ t.Errorf("Sin(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Sin(Conj(z)) == Conj(Sin(z))
+ if f := Sin(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Sinh(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ if math.IsNaN(real(v.in)) || math.IsNaN(real(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Sin(-z) == -Sin(z)
+ if f := Sin(-v.in); !cAlike(-v.want, f) && !cAlike(v.in, -v.in) {
+ t.Errorf("Sinh(%g) = %g, want %g", -v.in, f, -v.want)
+ }
+ }
+}
+func TestSinh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Sinh(vc[i]); !cSoclose(sinh[i], f, 2e-15) {
+ t.Errorf("Sinh(%g) = %g, want %g", vc[i], f, sinh[i])
+ }
+ }
+ for _, v := range sinhSC {
+ if f := Sinh(v.in); !cAlike(v.want, f) {
+ t.Errorf("Sinh(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Sinh(Conj(z)) == Conj(Sinh(z))
+ if f := Sinh(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Sinh(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ if math.IsNaN(real(v.in)) || math.IsNaN(real(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Sinh(-z) == -Sinh(z)
+ if f := Sinh(-v.in); !cAlike(-v.want, f) && !cAlike(v.in, -v.in) {
+ t.Errorf("Sinh(%g) = %g, want %g", -v.in, f, -v.want)
+ }
+ }
+}
+func TestSqrt(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Sqrt(vc[i]); !cVeryclose(sqrt[i], f) {
+ t.Errorf("Sqrt(%g) = %g, want %g", vc[i], f, sqrt[i])
+ }
+ }
+ for _, v := range sqrtSC {
+ if f := Sqrt(v.in); !cAlike(v.want, f) {
+ t.Errorf("Sqrt(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Sqrt(Conj(z)) == Conj(Sqrt(z))
+ if f := Sqrt(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Sqrt(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ }
+ for _, pt := range branchPoints {
+ if f0, f1 := Sqrt(pt[0]), Sqrt(pt[1]); !cVeryclose(f0, f1) {
+ t.Errorf("Sqrt(%g) not continuous, got %g want %g", pt[0], f0, f1)
+ }
+ }
+}
+func TestTan(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Tan(vc[i]); !cSoclose(tan[i], f, 3e-15) {
+ t.Errorf("Tan(%g) = %g, want %g", vc[i], f, tan[i])
+ }
+ }
+ for _, v := range tanSC {
+ if f := Tan(v.in); !cAlike(v.want, f) {
+ t.Errorf("Tan(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Tan(Conj(z)) == Conj(Tan(z))
+ if f := Tan(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Tan(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ if math.IsNaN(real(v.in)) || math.IsNaN(real(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Tan(-z) == -Tan(z)
+ if f := Tan(-v.in); !cAlike(-v.want, f) && !cAlike(v.in, -v.in) {
+ t.Errorf("Tan(%g) = %g, want %g", -v.in, f, -v.want)
+ }
+ }
+}
+func TestTanh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Tanh(vc[i]); !cSoclose(tanh[i], f, 2e-15) {
+ t.Errorf("Tanh(%g) = %g, want %g", vc[i], f, tanh[i])
+ }
+ }
+ for _, v := range tanhSC {
+ if f := Tanh(v.in); !cAlike(v.want, f) {
+ t.Errorf("Tanh(%g) = %g, want %g", v.in, f, v.want)
+ }
+ if math.IsNaN(imag(v.in)) || math.IsNaN(imag(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Tanh(Conj(z)) == Conj(Tanh(z))
+ if f := Tanh(Conj(v.in)); !cAlike(Conj(v.want), f) && !cAlike(v.in, Conj(v.in)) {
+ t.Errorf("Tanh(%g) = %g, want %g", Conj(v.in), f, Conj(v.want))
+ }
+ if math.IsNaN(real(v.in)) || math.IsNaN(real(v.want)) {
+ // Negating NaN is undefined with regard to the sign bit produced.
+ continue
+ }
+ // Tanh(-z) == -Tanh(z)
+ if f := Tanh(-v.in); !cAlike(-v.want, f) && !cAlike(v.in, -v.in) {
+ t.Errorf("Tanh(%g) = %g, want %g", -v.in, f, -v.want)
+ }
+ }
+}
+
+// See issue 17577
+func TestInfiniteLoopIntanSeries(t *testing.T) {
+ want := Inf()
+ if got := Cot(0); got != want {
+ t.Errorf("Cot(0): got %g, want %g", got, want)
+ }
+}
+
+func BenchmarkAbs(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Abs(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAcos(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Acos(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAcosh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Acosh(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAsin(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Asin(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAsinh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Asinh(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAtan(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Atan(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAtanh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Atanh(complex(2.5, 3.5))
+ }
+}
+func BenchmarkConj(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Conj(complex(2.5, 3.5))
+ }
+}
+func BenchmarkCos(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Cos(complex(2.5, 3.5))
+ }
+}
+func BenchmarkCosh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Cosh(complex(2.5, 3.5))
+ }
+}
+func BenchmarkExp(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Exp(complex(2.5, 3.5))
+ }
+}
+func BenchmarkLog(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Log(complex(2.5, 3.5))
+ }
+}
+func BenchmarkLog10(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Log10(complex(2.5, 3.5))
+ }
+}
+func BenchmarkPhase(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Phase(complex(2.5, 3.5))
+ }
+}
+func BenchmarkPolar(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Polar(complex(2.5, 3.5))
+ }
+}
+func BenchmarkPow(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Pow(complex(2.5, 3.5), complex(2.5, 3.5))
+ }
+}
+func BenchmarkRect(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Rect(2.5, 1.5)
+ }
+}
+func BenchmarkSin(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Sin(complex(2.5, 3.5))
+ }
+}
+func BenchmarkSinh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Sinh(complex(2.5, 3.5))
+ }
+}
+func BenchmarkSqrt(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Sqrt(complex(2.5, 3.5))
+ }
+}
+func BenchmarkTan(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Tan(complex(2.5, 3.5))
+ }
+}
+func BenchmarkTanh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Tanh(complex(2.5, 3.5))
+ }
+}
diff --git a/src/math/cmplx/conj.go b/src/math/cmplx/conj.go
new file mode 100644
index 0000000..34a4277
--- /dev/null
+++ b/src/math/cmplx/conj.go
@@ -0,0 +1,8 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+// Conj returns the complex conjugate of x.
+func Conj(x complex128) complex128 { return complex(real(x), -imag(x)) }
diff --git a/src/math/cmplx/example_test.go b/src/math/cmplx/example_test.go
new file mode 100644
index 0000000..f0ed963
--- /dev/null
+++ b/src/math/cmplx/example_test.go
@@ -0,0 +1,28 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx_test
+
+import (
+ "fmt"
+ "math"
+ "math/cmplx"
+)
+
+func ExampleAbs() {
+ fmt.Printf("%.1f", cmplx.Abs(3+4i))
+ // Output: 5.0
+}
+
+// ExampleExp computes Euler's identity.
+func ExampleExp() {
+ fmt.Printf("%.1f", cmplx.Exp(1i*math.Pi)+1)
+ // Output: (0.0+0.0i)
+}
+
+func ExamplePolar() {
+ r, theta := cmplx.Polar(2i)
+ fmt.Printf("r: %.1f, θ: %.1f*π", r, theta/math.Pi)
+ // Output: r: 2.0, θ: 0.5*π
+}
diff --git a/src/math/cmplx/exp.go b/src/math/cmplx/exp.go
new file mode 100644
index 0000000..d5d0a5d
--- /dev/null
+++ b/src/math/cmplx/exp.go
@@ -0,0 +1,72 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex exponential function
+//
+// DESCRIPTION:
+//
+// Returns the complex exponential of the complex argument z.
+//
+// If
+// z = x + iy,
+// r = exp(x),
+// then
+// w = r cos y + i r sin y.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 8700 3.7e-17 1.1e-17
+// IEEE -10,+10 30000 3.0e-16 8.7e-17
+
+// Exp returns e**x, the base-e exponential of x.
+func Exp(x complex128) complex128 {
+ switch re, im := real(x), imag(x); {
+ case math.IsInf(re, 0):
+ switch {
+ case re > 0 && im == 0:
+ return x
+ case math.IsInf(im, 0) || math.IsNaN(im):
+ if re < 0 {
+ return complex(0, math.Copysign(0, im))
+ } else {
+ return complex(math.Inf(1.0), math.NaN())
+ }
+ }
+ case math.IsNaN(re):
+ if im == 0 {
+ return complex(math.NaN(), im)
+ }
+ }
+ r := math.Exp(real(x))
+ s, c := math.Sincos(imag(x))
+ return complex(r*c, r*s)
+}
diff --git a/src/math/cmplx/huge_test.go b/src/math/cmplx/huge_test.go
new file mode 100644
index 0000000..f8e60c2
--- /dev/null
+++ b/src/math/cmplx/huge_test.go
@@ -0,0 +1,22 @@
+// Copyright 2020 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Disabled for s390x because it uses assembly routines that are not
+// accurate for huge arguments.
+
+// +build !s390x
+
+package cmplx
+
+import (
+ "testing"
+)
+
+func TestTanHuge(t *testing.T) {
+ for i, x := range hugeIn {
+ if f := Tan(x); !cSoclose(tanHuge[i], f, 3e-15) {
+ t.Errorf("Tan(%g) = %g, want %g", x, f, tanHuge[i])
+ }
+ }
+}
diff --git a/src/math/cmplx/isinf.go b/src/math/cmplx/isinf.go
new file mode 100644
index 0000000..6273cd3
--- /dev/null
+++ b/src/math/cmplx/isinf.go
@@ -0,0 +1,21 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// IsInf reports whether either real(x) or imag(x) is an infinity.
+func IsInf(x complex128) bool {
+ if math.IsInf(real(x), 0) || math.IsInf(imag(x), 0) {
+ return true
+ }
+ return false
+}
+
+// Inf returns a complex infinity, complex(+Inf, +Inf).
+func Inf() complex128 {
+ inf := math.Inf(1)
+ return complex(inf, inf)
+}
diff --git a/src/math/cmplx/isnan.go b/src/math/cmplx/isnan.go
new file mode 100644
index 0000000..d3382c0
--- /dev/null
+++ b/src/math/cmplx/isnan.go
@@ -0,0 +1,25 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// IsNaN reports whether either real(x) or imag(x) is NaN
+// and neither is an infinity.
+func IsNaN(x complex128) bool {
+ switch {
+ case math.IsInf(real(x), 0) || math.IsInf(imag(x), 0):
+ return false
+ case math.IsNaN(real(x)) || math.IsNaN(imag(x)):
+ return true
+ }
+ return false
+}
+
+// NaN returns a complex ``not-a-number'' value.
+func NaN() complex128 {
+ nan := math.NaN()
+ return complex(nan, nan)
+}
diff --git a/src/math/cmplx/log.go b/src/math/cmplx/log.go
new file mode 100644
index 0000000..fd39c76
--- /dev/null
+++ b/src/math/cmplx/log.go
@@ -0,0 +1,65 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex natural logarithm
+//
+// DESCRIPTION:
+//
+// Returns complex logarithm to the base e (2.718...) of
+// the complex argument z.
+//
+// If
+// z = x + iy, r = sqrt( x**2 + y**2 ),
+// then
+// w = log(r) + i arctan(y/x).
+//
+// The arctangent ranges from -PI to +PI.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 7000 8.5e-17 1.9e-17
+// IEEE -10,+10 30000 5.0e-15 1.1e-16
+//
+// Larger relative error can be observed for z near 1 +i0.
+// In IEEE arithmetic the peak absolute error is 5.2e-16, rms
+// absolute error 1.0e-16.
+
+// Log returns the natural logarithm of x.
+func Log(x complex128) complex128 {
+ return complex(math.Log(Abs(x)), Phase(x))
+}
+
+// Log10 returns the decimal logarithm of x.
+func Log10(x complex128) complex128 {
+ z := Log(x)
+ return complex(math.Log10E*real(z), math.Log10E*imag(z))
+}
diff --git a/src/math/cmplx/phase.go b/src/math/cmplx/phase.go
new file mode 100644
index 0000000..03cece8
--- /dev/null
+++ b/src/math/cmplx/phase.go
@@ -0,0 +1,11 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// Phase returns the phase (also called the argument) of x.
+// The returned value is in the range [-Pi, Pi].
+func Phase(x complex128) float64 { return math.Atan2(imag(x), real(x)) }
diff --git a/src/math/cmplx/polar.go b/src/math/cmplx/polar.go
new file mode 100644
index 0000000..9b192bc
--- /dev/null
+++ b/src/math/cmplx/polar.go
@@ -0,0 +1,12 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+// Polar returns the absolute value r and phase θ of x,
+// such that x = r * e**θi.
+// The phase is in the range [-Pi, Pi].
+func Polar(x complex128) (r, θ float64) {
+ return Abs(x), Phase(x)
+}
diff --git a/src/math/cmplx/pow.go b/src/math/cmplx/pow.go
new file mode 100644
index 0000000..5a405f8
--- /dev/null
+++ b/src/math/cmplx/pow.go
@@ -0,0 +1,81 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex power function
+//
+// DESCRIPTION:
+//
+// Raises complex A to the complex Zth power.
+// Definition is per AMS55 # 4.2.8,
+// analytically equivalent to cpow(a,z) = cexp(z clog(a)).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -10,+10 30000 9.4e-15 1.5e-15
+
+// Pow returns x**y, the base-x exponential of y.
+// For generalized compatibility with math.Pow:
+// Pow(0, ±0) returns 1+0i
+// Pow(0, c) for real(c)<0 returns Inf+0i if imag(c) is zero, otherwise Inf+Inf i.
+func Pow(x, y complex128) complex128 {
+ if x == 0 { // Guaranteed also true for x == -0.
+ if IsNaN(y) {
+ return NaN()
+ }
+ r, i := real(y), imag(y)
+ switch {
+ case r == 0:
+ return 1
+ case r < 0:
+ if i == 0 {
+ return complex(math.Inf(1), 0)
+ }
+ return Inf()
+ case r > 0:
+ return 0
+ }
+ panic("not reached")
+ }
+ modulus := Abs(x)
+ if modulus == 0 {
+ return complex(0, 0)
+ }
+ r := math.Pow(modulus, real(y))
+ arg := Phase(x)
+ theta := real(y) * arg
+ if imag(y) != 0 {
+ r *= math.Exp(-imag(y) * arg)
+ theta += imag(y) * math.Log(modulus)
+ }
+ s, c := math.Sincos(theta)
+ return complex(r*c, r*s)
+}
diff --git a/src/math/cmplx/rect.go b/src/math/cmplx/rect.go
new file mode 100644
index 0000000..bf94d78
--- /dev/null
+++ b/src/math/cmplx/rect.go
@@ -0,0 +1,13 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// Rect returns the complex number x with polar coordinates r, θ.
+func Rect(r, θ float64) complex128 {
+ s, c := math.Sincos(θ)
+ return complex(r*c, r*s)
+}
diff --git a/src/math/cmplx/sin.go b/src/math/cmplx/sin.go
new file mode 100644
index 0000000..febac0e
--- /dev/null
+++ b/src/math/cmplx/sin.go
@@ -0,0 +1,184 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex circular sine
+//
+// DESCRIPTION:
+//
+// If
+// z = x + iy,
+//
+// then
+//
+// w = sin x cosh y + i cos x sinh y.
+//
+// csin(z) = -i csinh(iz).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 8400 5.3e-17 1.3e-17
+// IEEE -10,+10 30000 3.8e-16 1.0e-16
+// Also tested by csin(casin(z)) = z.
+
+// Sin returns the sine of x.
+func Sin(x complex128) complex128 {
+ switch re, im := real(x), imag(x); {
+ case im == 0 && (math.IsInf(re, 0) || math.IsNaN(re)):
+ return complex(math.NaN(), im)
+ case math.IsInf(im, 0):
+ switch {
+ case re == 0:
+ return x
+ case math.IsInf(re, 0) || math.IsNaN(re):
+ return complex(math.NaN(), im)
+ }
+ case re == 0 && math.IsNaN(im):
+ return x
+ }
+ s, c := math.Sincos(real(x))
+ sh, ch := sinhcosh(imag(x))
+ return complex(s*ch, c*sh)
+}
+
+// Complex hyperbolic sine
+//
+// DESCRIPTION:
+//
+// csinh z = (cexp(z) - cexp(-z))/2
+// = sinh x * cos y + i cosh x * sin y .
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -10,+10 30000 3.1e-16 8.2e-17
+
+// Sinh returns the hyperbolic sine of x.
+func Sinh(x complex128) complex128 {
+ switch re, im := real(x), imag(x); {
+ case re == 0 && (math.IsInf(im, 0) || math.IsNaN(im)):
+ return complex(re, math.NaN())
+ case math.IsInf(re, 0):
+ switch {
+ case im == 0:
+ return complex(re, im)
+ case math.IsInf(im, 0) || math.IsNaN(im):
+ return complex(re, math.NaN())
+ }
+ case im == 0 && math.IsNaN(re):
+ return complex(math.NaN(), im)
+ }
+ s, c := math.Sincos(imag(x))
+ sh, ch := sinhcosh(real(x))
+ return complex(c*sh, s*ch)
+}
+
+// Complex circular cosine
+//
+// DESCRIPTION:
+//
+// If
+// z = x + iy,
+//
+// then
+//
+// w = cos x cosh y - i sin x sinh y.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 8400 4.5e-17 1.3e-17
+// IEEE -10,+10 30000 3.8e-16 1.0e-16
+
+// Cos returns the cosine of x.
+func Cos(x complex128) complex128 {
+ switch re, im := real(x), imag(x); {
+ case im == 0 && (math.IsInf(re, 0) || math.IsNaN(re)):
+ return complex(math.NaN(), -im*math.Copysign(0, re))
+ case math.IsInf(im, 0):
+ switch {
+ case re == 0:
+ return complex(math.Inf(1), -re*math.Copysign(0, im))
+ case math.IsInf(re, 0) || math.IsNaN(re):
+ return complex(math.Inf(1), math.NaN())
+ }
+ case re == 0 && math.IsNaN(im):
+ return complex(math.NaN(), 0)
+ }
+ s, c := math.Sincos(real(x))
+ sh, ch := sinhcosh(imag(x))
+ return complex(c*ch, -s*sh)
+}
+
+// Complex hyperbolic cosine
+//
+// DESCRIPTION:
+//
+// ccosh(z) = cosh x cos y + i sinh x sin y .
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -10,+10 30000 2.9e-16 8.1e-17
+
+// Cosh returns the hyperbolic cosine of x.
+func Cosh(x complex128) complex128 {
+ switch re, im := real(x), imag(x); {
+ case re == 0 && (math.IsInf(im, 0) || math.IsNaN(im)):
+ return complex(math.NaN(), re*math.Copysign(0, im))
+ case math.IsInf(re, 0):
+ switch {
+ case im == 0:
+ return complex(math.Inf(1), im*math.Copysign(0, re))
+ case math.IsInf(im, 0) || math.IsNaN(im):
+ return complex(math.Inf(1), math.NaN())
+ }
+ case im == 0 && math.IsNaN(re):
+ return complex(math.NaN(), im)
+ }
+ s, c := math.Sincos(imag(x))
+ sh, ch := sinhcosh(real(x))
+ return complex(c*ch, s*sh)
+}
+
+// calculate sinh and cosh
+func sinhcosh(x float64) (sh, ch float64) {
+ if math.Abs(x) <= 0.5 {
+ return math.Sinh(x), math.Cosh(x)
+ }
+ e := math.Exp(x)
+ ei := 0.5 / e
+ e *= 0.5
+ return e - ei, e + ei
+}
diff --git a/src/math/cmplx/sqrt.go b/src/math/cmplx/sqrt.go
new file mode 100644
index 0000000..eddce2f
--- /dev/null
+++ b/src/math/cmplx/sqrt.go
@@ -0,0 +1,107 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex square root
+//
+// DESCRIPTION:
+//
+// If z = x + iy, r = |z|, then
+//
+// 1/2
+// Re w = [ (r + x)/2 ] ,
+//
+// 1/2
+// Im w = [ (r - x)/2 ] .
+//
+// Cancellation error in r-x or r+x is avoided by using the
+// identity 2 Re w Im w = y.
+//
+// Note that -w is also a square root of z. The root chosen
+// is always in the right half plane and Im w has the same sign as y.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 25000 3.2e-17 9.6e-18
+// IEEE -10,+10 1,000,000 2.9e-16 6.1e-17
+
+// Sqrt returns the square root of x.
+// The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x).
+func Sqrt(x complex128) complex128 {
+ if imag(x) == 0 {
+ // Ensure that imag(r) has the same sign as imag(x) for imag(x) == signed zero.
+ if real(x) == 0 {
+ return complex(0, imag(x))
+ }
+ if real(x) < 0 {
+ return complex(0, math.Copysign(math.Sqrt(-real(x)), imag(x)))
+ }
+ return complex(math.Sqrt(real(x)), imag(x))
+ } else if math.IsInf(imag(x), 0) {
+ return complex(math.Inf(1.0), imag(x))
+ }
+ if real(x) == 0 {
+ if imag(x) < 0 {
+ r := math.Sqrt(-0.5 * imag(x))
+ return complex(r, -r)
+ }
+ r := math.Sqrt(0.5 * imag(x))
+ return complex(r, r)
+ }
+ a := real(x)
+ b := imag(x)
+ var scale float64
+ // Rescale to avoid internal overflow or underflow.
+ if math.Abs(a) > 4 || math.Abs(b) > 4 {
+ a *= 0.25
+ b *= 0.25
+ scale = 2
+ } else {
+ a *= 1.8014398509481984e16 // 2**54
+ b *= 1.8014398509481984e16
+ scale = 7.450580596923828125e-9 // 2**-27
+ }
+ r := math.Hypot(a, b)
+ var t float64
+ if a > 0 {
+ t = math.Sqrt(0.5*r + 0.5*a)
+ r = scale * math.Abs((0.5*b)/t)
+ t *= scale
+ } else {
+ r = math.Sqrt(0.5*r - 0.5*a)
+ t = scale * math.Abs((0.5*b)/r)
+ r *= scale
+ }
+ if b < 0 {
+ return complex(t, -r)
+ }
+ return complex(t, r)
+}
diff --git a/src/math/cmplx/tan.go b/src/math/cmplx/tan.go
new file mode 100644
index 0000000..67a1133
--- /dev/null
+++ b/src/math/cmplx/tan.go
@@ -0,0 +1,297 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import (
+ "math"
+ "math/bits"
+)
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex circular tangent
+//
+// DESCRIPTION:
+//
+// If
+// z = x + iy,
+//
+// then
+//
+// sin 2x + i sinh 2y
+// w = --------------------.
+// cos 2x + cosh 2y
+//
+// On the real axis the denominator is zero at odd multiples
+// of PI/2. The denominator is evaluated by its Taylor
+// series near these points.
+//
+// ctan(z) = -i ctanh(iz).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 5200 7.1e-17 1.6e-17
+// IEEE -10,+10 30000 7.2e-16 1.2e-16
+// Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
+
+// Tan returns the tangent of x.
+func Tan(x complex128) complex128 {
+ switch re, im := real(x), imag(x); {
+ case math.IsInf(im, 0):
+ switch {
+ case math.IsInf(re, 0) || math.IsNaN(re):
+ return complex(math.Copysign(0, re), math.Copysign(1, im))
+ }
+ return complex(math.Copysign(0, math.Sin(2*re)), math.Copysign(1, im))
+ case re == 0 && math.IsNaN(im):
+ return x
+ }
+ d := math.Cos(2*real(x)) + math.Cosh(2*imag(x))
+ if math.Abs(d) < 0.25 {
+ d = tanSeries(x)
+ }
+ if d == 0 {
+ return Inf()
+ }
+ return complex(math.Sin(2*real(x))/d, math.Sinh(2*imag(x))/d)
+}
+
+// Complex hyperbolic tangent
+//
+// DESCRIPTION:
+//
+// tanh z = (sinh 2x + i sin 2y) / (cosh 2x + cos 2y) .
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -10,+10 30000 1.7e-14 2.4e-16
+
+// Tanh returns the hyperbolic tangent of x.
+func Tanh(x complex128) complex128 {
+ switch re, im := real(x), imag(x); {
+ case math.IsInf(re, 0):
+ switch {
+ case math.IsInf(im, 0) || math.IsNaN(im):
+ return complex(math.Copysign(1, re), math.Copysign(0, im))
+ }
+ return complex(math.Copysign(1, re), math.Copysign(0, math.Sin(2*im)))
+ case im == 0 && math.IsNaN(re):
+ return x
+ }
+ d := math.Cosh(2*real(x)) + math.Cos(2*imag(x))
+ if d == 0 {
+ return Inf()
+ }
+ return complex(math.Sinh(2*real(x))/d, math.Sin(2*imag(x))/d)
+}
+
+// reducePi reduces the input argument x to the range (-Pi/2, Pi/2].
+// x must be greater than or equal to 0. For small arguments it
+// uses Cody-Waite reduction in 3 float64 parts based on:
+// "Elementary Function Evaluation: Algorithms and Implementation"
+// Jean-Michel Muller, 1997.
+// For very large arguments it uses Payne-Hanek range reduction based on:
+// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
+// K. C. Ng et al, March 24, 1992.
+func reducePi(x float64) float64 {
+ // reduceThreshold is the maximum value of x where the reduction using
+ // Cody-Waite reduction still gives accurate results. This threshold
+ // is set by t*PIn being representable as a float64 without error
+ // where t is given by t = floor(x * (1 / Pi)) and PIn are the leading partial
+ // terms of Pi. Since the leading terms, PI1 and PI2 below, have 30 and 32
+ // trailing zero bits respectively, t should have less than 30 significant bits.
+ // t < 1<<30 -> floor(x*(1/Pi)+0.5) < 1<<30 -> x < (1<<30-1) * Pi - 0.5
+ // So, conservatively we can take x < 1<<30.
+ const reduceThreshold float64 = 1 << 30
+ if math.Abs(x) < reduceThreshold {
+ // Use Cody-Waite reduction in three parts.
+ const (
+ // PI1, PI2 and PI3 comprise an extended precision value of PI
+ // such that PI ~= PI1 + PI2 + PI3. The parts are chosen so
+ // that PI1 and PI2 have an approximately equal number of trailing
+ // zero bits. This ensures that t*PI1 and t*PI2 are exact for
+ // large integer values of t. The full precision PI3 ensures the
+ // approximation of PI is accurate to 102 bits to handle cancellation
+ // during subtraction.
+ PI1 = 3.141592502593994 // 0x400921fb40000000
+ PI2 = 1.5099578831723193e-07 // 0x3e84442d00000000
+ PI3 = 1.0780605716316238e-14 // 0x3d08469898cc5170
+ )
+ t := x / math.Pi
+ t += 0.5
+ t = float64(int64(t)) // int64(t) = the multiple
+ return ((x - t*PI1) - t*PI2) - t*PI3
+ }
+ // Must apply Payne-Hanek range reduction
+ const (
+ mask = 0x7FF
+ shift = 64 - 11 - 1
+ bias = 1023
+ fracMask = 1<<shift - 1
+ )
+ // Extract out the integer and exponent such that,
+ // x = ix * 2 ** exp.
+ ix := math.Float64bits(x)
+ exp := int(ix>>shift&mask) - bias - shift
+ ix &= fracMask
+ ix |= 1 << shift
+
+ // mPi is the binary digits of 1/Pi as a uint64 array,
+ // that is, 1/Pi = Sum mPi[i]*2^(-64*i).
+ // 19 64-bit digits give 1216 bits of precision
+ // to handle the largest possible float64 exponent.
+ var mPi = [...]uint64{
+ 0x0000000000000000,
+ 0x517cc1b727220a94,
+ 0xfe13abe8fa9a6ee0,
+ 0x6db14acc9e21c820,
+ 0xff28b1d5ef5de2b0,
+ 0xdb92371d2126e970,
+ 0x0324977504e8c90e,
+ 0x7f0ef58e5894d39f,
+ 0x74411afa975da242,
+ 0x74ce38135a2fbf20,
+ 0x9cc8eb1cc1a99cfa,
+ 0x4e422fc5defc941d,
+ 0x8ffc4bffef02cc07,
+ 0xf79788c5ad05368f,
+ 0xb69b3f6793e584db,
+ 0xa7a31fb34f2ff516,
+ 0xba93dd63f5f2f8bd,
+ 0x9e839cfbc5294975,
+ 0x35fdafd88fc6ae84,
+ 0x2b0198237e3db5d5,
+ }
+ // Use the exponent to extract the 3 appropriate uint64 digits from mPi,
+ // B ~ (z0, z1, z2), such that the product leading digit has the exponent -64.
+ // Note, exp >= 50 since x >= reduceThreshold and exp < 971 for maximum float64.
+ digit, bitshift := uint(exp+64)/64, uint(exp+64)%64
+ z0 := (mPi[digit] << bitshift) | (mPi[digit+1] >> (64 - bitshift))
+ z1 := (mPi[digit+1] << bitshift) | (mPi[digit+2] >> (64 - bitshift))
+ z2 := (mPi[digit+2] << bitshift) | (mPi[digit+3] >> (64 - bitshift))
+ // Multiply mantissa by the digits and extract the upper two digits (hi, lo).
+ z2hi, _ := bits.Mul64(z2, ix)
+ z1hi, z1lo := bits.Mul64(z1, ix)
+ z0lo := z0 * ix
+ lo, c := bits.Add64(z1lo, z2hi, 0)
+ hi, _ := bits.Add64(z0lo, z1hi, c)
+ // Find the magnitude of the fraction.
+ lz := uint(bits.LeadingZeros64(hi))
+ e := uint64(bias - (lz + 1))
+ // Clear implicit mantissa bit and shift into place.
+ hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1)))
+ hi >>= 64 - shift
+ // Include the exponent and convert to a float.
+ hi |= e << shift
+ x = math.Float64frombits(hi)
+ // map to (-Pi/2, Pi/2]
+ if x > 0.5 {
+ x--
+ }
+ return math.Pi * x
+}
+
+// Taylor series expansion for cosh(2y) - cos(2x)
+func tanSeries(z complex128) float64 {
+ const MACHEP = 1.0 / (1 << 53)
+ x := math.Abs(2 * real(z))
+ y := math.Abs(2 * imag(z))
+ x = reducePi(x)
+ x = x * x
+ y = y * y
+ x2 := 1.0
+ y2 := 1.0
+ f := 1.0
+ rn := 0.0
+ d := 0.0
+ for {
+ rn++
+ f *= rn
+ rn++
+ f *= rn
+ x2 *= x
+ y2 *= y
+ t := y2 + x2
+ t /= f
+ d += t
+
+ rn++
+ f *= rn
+ rn++
+ f *= rn
+ x2 *= x
+ y2 *= y
+ t = y2 - x2
+ t /= f
+ d += t
+ if !(math.Abs(t/d) > MACHEP) {
+ // Caution: Use ! and > instead of <= for correct behavior if t/d is NaN.
+ // See issue 17577.
+ break
+ }
+ }
+ return d
+}
+
+// Complex circular cotangent
+//
+// DESCRIPTION:
+//
+// If
+// z = x + iy,
+//
+// then
+//
+// sin 2x - i sinh 2y
+// w = --------------------.
+// cosh 2y - cos 2x
+//
+// On the real axis, the denominator has zeros at even
+// multiples of PI/2. Near these points it is evaluated
+// by a Taylor series.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 3000 6.5e-17 1.6e-17
+// IEEE -10,+10 30000 9.2e-16 1.2e-16
+// Also tested by ctan * ccot = 1 + i0.
+
+// Cot returns the cotangent of x.
+func Cot(x complex128) complex128 {
+ d := math.Cosh(2*imag(x)) - math.Cos(2*real(x))
+ if math.Abs(d) < 0.25 {
+ d = tanSeries(x)
+ }
+ if d == 0 {
+ return Inf()
+ }
+ return complex(math.Sin(2*real(x))/d, -math.Sinh(2*imag(x))/d)
+}
diff --git a/src/math/const.go b/src/math/const.go
new file mode 100644
index 0000000..0fc8715
--- /dev/null
+++ b/src/math/const.go
@@ -0,0 +1,52 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package math provides basic constants and mathematical functions.
+//
+// This package does not guarantee bit-identical results across architectures.
+package math
+
+// Mathematical constants.
+const (
+ E = 2.71828182845904523536028747135266249775724709369995957496696763 // https://oeis.org/A001113
+ Pi = 3.14159265358979323846264338327950288419716939937510582097494459 // https://oeis.org/A000796
+ Phi = 1.61803398874989484820458683436563811772030917980576286213544862 // https://oeis.org/A001622
+
+ Sqrt2 = 1.41421356237309504880168872420969807856967187537694807317667974 // https://oeis.org/A002193
+ SqrtE = 1.64872127070012814684865078781416357165377610071014801157507931 // https://oeis.org/A019774
+ SqrtPi = 1.77245385090551602729816748334114518279754945612238712821380779 // https://oeis.org/A002161
+ SqrtPhi = 1.27201964951406896425242246173749149171560804184009624861664038 // https://oeis.org/A139339
+
+ Ln2 = 0.693147180559945309417232121458176568075500134360255254120680009 // https://oeis.org/A002162
+ Log2E = 1 / Ln2
+ Ln10 = 2.30258509299404568401799145468436420760110148862877297603332790 // https://oeis.org/A002392
+ Log10E = 1 / Ln10
+)
+
+// Floating-point limit values.
+// Max is the largest finite value representable by the type.
+// SmallestNonzero is the smallest positive, non-zero value representable by the type.
+const (
+ MaxFloat32 = 3.40282346638528859811704183484516925440e+38 // 2**127 * (2**24 - 1) / 2**23
+ SmallestNonzeroFloat32 = 1.401298464324817070923729583289916131280e-45 // 1 / 2**(127 - 1 + 23)
+
+ MaxFloat64 = 1.797693134862315708145274237317043567981e+308 // 2**1023 * (2**53 - 1) / 2**52
+ SmallestNonzeroFloat64 = 4.940656458412465441765687928682213723651e-324 // 1 / 2**(1023 - 1 + 52)
+)
+
+// Integer limit values.
+const (
+ MaxInt8 = 1<<7 - 1
+ MinInt8 = -1 << 7
+ MaxInt16 = 1<<15 - 1
+ MinInt16 = -1 << 15
+ MaxInt32 = 1<<31 - 1
+ MinInt32 = -1 << 31
+ MaxInt64 = 1<<63 - 1
+ MinInt64 = -1 << 63
+ MaxUint8 = 1<<8 - 1
+ MaxUint16 = 1<<16 - 1
+ MaxUint32 = 1<<32 - 1
+ MaxUint64 = 1<<64 - 1
+)
diff --git a/src/math/copysign.go b/src/math/copysign.go
new file mode 100644
index 0000000..719c64b
--- /dev/null
+++ b/src/math/copysign.go
@@ -0,0 +1,12 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Copysign returns a value with the magnitude
+// of x and the sign of y.
+func Copysign(x, y float64) float64 {
+ const sign = 1 << 63
+ return Float64frombits(Float64bits(x)&^sign | Float64bits(y)&sign)
+}
diff --git a/src/math/cosh_s390x.s b/src/math/cosh_s390x.s
new file mode 100644
index 0000000..ca1d86e
--- /dev/null
+++ b/src/math/cosh_s390x.s
@@ -0,0 +1,211 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Constants
+DATA coshrodataL23<>+0(SB)/8, $0.231904681384629956E-16
+DATA coshrodataL23<>+8(SB)/8, $0.693147180559945286E+00
+DATA coshrodataL23<>+16(SB)/8, $0.144269504088896339E+01
+DATA coshrodataL23<>+24(SB)/8, $704.E0
+GLOBL coshrodataL23<>+0(SB), RODATA, $32
+DATA coshxinf<>+0(SB)/8, $0x7FF0000000000000
+GLOBL coshxinf<>+0(SB), RODATA, $8
+DATA coshxlim1<>+0(SB)/8, $800.E0
+GLOBL coshxlim1<>+0(SB), RODATA, $8
+DATA coshxaddhy<>+0(SB)/8, $0xc2f0000100003fdf
+GLOBL coshxaddhy<>+0(SB), RODATA, $8
+DATA coshx4ff<>+0(SB)/8, $0x4ff0000000000000
+GLOBL coshx4ff<>+0(SB), RODATA, $8
+DATA coshe1<>+0(SB)/8, $0x3ff000000000000a
+GLOBL coshe1<>+0(SB), RODATA, $8
+
+// Log multiplier table
+DATA coshtab<>+0(SB)/8, $0.442737824274138381E-01
+DATA coshtab<>+8(SB)/8, $0.263602189790660309E-01
+DATA coshtab<>+16(SB)/8, $0.122565642281703586E-01
+DATA coshtab<>+24(SB)/8, $0.143757052860721398E-02
+DATA coshtab<>+32(SB)/8, $-.651375034121276075E-02
+DATA coshtab<>+40(SB)/8, $-.119317678849450159E-01
+DATA coshtab<>+48(SB)/8, $-.150868749549871069E-01
+DATA coshtab<>+56(SB)/8, $-.161992609578469234E-01
+DATA coshtab<>+64(SB)/8, $-.154492360403337917E-01
+DATA coshtab<>+72(SB)/8, $-.129850717389178721E-01
+DATA coshtab<>+80(SB)/8, $-.892902649276657891E-02
+DATA coshtab<>+88(SB)/8, $-.338202636596794887E-02
+DATA coshtab<>+96(SB)/8, $0.357266307045684762E-02
+DATA coshtab<>+104(SB)/8, $0.118665304327406698E-01
+DATA coshtab<>+112(SB)/8, $0.214434994118118914E-01
+DATA coshtab<>+120(SB)/8, $0.322580645161290314E-01
+GLOBL coshtab<>+0(SB), RODATA, $128
+
+// Minimax polynomial approximations
+DATA coshe2<>+0(SB)/8, $0.500000000000004237e+00
+GLOBL coshe2<>+0(SB), RODATA, $8
+DATA coshe3<>+0(SB)/8, $0.166666666630345592e+00
+GLOBL coshe3<>+0(SB), RODATA, $8
+DATA coshe4<>+0(SB)/8, $0.416666664838056960e-01
+GLOBL coshe4<>+0(SB), RODATA, $8
+DATA coshe5<>+0(SB)/8, $0.833349307718286047e-02
+GLOBL coshe5<>+0(SB), RODATA, $8
+DATA coshe6<>+0(SB)/8, $0.138926439368309441e-02
+GLOBL coshe6<>+0(SB), RODATA, $8
+
+// Cosh returns the hyperbolic cosine of x.
+//
+// Special cases are:
+// Cosh(±0) = 1
+// Cosh(±Inf) = +Inf
+// Cosh(NaN) = NaN
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·coshAsm(SB),NOSPLIT,$0-16
+ FMOVD x+0(FP), F0
+ MOVD $coshrodataL23<>+0(SB), R9
+ LTDBR F0, F0
+ MOVD $0x4086000000000000, R2
+ MOVD $0x4086000000000000, R3
+ BLTU L19
+ FMOVD F0, F4
+L2:
+ WORD $0xED409018 //cdb %f4,.L24-.L23(%r9)
+ BYTE $0x00
+ BYTE $0x19
+ BGE L14 //jnl .L14
+ BVS L14
+ WFCEDBS V4, V4, V2
+ BEQ L20
+L1:
+ FMOVD F0, ret+8(FP)
+ RET
+
+L14:
+ WFCEDBS V4, V4, V2
+ BVS L1
+ MOVD $coshxlim1<>+0(SB), R1
+ FMOVD 0(R1), F2
+ WFCHEDBS V4, V2, V2
+ BEQ L21
+ MOVD $coshxaddhy<>+0(SB), R1
+ FMOVD coshrodataL23<>+16(SB), F5
+ FMOVD 0(R1), F2
+ WFMSDB V0, V5, V2, V5
+ FMOVD coshrodataL23<>+8(SB), F3
+ FADD F5, F2
+ MOVD $coshe6<>+0(SB), R1
+ WFMSDB V2, V3, V0, V3
+ FMOVD 0(R1), F6
+ WFMDB V3, V3, V1
+ MOVD $coshe4<>+0(SB), R1
+ FMOVD coshrodataL23<>+0(SB), F7
+ WFMADB V2, V7, V3, V2
+ FMOVD 0(R1), F3
+ MOVD $coshe5<>+0(SB), R1
+ WFMADB V1, V6, V3, V6
+ FMOVD 0(R1), F7
+ MOVD $coshe3<>+0(SB), R1
+ FMOVD 0(R1), F3
+ WFMADB V1, V7, V3, V7
+ FNEG F2, F3
+ LGDR F5, R1
+ MOVD $coshe2<>+0(SB), R3
+ WFCEDBS V4, V0, V0
+ FMOVD 0(R3), F5
+ MOVD $coshe1<>+0(SB), R3
+ WFMADB V1, V6, V5, V6
+ FMOVD 0(R3), F5
+ RISBGN $0, $15, $48, R1, R2
+ WFMADB V1, V7, V5, V1
+ BVS L22
+ RISBGZ $57, $60, $3, R1, R4
+ MOVD $coshtab<>+0(SB), R3
+ WFMADB V3, V6, V1, V6
+ WORD $0x68043000 //ld %f0,0(%r4,%r3)
+ FMSUB F0, F3, F2
+ WORD $0xA71AF000 //ahi %r1,-4096
+ WFMADB V2, V6, V0, V6
+L17:
+ RISBGN $0, $15, $48, R1, R2
+ LDGR R2, F2
+ FMADD F2, F6, F2
+ MOVD $coshx4ff<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMUL F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L19:
+ FNEG F0, F4
+ BR L2
+L20:
+ MOVD $coshxaddhy<>+0(SB), R1
+ FMOVD coshrodataL23<>+16(SB), F3
+ FMOVD 0(R1), F2
+ WFMSDB V0, V3, V2, V3
+ FMOVD coshrodataL23<>+8(SB), F4
+ FADD F3, F2
+ MOVD $coshe6<>+0(SB), R1
+ FMSUB F4, F2, F0
+ FMOVD 0(R1), F6
+ WFMDB V0, V0, V1
+ MOVD $coshe4<>+0(SB), R1
+ FMOVD 0(R1), F4
+ MOVD $coshe5<>+0(SB), R1
+ FMOVD coshrodataL23<>+0(SB), F5
+ WFMADB V1, V6, V4, V6
+ FMADD F5, F2, F0
+ FMOVD 0(R1), F2
+ MOVD $coshe3<>+0(SB), R1
+ FMOVD 0(R1), F4
+ WFMADB V1, V2, V4, V2
+ MOVD $coshe2<>+0(SB), R1
+ FMOVD 0(R1), F5
+ FNEG F0, F4
+ WFMADB V1, V6, V5, V6
+ MOVD $coshe1<>+0(SB), R1
+ FMOVD 0(R1), F5
+ WFMADB V1, V2, V5, V1
+ LGDR F3, R1
+ MOVD $coshtab<>+0(SB), R5
+ WFMADB V4, V6, V1, V3
+ RISBGZ $57, $60, $3, R1, R4
+ WFMSDB V4, V6, V1, V6
+ WORD $0x68145000 //ld %f1,0(%r4,%r5)
+ WFMSDB V4, V1, V0, V2
+ WORD $0xA7487FBE //lhi %r4,32702
+ FMADD F3, F2, F1
+ SUBW R1, R4
+ RISBGZ $57, $60, $3, R4, R12
+ WORD $0x682C5000 //ld %f2,0(%r12,%r5)
+ FMSUB F2, F4, F0
+ RISBGN $0, $15, $48, R1, R2
+ WFMADB V0, V6, V2, V6
+ RISBGN $0, $15, $48, R4, R3
+ LDGR R2, F2
+ LDGR R3, F0
+ FMADD F2, F1, F2
+ FMADD F0, F6, F0
+ FADD F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L22:
+ WORD $0xA7387FBE //lhi %r3,32702
+ MOVD $coshtab<>+0(SB), R4
+ SUBW R1, R3
+ WFMSDB V3, V6, V1, V6
+ RISBGZ $57, $60, $3, R3, R3
+ WORD $0x68034000 //ld %f0,0(%r3,%r4)
+ FMSUB F0, F3, F2
+ WORD $0xA7386FBE //lhi %r3,28606
+ WFMADB V2, V6, V0, V6
+ SUBW R1, R3, R1
+ BR L17
+L21:
+ MOVD $coshxinf<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+
diff --git a/src/math/dim.go b/src/math/dim.go
new file mode 100644
index 0000000..d2e5d47
--- /dev/null
+++ b/src/math/dim.go
@@ -0,0 +1,81 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Dim returns the maximum of x-y or 0.
+//
+// Special cases are:
+// Dim(+Inf, +Inf) = NaN
+// Dim(-Inf, -Inf) = NaN
+// Dim(x, NaN) = Dim(NaN, x) = NaN
+func Dim(x, y float64) float64 {
+ // The special cases result in NaN after the subtraction:
+ // +Inf - +Inf = NaN
+ // -Inf - -Inf = NaN
+ // NaN - y = NaN
+ // x - NaN = NaN
+ v := x - y
+ if v <= 0 {
+ // v is negative or 0
+ return 0
+ }
+ // v is positive or NaN
+ return v
+}
+
+// Max returns the larger of x or y.
+//
+// Special cases are:
+// Max(x, +Inf) = Max(+Inf, x) = +Inf
+// Max(x, NaN) = Max(NaN, x) = NaN
+// Max(+0, ±0) = Max(±0, +0) = +0
+// Max(-0, -0) = -0
+func Max(x, y float64) float64
+
+func max(x, y float64) float64 {
+ // special cases
+ switch {
+ case IsInf(x, 1) || IsInf(y, 1):
+ return Inf(1)
+ case IsNaN(x) || IsNaN(y):
+ return NaN()
+ case x == 0 && x == y:
+ if Signbit(x) {
+ return y
+ }
+ return x
+ }
+ if x > y {
+ return x
+ }
+ return y
+}
+
+// Min returns the smaller of x or y.
+//
+// Special cases are:
+// Min(x, -Inf) = Min(-Inf, x) = -Inf
+// Min(x, NaN) = Min(NaN, x) = NaN
+// Min(-0, ±0) = Min(±0, -0) = -0
+func Min(x, y float64) float64
+
+func min(x, y float64) float64 {
+ // special cases
+ switch {
+ case IsInf(x, -1) || IsInf(y, -1):
+ return Inf(-1)
+ case IsNaN(x) || IsNaN(y):
+ return NaN()
+ case x == 0 && x == y:
+ if Signbit(x) {
+ return x
+ }
+ return y
+ }
+ if x < y {
+ return x
+ }
+ return y
+}
diff --git a/src/math/dim_amd64.s b/src/math/dim_amd64.s
new file mode 100644
index 0000000..85c02e6
--- /dev/null
+++ b/src/math/dim_amd64.s
@@ -0,0 +1,98 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define PosInf 0x7FF0000000000000
+#define NaN 0x7FF8000000000001
+#define NegInf 0xFFF0000000000000
+
+// func ·Max(x, y float64) float64
+TEXT ·Max(SB),NOSPLIT,$0
+ // +Inf special cases
+ MOVQ $PosInf, AX
+ MOVQ x+0(FP), R8
+ CMPQ AX, R8
+ JEQ isPosInf
+ MOVQ y+8(FP), R9
+ CMPQ AX, R9
+ JEQ isPosInf
+ // NaN special cases
+ MOVQ $~(1<<63), DX // bit mask
+ MOVQ $PosInf, AX
+ MOVQ R8, BX
+ ANDQ DX, BX // x = |x|
+ CMPQ AX, BX
+ JLT isMaxNaN
+ MOVQ R9, CX
+ ANDQ DX, CX // y = |y|
+ CMPQ AX, CX
+ JLT isMaxNaN
+ // ±0 special cases
+ ORQ CX, BX
+ JEQ isMaxZero
+
+ MOVQ R8, X0
+ MOVQ R9, X1
+ MAXSD X1, X0
+ MOVSD X0, ret+16(FP)
+ RET
+isMaxNaN: // return NaN
+ MOVQ $NaN, AX
+isPosInf: // return +Inf
+ MOVQ AX, ret+16(FP)
+ RET
+isMaxZero:
+ MOVQ $(1<<63), AX // -0.0
+ CMPQ AX, R8
+ JEQ +3(PC)
+ MOVQ R8, ret+16(FP) // return 0
+ RET
+ MOVQ R9, ret+16(FP) // return other 0
+ RET
+
+// func Min(x, y float64) float64
+TEXT ·Min(SB),NOSPLIT,$0
+ // -Inf special cases
+ MOVQ $NegInf, AX
+ MOVQ x+0(FP), R8
+ CMPQ AX, R8
+ JEQ isNegInf
+ MOVQ y+8(FP), R9
+ CMPQ AX, R9
+ JEQ isNegInf
+ // NaN special cases
+ MOVQ $~(1<<63), DX
+ MOVQ $PosInf, AX
+ MOVQ R8, BX
+ ANDQ DX, BX // x = |x|
+ CMPQ AX, BX
+ JLT isMinNaN
+ MOVQ R9, CX
+ ANDQ DX, CX // y = |y|
+ CMPQ AX, CX
+ JLT isMinNaN
+ // ±0 special cases
+ ORQ CX, BX
+ JEQ isMinZero
+
+ MOVQ R8, X0
+ MOVQ R9, X1
+ MINSD X1, X0
+ MOVSD X0, ret+16(FP)
+ RET
+isMinNaN: // return NaN
+ MOVQ $NaN, AX
+isNegInf: // return -Inf
+ MOVQ AX, ret+16(FP)
+ RET
+isMinZero:
+ MOVQ $(1<<63), AX // -0.0
+ CMPQ AX, R8
+ JEQ +3(PC)
+ MOVQ R9, ret+16(FP) // return other 0
+ RET
+ MOVQ R8, ret+16(FP) // return -0
+ RET
+
diff --git a/src/math/dim_arm64.s b/src/math/dim_arm64.s
new file mode 100644
index 0000000..2cb866f
--- /dev/null
+++ b/src/math/dim_arm64.s
@@ -0,0 +1,49 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define PosInf 0x7FF0000000000000
+#define NaN 0x7FF8000000000001
+#define NegInf 0xFFF0000000000000
+
+// func ·Max(x, y float64) float64
+TEXT ·Max(SB),NOSPLIT,$0
+ // +Inf special cases
+ MOVD $PosInf, R0
+ MOVD x+0(FP), R1
+ CMP R0, R1
+ BEQ isPosInf
+ MOVD y+8(FP), R2
+ CMP R0, R2
+ BEQ isPosInf
+ // normal case
+ FMOVD R1, F0
+ FMOVD R2, F1
+ FMAXD F0, F1, F0
+ FMOVD F0, ret+16(FP)
+ RET
+isPosInf: // return +Inf
+ MOVD R0, ret+16(FP)
+ RET
+
+// func Min(x, y float64) float64
+TEXT ·Min(SB),NOSPLIT,$0
+ // -Inf special cases
+ MOVD $NegInf, R0
+ MOVD x+0(FP), R1
+ CMP R0, R1
+ BEQ isNegInf
+ MOVD y+8(FP), R2
+ CMP R0, R2
+ BEQ isNegInf
+ // normal case
+ FMOVD R1, F0
+ FMOVD R2, F1
+ FMIND F0, F1, F0
+ FMOVD F0, ret+16(FP)
+ RET
+isNegInf: // return -Inf
+ MOVD R0, ret+16(FP)
+ RET
diff --git a/src/math/dim_riscv64.s b/src/math/dim_riscv64.s
new file mode 100644
index 0000000..38f5fe7
--- /dev/null
+++ b/src/math/dim_riscv64.s
@@ -0,0 +1,70 @@
+// Copyright 2020 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Values returned from an FCLASS instruction.
+#define NegInf 0x001
+#define PosInf 0x080
+#define NaN 0x200
+
+// func Max(x, y float64) float64
+TEXT ·Max(SB),NOSPLIT,$0
+ MOVD x+0(FP), F0
+ MOVD y+8(FP), F1
+ FCLASSD F0, X5
+ FCLASSD F1, X6
+
+ // +Inf special cases
+ MOV $PosInf, X7
+ BEQ X7, X5, isMaxX
+ BEQ X7, X6, isMaxY
+
+ // NaN special cases
+ MOV $NaN, X7
+ BEQ X7, X5, isMaxX
+ BEQ X7, X6, isMaxY
+
+ // normal case
+ FMAXD F0, F1, F0
+ MOVD F0, ret+16(FP)
+ RET
+
+isMaxX: // return x
+ MOVD F0, ret+16(FP)
+ RET
+
+isMaxY: // return y
+ MOVD F1, ret+16(FP)
+ RET
+
+// func Min(x, y float64) float64
+TEXT ·Min(SB),NOSPLIT,$0
+ MOVD x+0(FP), F0
+ MOVD y+8(FP), F1
+ FCLASSD F0, X5
+ FCLASSD F1, X6
+
+ // -Inf special cases
+ MOV $NegInf, X7
+ BEQ X7, X5, isMinX
+ BEQ X7, X6, isMinY
+
+ // NaN special cases
+ MOV $NaN, X7
+ BEQ X7, X5, isMinX
+ BEQ X7, X6, isMinY
+
+ // normal case
+ FMIND F0, F1, F0
+ MOVD F0, ret+16(FP)
+ RET
+
+isMinX: // return x
+ MOVD F0, ret+16(FP)
+ RET
+
+isMinY: // return y
+ MOVD F1, ret+16(FP)
+ RET
diff --git a/src/math/dim_s390x.s b/src/math/dim_s390x.s
new file mode 100644
index 0000000..74fdd75
--- /dev/null
+++ b/src/math/dim_s390x.s
@@ -0,0 +1,96 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Based on dim_amd64.s
+
+#include "textflag.h"
+
+#define PosInf 0x7FF0000000000000
+#define NaN 0x7FF8000000000001
+#define NegInf 0xFFF0000000000000
+
+// func ·Max(x, y float64) float64
+TEXT ·Max(SB),NOSPLIT,$0
+ // +Inf special cases
+ MOVD $PosInf, R4
+ MOVD x+0(FP), R8
+ CMPUBEQ R4, R8, isPosInf
+ MOVD y+8(FP), R9
+ CMPUBEQ R4, R9, isPosInf
+ // NaN special cases
+ MOVD $~(1<<63), R5 // bit mask
+ MOVD $PosInf, R4
+ MOVD R8, R2
+ AND R5, R2 // x = |x|
+ CMPUBLT R4, R2, isMaxNaN
+ MOVD R9, R3
+ AND R5, R3 // y = |y|
+ CMPUBLT R4, R3, isMaxNaN
+ // ±0 special cases
+ OR R3, R2
+ BEQ isMaxZero
+
+ FMOVD x+0(FP), F1
+ FMOVD y+8(FP), F2
+ FCMPU F2, F1
+ BGT +3(PC)
+ FMOVD F1, ret+16(FP)
+ RET
+ FMOVD F2, ret+16(FP)
+ RET
+isMaxNaN: // return NaN
+ MOVD $NaN, R4
+isPosInf: // return +Inf
+ MOVD R4, ret+16(FP)
+ RET
+isMaxZero:
+ MOVD $(1<<63), R4 // -0.0
+ CMPUBEQ R4, R8, +3(PC)
+ MOVD R8, ret+16(FP) // return 0
+ RET
+ MOVD R9, ret+16(FP) // return other 0
+ RET
+
+// func Min(x, y float64) float64
+TEXT ·Min(SB),NOSPLIT,$0
+ // -Inf special cases
+ MOVD $NegInf, R4
+ MOVD x+0(FP), R8
+ CMPUBEQ R4, R8, isNegInf
+ MOVD y+8(FP), R9
+ CMPUBEQ R4, R9, isNegInf
+ // NaN special cases
+ MOVD $~(1<<63), R5
+ MOVD $PosInf, R4
+ MOVD R8, R2
+ AND R5, R2 // x = |x|
+ CMPUBLT R4, R2, isMinNaN
+ MOVD R9, R3
+ AND R5, R3 // y = |y|
+ CMPUBLT R4, R3, isMinNaN
+ // ±0 special cases
+ OR R3, R2
+ BEQ isMinZero
+
+ FMOVD x+0(FP), F1
+ FMOVD y+8(FP), F2
+ FCMPU F2, F1
+ BLT +3(PC)
+ FMOVD F1, ret+16(FP)
+ RET
+ FMOVD F2, ret+16(FP)
+ RET
+isMinNaN: // return NaN
+ MOVD $NaN, R4
+isNegInf: // return -Inf
+ MOVD R4, ret+16(FP)
+ RET
+isMinZero:
+ MOVD $(1<<63), R4 // -0.0
+ CMPUBEQ R4, R8, +3(PC)
+ MOVD R9, ret+16(FP) // return other 0
+ RET
+ MOVD R8, ret+16(FP) // return -0
+ RET
+
diff --git a/src/math/erf.go b/src/math/erf.go
new file mode 100644
index 0000000..9b4048f
--- /dev/null
+++ b/src/math/erf.go
@@ -0,0 +1,339 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point error function and complementary error function.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/s_erf.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// double erf(double x)
+// double erfc(double x)
+// x
+// 2 |\
+// erf(x) = --------- | exp(-t*t)dt
+// sqrt(pi) \|
+// 0
+//
+// erfc(x) = 1-erf(x)
+// Note that
+// erf(-x) = -erf(x)
+// erfc(-x) = 2 - erfc(x)
+//
+// Method:
+// 1. For |x| in [0, 0.84375]
+// erf(x) = x + x*R(x**2)
+// erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
+// = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
+// where R = P/Q where P is an odd poly of degree 8 and
+// Q is an odd poly of degree 10.
+// -57.90
+// | R - (erf(x)-x)/x | <= 2
+//
+//
+// Remark. The formula is derived by noting
+// erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....)
+// and that
+// 2/sqrt(pi) = 1.128379167095512573896158903121545171688
+// is close to one. The interval is chosen because the fix
+// point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
+// near 0.6174), and by some experiment, 0.84375 is chosen to
+// guarantee the error is less than one ulp for erf.
+//
+// 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
+// c = 0.84506291151 rounded to single (24 bits)
+// erf(x) = sign(x) * (c + P1(s)/Q1(s))
+// erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
+// 1+(c+P1(s)/Q1(s)) if x < 0
+// |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
+// Remark: here we use the taylor series expansion at x=1.
+// erf(1+s) = erf(1) + s*Poly(s)
+// = 0.845.. + P1(s)/Q1(s)
+// That is, we use rational approximation to approximate
+// erf(1+s) - (c = (single)0.84506291151)
+// Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
+// where
+// P1(s) = degree 6 poly in s
+// Q1(s) = degree 6 poly in s
+//
+// 3. For x in [1.25,1/0.35(~2.857143)],
+// erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
+// erf(x) = 1 - erfc(x)
+// where
+// R1(z) = degree 7 poly in z, (z=1/x**2)
+// S1(z) = degree 8 poly in z
+//
+// 4. For x in [1/0.35,28]
+// erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
+// = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
+// = 2.0 - tiny (if x <= -6)
+// erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
+// erf(x) = sign(x)*(1.0 - tiny)
+// where
+// R2(z) = degree 6 poly in z, (z=1/x**2)
+// S2(z) = degree 7 poly in z
+//
+// Note1:
+// To compute exp(-x*x-0.5625+R/S), let s be a single
+// precision number and s := x; then
+// -x*x = -s*s + (s-x)*(s+x)
+// exp(-x*x-0.5626+R/S) =
+// exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
+// Note2:
+// Here 4 and 5 make use of the asymptotic series
+// exp(-x*x)
+// erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) )
+// x*sqrt(pi)
+// We use rational approximation to approximate
+// g(s)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625
+// Here is the error bound for R1/S1 and R2/S2
+// |R1/S1 - f(x)| < 2**(-62.57)
+// |R2/S2 - f(x)| < 2**(-61.52)
+//
+// 5. For inf > x >= 28
+// erf(x) = sign(x) *(1 - tiny) (raise inexact)
+// erfc(x) = tiny*tiny (raise underflow) if x > 0
+// = 2 - tiny if x<0
+//
+// 7. Special case:
+// erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
+// erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
+// erfc/erf(NaN) is NaN
+
+const (
+ erx = 8.45062911510467529297e-01 // 0x3FEB0AC160000000
+ // Coefficients for approximation to erf in [0, 0.84375]
+ efx = 1.28379167095512586316e-01 // 0x3FC06EBA8214DB69
+ efx8 = 1.02703333676410069053e+00 // 0x3FF06EBA8214DB69
+ pp0 = 1.28379167095512558561e-01 // 0x3FC06EBA8214DB68
+ pp1 = -3.25042107247001499370e-01 // 0xBFD4CD7D691CB913
+ pp2 = -2.84817495755985104766e-02 // 0xBF9D2A51DBD7194F
+ pp3 = -5.77027029648944159157e-03 // 0xBF77A291236668E4
+ pp4 = -2.37630166566501626084e-05 // 0xBEF8EAD6120016AC
+ qq1 = 3.97917223959155352819e-01 // 0x3FD97779CDDADC09
+ qq2 = 6.50222499887672944485e-02 // 0x3FB0A54C5536CEBA
+ qq3 = 5.08130628187576562776e-03 // 0x3F74D022C4D36B0F
+ qq4 = 1.32494738004321644526e-04 // 0x3F215DC9221C1A10
+ qq5 = -3.96022827877536812320e-06 // 0xBED09C4342A26120
+ // Coefficients for approximation to erf in [0.84375, 1.25]
+ pa0 = -2.36211856075265944077e-03 // 0xBF6359B8BEF77538
+ pa1 = 4.14856118683748331666e-01 // 0x3FDA8D00AD92B34D
+ pa2 = -3.72207876035701323847e-01 // 0xBFD7D240FBB8C3F1
+ pa3 = 3.18346619901161753674e-01 // 0x3FD45FCA805120E4
+ pa4 = -1.10894694282396677476e-01 // 0xBFBC63983D3E28EC
+ pa5 = 3.54783043256182359371e-02 // 0x3FA22A36599795EB
+ pa6 = -2.16637559486879084300e-03 // 0xBF61BF380A96073F
+ qa1 = 1.06420880400844228286e-01 // 0x3FBB3E6618EEE323
+ qa2 = 5.40397917702171048937e-01 // 0x3FE14AF092EB6F33
+ qa3 = 7.18286544141962662868e-02 // 0x3FB2635CD99FE9A7
+ qa4 = 1.26171219808761642112e-01 // 0x3FC02660E763351F
+ qa5 = 1.36370839120290507362e-02 // 0x3F8BEDC26B51DD1C
+ qa6 = 1.19844998467991074170e-02 // 0x3F888B545735151D
+ // Coefficients for approximation to erfc in [1.25, 1/0.35]
+ ra0 = -9.86494403484714822705e-03 // 0xBF843412600D6435
+ ra1 = -6.93858572707181764372e-01 // 0xBFE63416E4BA7360
+ ra2 = -1.05586262253232909814e+01 // 0xC0251E0441B0E726
+ ra3 = -6.23753324503260060396e+01 // 0xC04F300AE4CBA38D
+ ra4 = -1.62396669462573470355e+02 // 0xC0644CB184282266
+ ra5 = -1.84605092906711035994e+02 // 0xC067135CEBCCABB2
+ ra6 = -8.12874355063065934246e+01 // 0xC054526557E4D2F2
+ ra7 = -9.81432934416914548592e+00 // 0xC023A0EFC69AC25C
+ sa1 = 1.96512716674392571292e+01 // 0x4033A6B9BD707687
+ sa2 = 1.37657754143519042600e+02 // 0x4061350C526AE721
+ sa3 = 4.34565877475229228821e+02 // 0x407B290DD58A1A71
+ sa4 = 6.45387271733267880336e+02 // 0x40842B1921EC2868
+ sa5 = 4.29008140027567833386e+02 // 0x407AD02157700314
+ sa6 = 1.08635005541779435134e+02 // 0x405B28A3EE48AE2C
+ sa7 = 6.57024977031928170135e+00 // 0x401A47EF8E484A93
+ sa8 = -6.04244152148580987438e-02 // 0xBFAEEFF2EE749A62
+ // Coefficients for approximation to erfc in [1/.35, 28]
+ rb0 = -9.86494292470009928597e-03 // 0xBF84341239E86F4A
+ rb1 = -7.99283237680523006574e-01 // 0xBFE993BA70C285DE
+ rb2 = -1.77579549177547519889e+01 // 0xC031C209555F995A
+ rb3 = -1.60636384855821916062e+02 // 0xC064145D43C5ED98
+ rb4 = -6.37566443368389627722e+02 // 0xC083EC881375F228
+ rb5 = -1.02509513161107724954e+03 // 0xC09004616A2E5992
+ rb6 = -4.83519191608651397019e+02 // 0xC07E384E9BDC383F
+ sb1 = 3.03380607434824582924e+01 // 0x403E568B261D5190
+ sb2 = 3.25792512996573918826e+02 // 0x40745CAE221B9F0A
+ sb3 = 1.53672958608443695994e+03 // 0x409802EB189D5118
+ sb4 = 3.19985821950859553908e+03 // 0x40A8FFB7688C246A
+ sb5 = 2.55305040643316442583e+03 // 0x40A3F219CEDF3BE6
+ sb6 = 4.74528541206955367215e+02 // 0x407DA874E79FE763
+ sb7 = -2.24409524465858183362e+01 // 0xC03670E242712D62
+)
+
+// Erf returns the error function of x.
+//
+// Special cases are:
+// Erf(+Inf) = 1
+// Erf(-Inf) = -1
+// Erf(NaN) = NaN
+func Erf(x float64) float64
+
+func erf(x float64) float64 {
+ const (
+ VeryTiny = 2.848094538889218e-306 // 0x0080000000000000
+ Small = 1.0 / (1 << 28) // 2**-28
+ )
+ // special cases
+ switch {
+ case IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return 1
+ case IsInf(x, -1):
+ return -1
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ if x < 0.84375 { // |x| < 0.84375
+ var temp float64
+ if x < Small { // |x| < 2**-28
+ if x < VeryTiny {
+ temp = 0.125 * (8.0*x + efx8*x) // avoid underflow
+ } else {
+ temp = x + efx*x
+ }
+ } else {
+ z := x * x
+ r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
+ s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
+ y := r / s
+ temp = x + x*y
+ }
+ if sign {
+ return -temp
+ }
+ return temp
+ }
+ if x < 1.25 { // 0.84375 <= |x| < 1.25
+ s := x - 1
+ P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
+ Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
+ if sign {
+ return -erx - P/Q
+ }
+ return erx + P/Q
+ }
+ if x >= 6 { // inf > |x| >= 6
+ if sign {
+ return -1
+ }
+ return 1
+ }
+ s := 1 / (x * x)
+ var R, S float64
+ if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
+ R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
+ S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
+ } else { // |x| >= 1 / 0.35 ~ 2.857143
+ R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
+ S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
+ }
+ z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
+ r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
+ if sign {
+ return r/x - 1
+ }
+ return 1 - r/x
+}
+
+// Erfc returns the complementary error function of x.
+//
+// Special cases are:
+// Erfc(+Inf) = 0
+// Erfc(-Inf) = 2
+// Erfc(NaN) = NaN
+func Erfc(x float64) float64
+
+func erfc(x float64) float64 {
+ const Tiny = 1.0 / (1 << 56) // 2**-56
+ // special cases
+ switch {
+ case IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return 0
+ case IsInf(x, -1):
+ return 2
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ if x < 0.84375 { // |x| < 0.84375
+ var temp float64
+ if x < Tiny { // |x| < 2**-56
+ temp = x
+ } else {
+ z := x * x
+ r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
+ s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
+ y := r / s
+ if x < 0.25 { // |x| < 1/4
+ temp = x + x*y
+ } else {
+ temp = 0.5 + (x*y + (x - 0.5))
+ }
+ }
+ if sign {
+ return 1 + temp
+ }
+ return 1 - temp
+ }
+ if x < 1.25 { // 0.84375 <= |x| < 1.25
+ s := x - 1
+ P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
+ Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
+ if sign {
+ return 1 + erx + P/Q
+ }
+ return 1 - erx - P/Q
+
+ }
+ if x < 28 { // |x| < 28
+ s := 1 / (x * x)
+ var R, S float64
+ if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
+ R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
+ S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
+ } else { // |x| >= 1 / 0.35 ~ 2.857143
+ if sign && x > 6 {
+ return 2 // x < -6
+ }
+ R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
+ S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
+ }
+ z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
+ r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
+ if sign {
+ return 2 - r/x
+ }
+ return r / x
+ }
+ if sign {
+ return 2
+ }
+ return 0
+}
diff --git a/src/math/erf_s390x.s b/src/math/erf_s390x.s
new file mode 100644
index 0000000..99ab436
--- /dev/null
+++ b/src/math/erf_s390x.s
@@ -0,0 +1,293 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial coefficients and other constants
+DATA ·erfrodataL13<> + 0(SB)/8, $0.243673229298474689E+01
+DATA ·erfrodataL13<> + 8(SB)/8, $-.654905018503145600E+00
+DATA ·erfrodataL13<> + 16(SB)/8, $0.404669310217538718E+01
+DATA ·erfrodataL13<> + 24(SB)/8, $-.564189219162765367E+00
+DATA ·erfrodataL13<> + 32(SB)/8, $-.200104300906596851E+01
+DATA ·erfrodataL13<> + 40(SB)/8, $0.5
+DATA ·erfrodataL13<> + 48(SB)/8, $0.144070097650207154E+00
+DATA ·erfrodataL13<> + 56(SB)/8, $-.116697735205906191E+00
+DATA ·erfrodataL13<> + 64(SB)/8, $0.256847684882319665E-01
+DATA ·erfrodataL13<> + 72(SB)/8, $-.510805169106229148E-02
+DATA ·erfrodataL13<> + 80(SB)/8, $0.885258164825590267E-03
+DATA ·erfrodataL13<> + 88(SB)/8, $-.133861989591931411E-03
+DATA ·erfrodataL13<> + 96(SB)/8, $0.178294867340272534E-04
+DATA ·erfrodataL13<> + 104(SB)/8, $-.211436095674019218E-05
+DATA ·erfrodataL13<> + 112(SB)/8, $0.225503753499344434E-06
+DATA ·erfrodataL13<> + 120(SB)/8, $-.218247939190783624E-07
+DATA ·erfrodataL13<> + 128(SB)/8, $0.193179206264594029E-08
+DATA ·erfrodataL13<> + 136(SB)/8, $-.157440643541715319E-09
+DATA ·erfrodataL13<> + 144(SB)/8, $0.118878583237342616E-10
+DATA ·erfrodataL13<> + 152(SB)/8, $0.554289288424588473E-13
+DATA ·erfrodataL13<> + 160(SB)/8, $-.277649758489502214E-14
+DATA ·erfrodataL13<> + 168(SB)/8, $-.839318416990049443E-12
+DATA ·erfrodataL13<> + 176(SB)/8, $-2.25
+DATA ·erfrodataL13<> + 184(SB)/8, $.12837916709551258632
+DATA ·erfrodataL13<> + 192(SB)/8, $1.0
+DATA ·erfrodataL13<> + 200(SB)/8, $0.500000000000004237e+00
+DATA ·erfrodataL13<> + 208(SB)/8, $1.0
+DATA ·erfrodataL13<> + 216(SB)/8, $0.416666664838056960e-01
+DATA ·erfrodataL13<> + 224(SB)/8, $0.166666666630345592e+00
+DATA ·erfrodataL13<> + 232(SB)/8, $0.138926439368309441e-02
+DATA ·erfrodataL13<> + 240(SB)/8, $0.833349307718286047e-02
+DATA ·erfrodataL13<> + 248(SB)/8, $-.693147180559945286e+00
+DATA ·erfrodataL13<> + 256(SB)/8, $-.144269504088896339e+01
+DATA ·erfrodataL13<> + 264(SB)/8, $281475245147134.9375
+DATA ·erfrodataL13<> + 272(SB)/8, $0.358256136398192529E+01
+DATA ·erfrodataL13<> + 280(SB)/8, $-.554084396500738270E+00
+DATA ·erfrodataL13<> + 288(SB)/8, $0.203630123025312046E+02
+DATA ·erfrodataL13<> + 296(SB)/8, $-.735750304705934424E+01
+DATA ·erfrodataL13<> + 304(SB)/8, $0.250491598091071797E+02
+DATA ·erfrodataL13<> + 312(SB)/8, $-.118955882760959931E+02
+DATA ·erfrodataL13<> + 320(SB)/8, $0.942903335085524187E+01
+DATA ·erfrodataL13<> + 328(SB)/8, $-.564189522219085689E+00
+DATA ·erfrodataL13<> + 336(SB)/8, $-.503767199403555540E+01
+DATA ·erfrodataL13<> + 344(SB)/8, $0xbbc79ca10c924223
+DATA ·erfrodataL13<> + 352(SB)/8, $0.004099975562609307E+01
+DATA ·erfrodataL13<> + 360(SB)/8, $-.324434353381296556E+00
+DATA ·erfrodataL13<> + 368(SB)/8, $0.945204812084476250E-01
+DATA ·erfrodataL13<> + 376(SB)/8, $-.221407443830058214E-01
+DATA ·erfrodataL13<> + 384(SB)/8, $0.426072376238804349E-02
+DATA ·erfrodataL13<> + 392(SB)/8, $-.692229229127016977E-03
+DATA ·erfrodataL13<> + 400(SB)/8, $0.971111253652087188E-04
+DATA ·erfrodataL13<> + 408(SB)/8, $-.119752226272050504E-04
+DATA ·erfrodataL13<> + 416(SB)/8, $0.131662993588532278E-05
+DATA ·erfrodataL13<> + 424(SB)/8, $0.115776482315851236E-07
+DATA ·erfrodataL13<> + 432(SB)/8, $-.780118522218151687E-09
+DATA ·erfrodataL13<> + 440(SB)/8, $-.130465975877241088E-06
+DATA ·erfrodataL13<> + 448(SB)/8, $-0.25
+GLOBL ·erfrodataL13<> + 0(SB), RODATA, $456
+
+// Table of log correction terms
+DATA ·erftab2066<> + 0(SB)/8, $0.442737824274138381e-01
+DATA ·erftab2066<> + 8(SB)/8, $0.263602189790660309e-01
+DATA ·erftab2066<> + 16(SB)/8, $0.122565642281703586e-01
+DATA ·erftab2066<> + 24(SB)/8, $0.143757052860721398e-02
+DATA ·erftab2066<> + 32(SB)/8, $-.651375034121276075e-02
+DATA ·erftab2066<> + 40(SB)/8, $-.119317678849450159e-01
+DATA ·erftab2066<> + 48(SB)/8, $-.150868749549871069e-01
+DATA ·erftab2066<> + 56(SB)/8, $-.161992609578469234e-01
+DATA ·erftab2066<> + 64(SB)/8, $-.154492360403337917e-01
+DATA ·erftab2066<> + 72(SB)/8, $-.129850717389178721e-01
+DATA ·erftab2066<> + 80(SB)/8, $-.892902649276657891e-02
+DATA ·erftab2066<> + 88(SB)/8, $-.338202636596794887e-02
+DATA ·erftab2066<> + 96(SB)/8, $0.357266307045684762e-02
+DATA ·erftab2066<> + 104(SB)/8, $0.118665304327406698e-01
+DATA ·erftab2066<> + 112(SB)/8, $0.214434994118118914e-01
+DATA ·erftab2066<> + 120(SB)/8, $0.322580645161290314e-01
+GLOBL ·erftab2066<> + 0(SB), RODATA, $128
+
+// Table of +/- 1.0
+DATA ·erftab12067<> + 0(SB)/8, $1.0
+DATA ·erftab12067<> + 8(SB)/8, $-1.0
+GLOBL ·erftab12067<> + 0(SB), RODATA, $16
+
+// Erf returns the error function of the argument.
+//
+// Special cases are:
+// Erf(+Inf) = 1
+// Erf(-Inf) = -1
+// Erf(NaN) = NaN
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·erfAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·erfrodataL13<>+0(SB), R5
+ LGDR F0, R1
+ FMOVD F0, F6
+ SRAD $48, R1
+ MOVH $16383, R3
+ RISBGZ $49, $63, $0, R1, R2
+ MOVW R2, R6
+ MOVW R3, R7
+ CMPBGT R6, R7, L2
+ MOVH $12287, R1
+ MOVW R1, R7
+ CMPBLE R6, R7 ,L12
+ MOVH $16367, R1
+ MOVW R1, R7
+ CMPBGT R6, R7, L5
+ FMOVD 448(R5), F4
+ FMADD F0, F0, F4
+ FMOVD 440(R5), F3
+ WFMDB V4, V4, V2
+ FMOVD 432(R5), F0
+ FMOVD 424(R5), F1
+ WFMADB V2, V0, V3, V0
+ FMOVD 416(R5), F3
+ WFMADB V2, V1, V3, V1
+ FMOVD 408(R5), F5
+ FMOVD 400(R5), F3
+ WFMADB V2, V0, V5, V0
+ WFMADB V2, V1, V3, V1
+ FMOVD 392(R5), F5
+ FMOVD 384(R5), F3
+ WFMADB V2, V0, V5, V0
+ WFMADB V2, V1, V3, V1
+ FMOVD 376(R5), F5
+ FMOVD 368(R5), F3
+ WFMADB V2, V0, V5, V0
+ WFMADB V2, V1, V3, V1
+ FMOVD 360(R5), F5
+ FMOVD 352(R5), F3
+ WFMADB V2, V0, V5, V0
+ WFMADB V2, V1, V3, V2
+ WFMADB V4, V0, V2, V0
+ WFMADB V6, V0, V6, V0
+L1:
+ FMOVD F0, ret+8(FP)
+ RET
+L2:
+ MOVH R1, R1
+ MOVH $16407, R3
+ SRW $31, R1, R1
+ MOVW R2, R6
+ MOVW R3, R7
+ CMPBLE R6, R7, L6
+ MOVW R1, R1
+ SLD $3, R1, R1
+ MOVD $·erftab12067<>+0(SB), R3
+ WORD $0x68013000 //ld %f0,0(%r1,%r3)
+ MOVH $32751, R1
+ MOVW R1, R7
+ CMPBGT R6, R7, L7
+ FMOVD 344(R5), F2
+ FMADD F2, F0, F0
+L7:
+ WFCEDBS V6, V6, V2
+ BEQ L1
+ FMOVD F6, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L6:
+ MOVW R1, R1
+ SLD $3, R1, R1
+ MOVD $·erftab12067<>+0(SB), R4
+ WFMDB V0, V0, V1
+ MOVH $0x0, R3
+ WORD $0x68014000 //ld %f0,0(%r1,%r4)
+ MOVH $16399, R1
+ MOVW R2, R6
+ MOVW R1, R7
+ CMPBGT R6, R7, L8
+ FMOVD 336(R5), F3
+ FMOVD 328(R5), F2
+ FMOVD F1, F4
+ WFMADB V1, V2, V3, V2
+ WORD $0xED405140 //adb %f4,.L30-.L13(%r5)
+ BYTE $0x00
+ BYTE $0x1A
+ FMOVD 312(R5), F3
+ WFMADB V1, V2, V3, V2
+ FMOVD 304(R5), F3
+ WFMADB V1, V4, V3, V4
+ FMOVD 296(R5), F3
+ WFMADB V1, V2, V3, V2
+ FMOVD 288(R5), F3
+ WFMADB V1, V4, V3, V4
+ FMOVD 280(R5), F3
+ WFMADB V1, V2, V3, V2
+ FMOVD 272(R5), F3
+ WFMADB V1, V4, V3, V4
+L9:
+ FMOVD 264(R5), F3
+ FMUL F4, F6
+ FMOVD 256(R5), F4
+ WFMADB V1, V4, V3, V4
+ FDIV F6, F2
+ LGDR F4, R1
+ FSUB F3, F4
+ FMOVD 248(R5), F6
+ WFMSDB V4, V6, V1, V4
+ FMOVD 240(R5), F1
+ FMOVD 232(R5), F6
+ WFMADB V4, V6, V1, V6
+ FMOVD 224(R5), F1
+ FMOVD 216(R5), F3
+ WFMADB V4, V3, V1, V3
+ WFMDB V4, V4, V1
+ FMOVD 208(R5), F5
+ WFMADB V6, V1, V3, V6
+ FMOVD 200(R5), F3
+ MOVH R1,R1
+ WFMADB V4, V3, V5, V3
+ RISBGZ $57, $60, $3, R1, R2
+ WFMADB V1, V6, V3, V6
+ RISBGN $0, $15, $48, R1, R3
+ MOVD $·erftab2066<>+0(SB), R1
+ FMOVD 192(R5), F1
+ LDGR R3, F3
+ WORD $0xED221000 //madb %f2,%f2,0(%r2,%r1)
+ BYTE $0x20
+ BYTE $0x1E
+ WFMADB V4, V6, V1, V4
+ FMUL F3, F2
+ FMADD F4, F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+L12:
+ FMOVD 184(R5), F0
+ WFMADB V6, V0, V6, V0
+ FMOVD F0, ret+8(FP)
+ RET
+L5:
+ FMOVD 176(R5), F1
+ FMADD F0, F0, F1
+ FMOVD 168(R5), F3
+ WFMDB V1, V1, V2
+ FMOVD 160(R5), F0
+ FMOVD 152(R5), F4
+ WFMADB V2, V0, V3, V0
+ FMOVD 144(R5), F3
+ WFMADB V2, V4, V3, V4
+ FMOVD 136(R5), F5
+ FMOVD 128(R5), F3
+ WFMADB V2, V0, V5, V0
+ WFMADB V2, V4, V3, V4
+ FMOVD 120(R5), F5
+ FMOVD 112(R5), F3
+ WFMADB V2, V0, V5, V0
+ WFMADB V2, V4, V3, V4
+ FMOVD 104(R5), F5
+ FMOVD 96(R5), F3
+ WFMADB V2, V0, V5, V0
+ WFMADB V2, V4, V3, V4
+ FMOVD 88(R5), F5
+ FMOVD 80(R5), F3
+ WFMADB V2, V0, V5, V0
+ WFMADB V2, V4, V3, V4
+ FMOVD 72(R5), F5
+ FMOVD 64(R5), F3
+ WFMADB V2, V0, V5, V0
+ WFMADB V2, V4, V3, V4
+ FMOVD 56(R5), F5
+ FMOVD 48(R5), F3
+ WFMADB V2, V0, V5, V0
+ WFMADB V2, V4, V3, V2
+ FMOVD 40(R5), F4
+ WFMADB V1, V0, V2, V0
+ FMUL F6, F0
+ FMADD F4, F6, F0
+ FMOVD F0, ret+8(FP)
+ RET
+L8:
+ FMOVD 32(R5), F3
+ FMOVD 24(R5), F2
+ FMOVD F1, F4
+ WFMADB V1, V2, V3, V2
+ WORD $0xED405010 //adb %f4,.L68-.L13(%r5)
+ BYTE $0x00
+ BYTE $0x1A
+ FMOVD 8(R5), F3
+ WFMADB V1, V2, V3, V2
+ FMOVD ·erfrodataL13<>+0(SB), F3
+ WFMADB V1, V4, V3, V4
+ BR L9
diff --git a/src/math/erfc_s390x.s b/src/math/erfc_s390x.s
new file mode 100644
index 0000000..7e9d469
--- /dev/null
+++ b/src/math/erfc_s390x.s
@@ -0,0 +1,527 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define Neg2p11 0xC000E147AE147AE1
+#define Pos15 0x402E
+
+// Minimax polynomial coefficients and other constants
+DATA ·erfcrodataL38<> + 0(SB)/8, $.234875460637085087E-01
+DATA ·erfcrodataL38<> + 8(SB)/8, $.234469449299256284E-01
+DATA ·erfcrodataL38<> + 16(SB)/8, $-.606918710392844955E-04
+DATA ·erfcrodataL38<> + 24(SB)/8, $-.198827088077636213E-04
+DATA ·erfcrodataL38<> + 32(SB)/8, $.257805645845475331E-06
+DATA ·erfcrodataL38<> + 40(SB)/8, $-.184427218110620284E-09
+DATA ·erfcrodataL38<> + 48(SB)/8, $.122408098288933181E-10
+DATA ·erfcrodataL38<> + 56(SB)/8, $.484691106751495392E-07
+DATA ·erfcrodataL38<> + 64(SB)/8, $-.150147637632890281E-08
+DATA ·erfcrodataL38<> + 72(SB)/8, $23.999999999973521625
+DATA ·erfcrodataL38<> + 80(SB)/8, $27.226017111108365754
+DATA ·erfcrodataL38<> + 88(SB)/8, $-2.0
+DATA ·erfcrodataL38<> + 96(SB)/8, $0.100108802034478228E+00
+DATA ·erfcrodataL38<> + 104(SB)/8, $0.244588413746558125E+00
+DATA ·erfcrodataL38<> + 112(SB)/8, $-.669188879646637174E-01
+DATA ·erfcrodataL38<> + 120(SB)/8, $0.151311447000953551E-01
+DATA ·erfcrodataL38<> + 128(SB)/8, $-.284720833493302061E-02
+DATA ·erfcrodataL38<> + 136(SB)/8, $0.455491239358743212E-03
+DATA ·erfcrodataL38<> + 144(SB)/8, $-.631850539280720949E-04
+DATA ·erfcrodataL38<> + 152(SB)/8, $0.772532660726086679E-05
+DATA ·erfcrodataL38<> + 160(SB)/8, $-.843706007150936940E-06
+DATA ·erfcrodataL38<> + 168(SB)/8, $-.735330214904227472E-08
+DATA ·erfcrodataL38<> + 176(SB)/8, $0.753002008837084967E-09
+DATA ·erfcrodataL38<> + 184(SB)/8, $0.832482036660624637E-07
+DATA ·erfcrodataL38<> + 192(SB)/8, $-0.75
+DATA ·erfcrodataL38<> + 200(SB)/8, $.927765678007128609E-01
+DATA ·erfcrodataL38<> + 208(SB)/8, $.903621209344751506E-01
+DATA ·erfcrodataL38<> + 216(SB)/8, $-.344203375025257265E-02
+DATA ·erfcrodataL38<> + 224(SB)/8, $-.869243428221791329E-03
+DATA ·erfcrodataL38<> + 232(SB)/8, $.174699813107105603E-03
+DATA ·erfcrodataL38<> + 240(SB)/8, $.649481036316130000E-05
+DATA ·erfcrodataL38<> + 248(SB)/8, $-.895265844897118382E-05
+DATA ·erfcrodataL38<> + 256(SB)/8, $.135970046909529513E-05
+DATA ·erfcrodataL38<> + 264(SB)/8, $.277617717014748015E-06
+DATA ·erfcrodataL38<> + 272(SB)/8, $.810628018408232910E-08
+DATA ·erfcrodataL38<> + 280(SB)/8, $.210430084693497985E-07
+DATA ·erfcrodataL38<> + 288(SB)/8, $-.342138077525615091E-08
+DATA ·erfcrodataL38<> + 296(SB)/8, $-.165467946798610800E-06
+DATA ·erfcrodataL38<> + 304(SB)/8, $5.999999999988412824
+DATA ·erfcrodataL38<> + 312(SB)/8, $.468542210149072159E-01
+DATA ·erfcrodataL38<> + 320(SB)/8, $.465343528567604256E-01
+DATA ·erfcrodataL38<> + 328(SB)/8, $-.473338083650201733E-03
+DATA ·erfcrodataL38<> + 336(SB)/8, $-.147220659069079156E-03
+DATA ·erfcrodataL38<> + 344(SB)/8, $.755284723554388339E-05
+DATA ·erfcrodataL38<> + 352(SB)/8, $.116158570631428789E-05
+DATA ·erfcrodataL38<> + 360(SB)/8, $-.155445501551602389E-06
+DATA ·erfcrodataL38<> + 368(SB)/8, $-.616940119847805046E-10
+DATA ·erfcrodataL38<> + 376(SB)/8, $-.728705590727563158E-10
+DATA ·erfcrodataL38<> + 384(SB)/8, $-.983452460354586779E-08
+DATA ·erfcrodataL38<> + 392(SB)/8, $.365156164194346316E-08
+DATA ·erfcrodataL38<> + 400(SB)/8, $11.999999999996530775
+DATA ·erfcrodataL38<> + 408(SB)/8, $0.467773498104726584E-02
+DATA ·erfcrodataL38<> + 416(SB)/8, $0.206669853540920535E-01
+DATA ·erfcrodataL38<> + 424(SB)/8, $0.413339707081841473E-01
+DATA ·erfcrodataL38<> + 432(SB)/8, $0.482229658262131320E-01
+DATA ·erfcrodataL38<> + 440(SB)/8, $0.344449755901841897E-01
+DATA ·erfcrodataL38<> + 448(SB)/8, $0.130890907240765465E-01
+DATA ·erfcrodataL38<> + 456(SB)/8, $-.459266344100642687E-03
+DATA ·erfcrodataL38<> + 464(SB)/8, $-.337888800856913728E-02
+DATA ·erfcrodataL38<> + 472(SB)/8, $-.159103061687062373E-02
+DATA ·erfcrodataL38<> + 480(SB)/8, $-.501128905515922644E-04
+DATA ·erfcrodataL38<> + 488(SB)/8, $0.262775855852903132E-03
+DATA ·erfcrodataL38<> + 496(SB)/8, $0.103860982197462436E-03
+DATA ·erfcrodataL38<> + 504(SB)/8, $-.548835785414200775E-05
+DATA ·erfcrodataL38<> + 512(SB)/8, $-.157075054646618214E-04
+DATA ·erfcrodataL38<> + 520(SB)/8, $-.480056366276045110E-05
+DATA ·erfcrodataL38<> + 528(SB)/8, $0.198263013759701555E-05
+DATA ·erfcrodataL38<> + 536(SB)/8, $-.224394262958888780E-06
+DATA ·erfcrodataL38<> + 544(SB)/8, $-.321853693146683428E-06
+DATA ·erfcrodataL38<> + 552(SB)/8, $0.445073894984683537E-07
+DATA ·erfcrodataL38<> + 560(SB)/8, $0.660425940000555729E-06
+DATA ·erfcrodataL38<> + 568(SB)/8, $2.0
+DATA ·erfcrodataL38<> + 576(SB)/8, $8.63616855509444462538e-78
+DATA ·erfcrodataL38<> + 584(SB)/8, $1.00000000000000222044
+DATA ·erfcrodataL38<> + 592(SB)/8, $0.500000000000004237e+00
+DATA ·erfcrodataL38<> + 600(SB)/8, $0.416666664838056960e-01
+DATA ·erfcrodataL38<> + 608(SB)/8, $0.166666666630345592e+00
+DATA ·erfcrodataL38<> + 616(SB)/8, $0.138926439368309441e-02
+DATA ·erfcrodataL38<> + 624(SB)/8, $0.833349307718286047e-02
+DATA ·erfcrodataL38<> + 632(SB)/8, $-.693147180558298714e+00
+DATA ·erfcrodataL38<> + 640(SB)/8, $-.164659495826017651e-11
+DATA ·erfcrodataL38<> + 648(SB)/8, $.179001151181866548E+00
+DATA ·erfcrodataL38<> + 656(SB)/8, $-.144269504088896339e+01
+DATA ·erfcrodataL38<> + 664(SB)/8, $+281475245147134.9375
+DATA ·erfcrodataL38<> + 672(SB)/8, $.163116780021877404E+00
+DATA ·erfcrodataL38<> + 680(SB)/8, $-.201574395828120710E-01
+DATA ·erfcrodataL38<> + 688(SB)/8, $-.185726336009394125E-02
+DATA ·erfcrodataL38<> + 696(SB)/8, $.199349204957273749E-02
+DATA ·erfcrodataL38<> + 704(SB)/8, $-.554902415532606242E-03
+DATA ·erfcrodataL38<> + 712(SB)/8, $-.638914789660242846E-05
+DATA ·erfcrodataL38<> + 720(SB)/8, $-.424441522653742898E-04
+DATA ·erfcrodataL38<> + 728(SB)/8, $.827967511921486190E-04
+DATA ·erfcrodataL38<> + 736(SB)/8, $.913965446284062654E-05
+DATA ·erfcrodataL38<> + 744(SB)/8, $.277344791076320853E-05
+DATA ·erfcrodataL38<> + 752(SB)/8, $-.467239678927239526E-06
+DATA ·erfcrodataL38<> + 760(SB)/8, $.344814065920419986E-07
+DATA ·erfcrodataL38<> + 768(SB)/8, $-.366013491552527132E-05
+DATA ·erfcrodataL38<> + 776(SB)/8, $.181242810023783439E-05
+DATA ·erfcrodataL38<> + 784(SB)/8, $2.999999999991234567
+DATA ·erfcrodataL38<> + 792(SB)/8, $1.0
+GLOBL ·erfcrodataL38<> + 0(SB), RODATA, $800
+
+// Table of log correction terms
+DATA ·erfctab2069<> + 0(SB)/8, $0.442737824274138381e-01
+DATA ·erfctab2069<> + 8(SB)/8, $0.263602189790660309e-01
+DATA ·erfctab2069<> + 16(SB)/8, $0.122565642281703586e-01
+DATA ·erfctab2069<> + 24(SB)/8, $0.143757052860721398e-02
+DATA ·erfctab2069<> + 32(SB)/8, $-.651375034121276075e-02
+DATA ·erfctab2069<> + 40(SB)/8, $-.119317678849450159e-01
+DATA ·erfctab2069<> + 48(SB)/8, $-.150868749549871069e-01
+DATA ·erfctab2069<> + 56(SB)/8, $-.161992609578469234e-01
+DATA ·erfctab2069<> + 64(SB)/8, $-.154492360403337917e-01
+DATA ·erfctab2069<> + 72(SB)/8, $-.129850717389178721e-01
+DATA ·erfctab2069<> + 80(SB)/8, $-.892902649276657891e-02
+DATA ·erfctab2069<> + 88(SB)/8, $-.338202636596794887e-02
+DATA ·erfctab2069<> + 96(SB)/8, $0.357266307045684762e-02
+DATA ·erfctab2069<> + 104(SB)/8, $0.118665304327406698e-01
+DATA ·erfctab2069<> + 112(SB)/8, $0.214434994118118914e-01
+DATA ·erfctab2069<> + 120(SB)/8, $0.322580645161290314e-01
+GLOBL ·erfctab2069<> + 0(SB), RODATA, $128
+
+// Erfc returns the complementary error function of the argument.
+//
+// Special cases are:
+// Erfc(+Inf) = 0
+// Erfc(-Inf) = 2
+// Erfc(NaN) = NaN
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+// This assembly implementation handles inputs in the range [-2.11, +15].
+// For all other inputs we call the generic Go implementation.
+
+TEXT ·erfcAsm(SB), NOSPLIT|NOFRAME, $0-16
+ MOVD x+0(FP), R1
+ MOVD $Neg2p11, R2
+ CMPUBGT R1, R2, usego
+
+ FMOVD x+0(FP), F0
+ MOVD $·erfcrodataL38<>+0(SB), R9
+ FMOVD F0, F2
+ SRAD $48, R1
+ MOVH R1, R2
+ ANDW $0x7FFF, R1
+ MOVH $Pos15, R3
+ CMPW R1, R3
+ BGT usego
+ MOVH $0x3FFF, R3
+ MOVW R1, R6
+ MOVW R3, R7
+ CMPBGT R6, R7, L2
+ MOVH $0x3FEF, R3
+ MOVW R3, R7
+ CMPBGT R6, R7, L3
+ MOVH $0x2FFF, R2
+ MOVW R2, R7
+ CMPBGT R6, R7, L4
+ FMOVD 792(R9), F0
+ WFSDB V2, V0, V2
+ FMOVD F2, ret+8(FP)
+ RET
+
+L2:
+ LTDBR F0, F0
+ MOVH $0x0, R4
+ BLTU L3
+ FMOVD F0, F1
+L9:
+ MOVH $0x400F, R3
+ MOVW R1, R6
+ MOVW R3, R7
+ CMPBGT R6, R7, L10
+ FMOVD 784(R9), F3
+ FSUB F1, F3
+ VLEG $0, 776(R9), V20
+ WFDDB V1, V3, V6
+ VLEG $0, 768(R9), V18
+ FMOVD 760(R9), F7
+ FMOVD 752(R9), F5
+ VLEG $0, 744(R9), V16
+ FMOVD 736(R9), F3
+ FMOVD 728(R9), F2
+ FMOVD 720(R9), F4
+ WFMDB V6, V6, V1
+ FMUL F0, F0
+ MOVH $0x0, R3
+ WFMADB V1, V7, V20, V7
+ WFMADB V1, V5, V18, V5
+ WFMADB V1, V7, V16, V7
+ WFMADB V1, V5, V3, V5
+ WFMADB V1, V7, V4, V7
+ WFMADB V1, V5, V2, V5
+ FMOVD 712(R9), F2
+ WFMADB V1, V7, V2, V7
+ FMOVD 704(R9), F2
+ WFMADB V1, V5, V2, V5
+ FMOVD 696(R9), F2
+ WFMADB V1, V7, V2, V7
+ FMOVD 688(R9), F2
+ MOVH $0x0, R1
+ WFMADB V1, V5, V2, V5
+ FMOVD 680(R9), F2
+ WFMADB V1, V7, V2, V7
+ FMOVD 672(R9), F2
+ WFMADB V1, V5, V2, V1
+ FMOVD 664(R9), F3
+ WFMADB V6, V7, V1, V7
+ FMOVD 656(R9), F5
+ FMOVD 648(R9), F2
+ WFMADB V0, V5, V3, V5
+ WFMADB V6, V7, V2, V7
+L11:
+ LGDR F5, R6
+ WFSDB V0, V0, V2
+ WORD $0xED509298 //sdb %f5,.L55-.L38(%r9)
+ BYTE $0x00
+ BYTE $0x1B
+ FMOVD 640(R9), F6
+ FMOVD 632(R9), F4
+ WFMSDB V5, V6, V2, V6
+ WFMSDB V5, V4, V0, V4
+ FMOVD 624(R9), F2
+ FADD F6, F4
+ FMOVD 616(R9), F0
+ FMOVD 608(R9), F6
+ WFMADB V4, V0, V2, V0
+ FMOVD 600(R9), F3
+ WFMDB V4, V4, V2
+ MOVH R6,R6
+ ADD R6, R3
+ WFMADB V4, V3, V6, V3
+ FMOVD 592(R9), F6
+ WFMADB V0, V2, V3, V0
+ FMOVD 584(R9), F3
+ WFMADB V4, V6, V3, V6
+ RISBGZ $57, $60, $3, R3, R12
+ WFMADB V2, V0, V6, V0
+ MOVD $·erfctab2069<>+0(SB), R5
+ WORD $0x682C5000 //ld %f2,0(%r12,%r5)
+ FMADD F2, F4, F4
+ RISBGN $0, $15, $48, R3, R4
+ WFMADB V4, V0, V2, V4
+ LDGR R4, F2
+ FMADD F4, F2, F2
+ MOVW R2, R6
+ CMPBLE R6, $0, L20
+ MOVW R1, R6
+ CMPBEQ R6, $0, L21
+ WORD $0xED709240 //mdb %f7,.L66-.L38(%r9)
+ BYTE $0x00
+ BYTE $0x1C
+L21:
+ FMUL F7, F2
+L1:
+ FMOVD F2, ret+8(FP)
+ RET
+L3:
+ LTDBR F0, F0
+ BLTU L30
+ FMOVD 568(R9), F2
+ WFSDB V0, V2, V0
+L8:
+ WFMDB V0, V0, V4
+ FMOVD 560(R9), F2
+ FMOVD 552(R9), F6
+ FMOVD 544(R9), F1
+ WFMADB V4, V6, V2, V6
+ FMOVD 536(R9), F2
+ WFMADB V4, V1, V2, V1
+ FMOVD 528(R9), F3
+ FMOVD 520(R9), F2
+ WFMADB V4, V6, V3, V6
+ WFMADB V4, V1, V2, V1
+ FMOVD 512(R9), F3
+ FMOVD 504(R9), F2
+ WFMADB V4, V6, V3, V6
+ WFMADB V4, V1, V2, V1
+ FMOVD 496(R9), F3
+ FMOVD 488(R9), F2
+ WFMADB V4, V6, V3, V6
+ WFMADB V4, V1, V2, V1
+ FMOVD 480(R9), F3
+ FMOVD 472(R9), F2
+ WFMADB V4, V6, V3, V6
+ WFMADB V4, V1, V2, V1
+ FMOVD 464(R9), F3
+ FMOVD 456(R9), F2
+ WFMADB V4, V6, V3, V6
+ WFMADB V4, V1, V2, V1
+ FMOVD 448(R9), F3
+ FMOVD 440(R9), F2
+ WFMADB V4, V6, V3, V6
+ WFMADB V4, V1, V2, V1
+ FMOVD 432(R9), F3
+ FMOVD 424(R9), F2
+ WFMADB V4, V6, V3, V6
+ WFMADB V4, V1, V2, V1
+ FMOVD 416(R9), F3
+ FMOVD 408(R9), F2
+ WFMADB V4, V6, V3, V6
+ FMADD F1, F4, F2
+ FMADD F6, F0, F2
+ MOVW R2, R6
+ CMPBGE R6, $0, L1
+ FMOVD 568(R9), F0
+ WFSDB V2, V0, V2
+ BR L1
+L10:
+ MOVH $0x401F, R3
+ MOVW R1, R6
+ MOVW R3, R7
+ CMPBLE R6, R7, L36
+ MOVH $0x402F, R3
+ MOVW R3, R7
+ CMPBGT R6, R7, L13
+ FMOVD 400(R9), F3
+ FSUB F1, F3
+ VLEG $0, 392(R9), V20
+ WFDDB V1, V3, V6
+ VLEG $0, 384(R9), V18
+ FMOVD 376(R9), F2
+ FMOVD 368(R9), F4
+ VLEG $0, 360(R9), V16
+ FMOVD 352(R9), F7
+ FMOVD 344(R9), F3
+ FMUL F0, F0
+ WFMDB V6, V6, V1
+ FMOVD 656(R9), F5
+ MOVH $0x0, R3
+ WFMADB V1, V2, V20, V2
+ WFMADB V1, V4, V18, V4
+ WFMADB V1, V2, V16, V2
+ WFMADB V1, V4, V7, V4
+ WFMADB V1, V2, V3, V2
+ FMOVD 336(R9), F3
+ WFMADB V1, V4, V3, V4
+ FMOVD 328(R9), F3
+ WFMADB V1, V2, V3, V2
+ FMOVD 320(R9), F3
+ WFMADB V1, V4, V3, V1
+ FMOVD 312(R9), F7
+ WFMADB V6, V2, V1, V2
+ MOVH $0x0, R1
+ FMOVD 664(R9), F3
+ FMADD F2, F6, F7
+ WFMADB V0, V5, V3, V5
+ BR L11
+L35:
+ WORD $0xB3130010 //lcdbr %f1,%f0
+ BR L9
+L36:
+ FMOVD 304(R9), F3
+ FSUB F1, F3
+ VLEG $0, 296(R9), V20
+ WFDDB V1, V3, V6
+ FMOVD 288(R9), F5
+ FMOVD 280(R9), F1
+ FMOVD 272(R9), F2
+ VLEG $0, 264(R9), V18
+ VLEG $0, 256(R9), V16
+ FMOVD 248(R9), F3
+ FMOVD 240(R9), F4
+ WFMDB V6, V6, V7
+ FMUL F0, F0
+ MOVH $0x0, R3
+ FMADD F5, F7, F1
+ WFMADB V7, V2, V20, V2
+ WFMADB V7, V1, V18, V1
+ WFMADB V7, V2, V16, V2
+ WFMADB V7, V1, V3, V1
+ WFMADB V7, V2, V4, V2
+ FMOVD 232(R9), F4
+ WFMADB V7, V1, V4, V1
+ FMOVD 224(R9), F4
+ WFMADB V7, V2, V4, V2
+ FMOVD 216(R9), F4
+ WFMADB V7, V1, V4, V1
+ FMOVD 208(R9), F4
+ MOVH $0x0, R1
+ WFMADB V7, V2, V4, V7
+ FMOVD 656(R9), F5
+ WFMADB V6, V1, V7, V1
+ FMOVD 664(R9), F3
+ FMOVD 200(R9), F7
+ WFMADB V0, V5, V3, V5
+ FMADD F1, F6, F7
+ BR L11
+L4:
+ FMOVD 192(R9), F1
+ FMADD F0, F0, F1
+ FMOVD 184(R9), F3
+ WFMDB V1, V1, V0
+ FMOVD 176(R9), F4
+ FMOVD 168(R9), F6
+ WFMADB V0, V4, V3, V4
+ FMOVD 160(R9), F3
+ WFMADB V0, V6, V3, V6
+ FMOVD 152(R9), F5
+ FMOVD 144(R9), F3
+ WFMADB V0, V4, V5, V4
+ WFMADB V0, V6, V3, V6
+ FMOVD 136(R9), F5
+ FMOVD 128(R9), F3
+ WFMADB V0, V4, V5, V4
+ WFMADB V0, V6, V3, V6
+ FMOVD 120(R9), F5
+ FMOVD 112(R9), F3
+ WFMADB V0, V4, V5, V4
+ WFMADB V0, V6, V3, V6
+ FMOVD 104(R9), F5
+ FMOVD 96(R9), F3
+ WFMADB V0, V4, V5, V4
+ WFMADB V0, V6, V3, V0
+ FMOVD F2, F6
+ FMADD F4, F1, F0
+ WORD $0xED609318 //sdb %f6,.L39-.L38(%r9)
+ BYTE $0x00
+ BYTE $0x1B
+ WFMSDB V2, V0, V6, V2
+ FMOVD F2, ret+8(FP)
+ RET
+L30:
+ WORD $0xED009238 //adb %f0,.L67-.L38(%r9)
+ BYTE $0x00
+ BYTE $0x1A
+ BR L8
+L20:
+ FMOVD 88(R9), F0
+ WFMADB V7, V2, V0, V2
+ WORD $0xB3130022 //lcdbr %f2,%f2
+ FMOVD F2, ret+8(FP)
+ RET
+L13:
+ MOVH $0x403A, R3
+ MOVW R1, R6
+ MOVW R3, R7
+ CMPBLE R6, R7, L4
+ WORD $0xED109050 //cdb %f1,.L128-.L38(%r9)
+ BYTE $0x00
+ BYTE $0x19
+ BGE L37
+ BVS L37
+ FMOVD 72(R9), F6
+ FSUB F1, F6
+ MOVH $0x1000, R3
+ FDIV F1, F6
+ MOVH $0x1000, R1
+L17:
+ WFMDB V6, V6, V1
+ FMOVD 64(R9), F2
+ FMOVD 56(R9), F4
+ FMOVD 48(R9), F3
+ WFMADB V1, V3, V2, V3
+ FMOVD 40(R9), F2
+ WFMADB V1, V2, V4, V2
+ FMOVD 32(R9), F4
+ WFMADB V1, V3, V4, V3
+ FMOVD 24(R9), F4
+ WFMADB V1, V2, V4, V2
+ FMOVD 16(R9), F4
+ WFMADB V1, V3, V4, V3
+ FMOVD 8(R9), F4
+ WFMADB V1, V2, V4, V1
+ FMUL F0, F0
+ WFMADB V3, V6, V1, V3
+ FMOVD 656(R9), F5
+ FMOVD 664(R9), F4
+ FMOVD 0(R9), F7
+ WFMADB V0, V5, V4, V5
+ FMADD F6, F3, F7
+ BR L11
+L14:
+ FMOVD 72(R9), F6
+ FSUB F1, F6
+ MOVH $0x403A, R3
+ FDIV F1, F6
+ MOVW R1, R6
+ MOVW R3, R7
+ CMPBEQ R6, R7, L23
+ MOVH $0x0, R3
+ MOVH $0x0, R1
+ BR L17
+L37:
+ WFCEDBS V0, V0, V0
+ BVS L1
+ MOVW R2, R6
+ CMPBLE R6, $0, L18
+ MOVH $0x7FEF, R2
+ MOVW R1, R6
+ MOVW R2, R7
+ CMPBGT R6, R7, L24
+
+ WORD $0xA5400010 //iihh %r4,16
+ LDGR R4, F2
+ FMUL F2, F2
+ BR L1
+L23:
+ MOVH $0x1000, R3
+ MOVH $0x1000, R1
+ BR L17
+L24:
+ FMOVD $0, F2
+ BR L1
+L18:
+ MOVH $0x7FEF, R2
+ MOVW R1, R6
+ MOVW R2, R7
+ CMPBGT R6, R7, L25
+ WORD $0xA5408010 //iihh %r4,32784
+ FMOVD 568(R9), F2
+ LDGR R4, F0
+ FMADD F2, F0, F2
+ BR L1
+L25:
+ FMOVD 568(R9), F2
+ BR L1
+usego:
+ BR ·erfc(SB)
diff --git a/src/math/erfinv.go b/src/math/erfinv.go
new file mode 100644
index 0000000..ee423d3
--- /dev/null
+++ b/src/math/erfinv.go
@@ -0,0 +1,127 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Inverse of the floating-point error function.
+*/
+
+// This implementation is based on the rational approximation
+// of percentage points of normal distribution available from
+// https://www.jstor.org/stable/2347330.
+
+const (
+ // Coefficients for approximation to erf in |x| <= 0.85
+ a0 = 1.1975323115670912564578e0
+ a1 = 4.7072688112383978012285e1
+ a2 = 6.9706266534389598238465e2
+ a3 = 4.8548868893843886794648e3
+ a4 = 1.6235862515167575384252e4
+ a5 = 2.3782041382114385731252e4
+ a6 = 1.1819493347062294404278e4
+ a7 = 8.8709406962545514830200e2
+ b0 = 1.0000000000000000000e0
+ b1 = 4.2313330701600911252e1
+ b2 = 6.8718700749205790830e2
+ b3 = 5.3941960214247511077e3
+ b4 = 2.1213794301586595867e4
+ b5 = 3.9307895800092710610e4
+ b6 = 2.8729085735721942674e4
+ b7 = 5.2264952788528545610e3
+ // Coefficients for approximation to erf in 0.85 < |x| <= 1-2*exp(-25)
+ c0 = 1.42343711074968357734e0
+ c1 = 4.63033784615654529590e0
+ c2 = 5.76949722146069140550e0
+ c3 = 3.64784832476320460504e0
+ c4 = 1.27045825245236838258e0
+ c5 = 2.41780725177450611770e-1
+ c6 = 2.27238449892691845833e-2
+ c7 = 7.74545014278341407640e-4
+ d0 = 1.4142135623730950488016887e0
+ d1 = 2.9036514445419946173133295e0
+ d2 = 2.3707661626024532365971225e0
+ d3 = 9.7547832001787427186894837e-1
+ d4 = 2.0945065210512749128288442e-1
+ d5 = 2.1494160384252876777097297e-2
+ d6 = 7.7441459065157709165577218e-4
+ d7 = 1.4859850019840355905497876e-9
+ // Coefficients for approximation to erf in 1-2*exp(-25) < |x| < 1
+ e0 = 6.65790464350110377720e0
+ e1 = 5.46378491116411436990e0
+ e2 = 1.78482653991729133580e0
+ e3 = 2.96560571828504891230e-1
+ e4 = 2.65321895265761230930e-2
+ e5 = 1.24266094738807843860e-3
+ e6 = 2.71155556874348757815e-5
+ e7 = 2.01033439929228813265e-7
+ f0 = 1.414213562373095048801689e0
+ f1 = 8.482908416595164588112026e-1
+ f2 = 1.936480946950659106176712e-1
+ f3 = 2.103693768272068968719679e-2
+ f4 = 1.112800997078859844711555e-3
+ f5 = 2.611088405080593625138020e-5
+ f6 = 2.010321207683943062279931e-7
+ f7 = 2.891024605872965461538222e-15
+)
+
+// Erfinv returns the inverse error function of x.
+//
+// Special cases are:
+// Erfinv(1) = +Inf
+// Erfinv(-1) = -Inf
+// Erfinv(x) = NaN if x < -1 or x > 1
+// Erfinv(NaN) = NaN
+func Erfinv(x float64) float64 {
+ // special cases
+ if IsNaN(x) || x <= -1 || x >= 1 {
+ if x == -1 || x == 1 {
+ return Inf(int(x))
+ }
+ return NaN()
+ }
+
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+
+ var ans float64
+ if x <= 0.85 { // |x| <= 0.85
+ r := 0.180625 - 0.25*x*x
+ z1 := ((((((a7*r+a6)*r+a5)*r+a4)*r+a3)*r+a2)*r+a1)*r + a0
+ z2 := ((((((b7*r+b6)*r+b5)*r+b4)*r+b3)*r+b2)*r+b1)*r + b0
+ ans = (x * z1) / z2
+ } else {
+ var z1, z2 float64
+ r := Sqrt(Ln2 - Log(1.0-x))
+ if r <= 5.0 {
+ r -= 1.6
+ z1 = ((((((c7*r+c6)*r+c5)*r+c4)*r+c3)*r+c2)*r+c1)*r + c0
+ z2 = ((((((d7*r+d6)*r+d5)*r+d4)*r+d3)*r+d2)*r+d1)*r + d0
+ } else {
+ r -= 5.0
+ z1 = ((((((e7*r+e6)*r+e5)*r+e4)*r+e3)*r+e2)*r+e1)*r + e0
+ z2 = ((((((f7*r+f6)*r+f5)*r+f4)*r+f3)*r+f2)*r+f1)*r + f0
+ }
+ ans = z1 / z2
+ }
+
+ if sign {
+ return -ans
+ }
+ return ans
+}
+
+// Erfcinv returns the inverse of Erfc(x).
+//
+// Special cases are:
+// Erfcinv(0) = +Inf
+// Erfcinv(2) = -Inf
+// Erfcinv(x) = NaN if x < 0 or x > 2
+// Erfcinv(NaN) = NaN
+func Erfcinv(x float64) float64 {
+ return Erfinv(1 - x)
+}
diff --git a/src/math/example_test.go b/src/math/example_test.go
new file mode 100644
index 0000000..9fc1967
--- /dev/null
+++ b/src/math/example_test.go
@@ -0,0 +1,240 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math_test
+
+import (
+ "fmt"
+ "math"
+)
+
+func ExampleAcos() {
+ fmt.Printf("%.2f", math.Acos(1))
+ // Output: 0.00
+}
+
+func ExampleAcosh() {
+ fmt.Printf("%.2f", math.Acosh(1))
+ // Output: 0.00
+}
+
+func ExampleAsin() {
+ fmt.Printf("%.2f", math.Asin(0))
+ // Output: 0.00
+}
+
+func ExampleAsinh() {
+ fmt.Printf("%.2f", math.Asinh(0))
+ // Output: 0.00
+}
+
+func ExampleAtan() {
+ fmt.Printf("%.2f", math.Atan(0))
+ // Output: 0.00
+}
+
+func ExampleAtan2() {
+ fmt.Printf("%.2f", math.Atan2(0, 0))
+ // Output: 0.00
+}
+
+func ExampleAtanh() {
+ fmt.Printf("%.2f", math.Atanh(0))
+ // Output: 0.00
+}
+
+func ExampleCopysign() {
+ fmt.Printf("%.2f", math.Copysign(3.2, -1))
+ // Output: -3.20
+}
+
+func ExampleCos() {
+ fmt.Printf("%.2f", math.Cos(math.Pi/2))
+ // Output: 0.00
+}
+
+func ExampleCosh() {
+ fmt.Printf("%.2f", math.Cosh(0))
+ // Output: 1.00
+}
+
+func ExampleSin() {
+ fmt.Printf("%.2f", math.Sin(math.Pi))
+ // Output: 0.00
+}
+
+func ExampleSincos() {
+ sin, cos := math.Sincos(0)
+ fmt.Printf("%.2f, %.2f", sin, cos)
+ // Output: 0.00, 1.00
+}
+
+func ExampleSinh() {
+ fmt.Printf("%.2f", math.Sinh(0))
+ // Output: 0.00
+}
+
+func ExampleTan() {
+ fmt.Printf("%.2f", math.Tan(0))
+ // Output: 0.00
+}
+
+func ExampleTanh() {
+ fmt.Printf("%.2f", math.Tanh(0))
+ // Output: 0.00
+}
+
+func ExampleSqrt() {
+ const (
+ a = 3
+ b = 4
+ )
+ c := math.Sqrt(a*a + b*b)
+ fmt.Printf("%.1f", c)
+ // Output: 5.0
+}
+
+func ExampleCeil() {
+ c := math.Ceil(1.49)
+ fmt.Printf("%.1f", c)
+ // Output: 2.0
+}
+
+func ExampleFloor() {
+ c := math.Floor(1.51)
+ fmt.Printf("%.1f", c)
+ // Output: 1.0
+}
+
+func ExamplePow() {
+ c := math.Pow(2, 3)
+ fmt.Printf("%.1f", c)
+ // Output: 8.0
+}
+
+func ExamplePow10() {
+ c := math.Pow10(2)
+ fmt.Printf("%.1f", c)
+ // Output: 100.0
+}
+
+func ExampleRound() {
+ p := math.Round(10.5)
+ fmt.Printf("%.1f\n", p)
+
+ n := math.Round(-10.5)
+ fmt.Printf("%.1f\n", n)
+ // Output:
+ // 11.0
+ // -11.0
+}
+
+func ExampleRoundToEven() {
+ u := math.RoundToEven(11.5)
+ fmt.Printf("%.1f\n", u)
+
+ d := math.RoundToEven(12.5)
+ fmt.Printf("%.1f\n", d)
+ // Output:
+ // 12.0
+ // 12.0
+}
+
+func ExampleLog() {
+ x := math.Log(1)
+ fmt.Printf("%.1f\n", x)
+
+ y := math.Log(2.7183)
+ fmt.Printf("%.1f\n", y)
+ // Output:
+ // 0.0
+ // 1.0
+}
+
+func ExampleLog2() {
+ fmt.Printf("%.1f", math.Log2(256))
+ // Output: 8.0
+}
+
+func ExampleLog10() {
+ fmt.Printf("%.1f", math.Log10(100))
+ // Output: 2.0
+}
+
+func ExampleMod() {
+ c := math.Mod(7, 4)
+ fmt.Printf("%.1f", c)
+ // Output: 3.0
+}
+
+func ExampleAbs() {
+ x := math.Abs(-2)
+ fmt.Printf("%.1f\n", x)
+
+ y := math.Abs(2)
+ fmt.Printf("%.1f\n", y)
+ // Output:
+ // 2.0
+ // 2.0
+}
+func ExampleDim() {
+ fmt.Printf("%.2f\n", math.Dim(4, -2))
+ fmt.Printf("%.2f\n", math.Dim(-4, 2))
+ // Output:
+ // 6.00
+ // 0.00
+}
+
+func ExampleExp() {
+ fmt.Printf("%.2f\n", math.Exp(1))
+ fmt.Printf("%.2f\n", math.Exp(2))
+ fmt.Printf("%.2f\n", math.Exp(-1))
+ // Output:
+ // 2.72
+ // 7.39
+ // 0.37
+}
+
+func ExampleExp2() {
+ fmt.Printf("%.2f\n", math.Exp2(1))
+ fmt.Printf("%.2f\n", math.Exp2(-3))
+ // Output:
+ // 2.00
+ // 0.12
+}
+
+func ExampleExpm1() {
+ fmt.Printf("%.6f\n", math.Expm1(0.01))
+ fmt.Printf("%.6f\n", math.Expm1(-1))
+ // Output:
+ // 0.010050
+ // -0.632121
+}
+
+func ExampleTrunc() {
+ fmt.Printf("%.2f\n", math.Trunc(math.Pi))
+ fmt.Printf("%.2f\n", math.Trunc(-1.2345))
+ // Output:
+ // 3.00
+ // -1.00
+}
+
+func ExampleCbrt() {
+ fmt.Printf("%.2f\n", math.Cbrt(8))
+ fmt.Printf("%.2f\n", math.Cbrt(27))
+ // Output:
+ // 2.00
+ // 3.00
+}
+
+func ExampleModf() {
+ int, frac := math.Modf(3.14)
+ fmt.Printf("%.2f, %.2f\n", int, frac)
+
+ int, frac = math.Modf(-2.71)
+ fmt.Printf("%.2f, %.2f\n", int, frac)
+ // Output:
+ // 3.00, 0.14
+ // -2.00, -0.71
+}
diff --git a/src/math/exp.go b/src/math/exp.go
new file mode 100644
index 0000000..bd4c5c9
--- /dev/null
+++ b/src/math/exp.go
@@ -0,0 +1,191 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Exp returns e**x, the base-e exponential of x.
+//
+// Special cases are:
+// Exp(+Inf) = +Inf
+// Exp(NaN) = NaN
+// Very large values overflow to 0 or +Inf.
+// Very small values underflow to 1.
+func Exp(x float64) float64
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+//
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// exp(x)
+// Returns the exponential of x.
+//
+// Method
+// 1. Argument reduction:
+// Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
+// Given x, find r and integer k such that
+//
+// x = k*ln2 + r, |r| <= 0.5*ln2.
+//
+// Here r will be represented as r = hi-lo for better
+// accuracy.
+//
+// 2. Approximation of exp(r) by a special rational function on
+// the interval [0,0.34658]:
+// Write
+// R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
+// We use a special Remez algorithm on [0,0.34658] to generate
+// a polynomial of degree 5 to approximate R. The maximum error
+// of this polynomial approximation is bounded by 2**-59. In
+// other words,
+// R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
+// (where z=r*r, and the values of P1 to P5 are listed below)
+// and
+// | 5 | -59
+// | 2.0+P1*z+...+P5*z - R(z) | <= 2
+// | |
+// The computation of exp(r) thus becomes
+// 2*r
+// exp(r) = 1 + -------
+// R - r
+// r*R1(r)
+// = 1 + r + ----------- (for better accuracy)
+// 2 - R1(r)
+// where
+// 2 4 10
+// R1(r) = r - (P1*r + P2*r + ... + P5*r ).
+//
+// 3. Scale back to obtain exp(x):
+// From step 1, we have
+// exp(x) = 2**k * exp(r)
+//
+// Special cases:
+// exp(INF) is INF, exp(NaN) is NaN;
+// exp(-INF) is 0, and
+// for finite argument, only exp(0)=1 is exact.
+//
+// Accuracy:
+// according to an error analysis, the error is always less than
+// 1 ulp (unit in the last place).
+//
+// Misc. info.
+// For IEEE double
+// if x > 7.09782712893383973096e+02 then exp(x) overflow
+// if x < -7.45133219101941108420e+02 then exp(x) underflow
+//
+// Constants:
+// The hexadecimal values are the intended ones for the following
+// constants. The decimal values may be used, provided that the
+// compiler will convert from decimal to binary accurately enough
+// to produce the hexadecimal values shown.
+
+func exp(x float64) float64 {
+ const (
+ Ln2Hi = 6.93147180369123816490e-01
+ Ln2Lo = 1.90821492927058770002e-10
+ Log2e = 1.44269504088896338700e+00
+
+ Overflow = 7.09782712893383973096e+02
+ Underflow = -7.45133219101941108420e+02
+ NearZero = 1.0 / (1 << 28) // 2**-28
+ )
+
+ // special cases
+ switch {
+ case IsNaN(x) || IsInf(x, 1):
+ return x
+ case IsInf(x, -1):
+ return 0
+ case x > Overflow:
+ return Inf(1)
+ case x < Underflow:
+ return 0
+ case -NearZero < x && x < NearZero:
+ return 1 + x
+ }
+
+ // reduce; computed as r = hi - lo for extra precision.
+ var k int
+ switch {
+ case x < 0:
+ k = int(Log2e*x - 0.5)
+ case x > 0:
+ k = int(Log2e*x + 0.5)
+ }
+ hi := x - float64(k)*Ln2Hi
+ lo := float64(k) * Ln2Lo
+
+ // compute
+ return expmulti(hi, lo, k)
+}
+
+// Exp2 returns 2**x, the base-2 exponential of x.
+//
+// Special cases are the same as Exp.
+func Exp2(x float64) float64
+
+func exp2(x float64) float64 {
+ const (
+ Ln2Hi = 6.93147180369123816490e-01
+ Ln2Lo = 1.90821492927058770002e-10
+
+ Overflow = 1.0239999999999999e+03
+ Underflow = -1.0740e+03
+ )
+
+ // special cases
+ switch {
+ case IsNaN(x) || IsInf(x, 1):
+ return x
+ case IsInf(x, -1):
+ return 0
+ case x > Overflow:
+ return Inf(1)
+ case x < Underflow:
+ return 0
+ }
+
+ // argument reduction; x = r×lg(e) + k with |r| ≤ ln(2)/2.
+ // computed as r = hi - lo for extra precision.
+ var k int
+ switch {
+ case x > 0:
+ k = int(x + 0.5)
+ case x < 0:
+ k = int(x - 0.5)
+ }
+ t := x - float64(k)
+ hi := t * Ln2Hi
+ lo := -t * Ln2Lo
+
+ // compute
+ return expmulti(hi, lo, k)
+}
+
+// exp1 returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2.
+func expmulti(hi, lo float64, k int) float64 {
+ const (
+ P1 = 1.66666666666666657415e-01 /* 0x3FC55555; 0x55555555 */
+ P2 = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */
+ P3 = 6.61375632143793436117e-05 /* 0x3F11566A; 0xAF25DE2C */
+ P4 = -1.65339022054652515390e-06 /* 0xBEBBBD41; 0xC5D26BF1 */
+ P5 = 4.13813679705723846039e-08 /* 0x3E663769; 0x72BEA4D0 */
+ )
+
+ r := hi - lo
+ t := r * r
+ c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))))
+ y := 1 - ((lo - (r*c)/(2-c)) - hi)
+ // TODO(rsc): make sure Ldexp can handle boundary k
+ return Ldexp(y, k)
+}
diff --git a/src/math/exp2_386.s b/src/math/exp2_386.s
new file mode 100644
index 0000000..d04cad6
--- /dev/null
+++ b/src/math/exp2_386.s
@@ -0,0 +1,40 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Exp2(x float64) float64
+TEXT ·Exp2(SB),NOSPLIT,$0
+// test bits for not-finite
+ MOVL x_hi+4(FP), AX
+ ANDL $0x7ff00000, AX
+ CMPL AX, $0x7ff00000
+ JEQ not_finite
+ FMOVD x+0(FP), F0 // F0=x
+ FMOVD F0, F1 // F0=x, F1=x
+ FRNDINT // F0=int(x), F1=x
+ FSUBD F0, F1 // F0=int(x), F1=x-int(x)
+ FXCHD F0, F1 // F0=x-int(x), F1=int(x)
+ F2XM1 // F0=2**(x-int(x))-1, F1=int(x)
+ FLD1 // F0=1, F1=2**(x-int(x))-1, F2=int(x)
+ FADDDP F0, F1 // F0=2**(x-int(x)), F1=int(x)
+ FSCALE // F0=2**x, F1=int(x)
+ FMOVDP F0, F1 // F0=2**x
+ FMOVDP F0, ret+8(FP)
+ RET
+not_finite:
+// test bits for -Inf
+ MOVL x_hi+4(FP), BX
+ MOVL x_lo+0(FP), CX
+ CMPL BX, $0xfff00000
+ JNE not_neginf
+ CMPL CX, $0
+ JNE not_neginf
+ MOVL $0, ret_lo+8(FP)
+ MOVL $0, ret_hi+12(FP)
+ RET
+not_neginf:
+ MOVL CX, ret_lo+8(FP)
+ MOVL BX, ret_hi+12(FP)
+ RET
diff --git a/src/math/exp_amd64.s b/src/math/exp_amd64.s
new file mode 100644
index 0000000..b3e1c22
--- /dev/null
+++ b/src/math/exp_amd64.s
@@ -0,0 +1,159 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// The method is based on a paper by Naoki Shibata: "Efficient evaluation
+// methods of elementary functions suitable for SIMD computation", Proc.
+// of International Supercomputing Conference 2010 (ISC'10), pp. 25 -- 32
+// (May 2010). The paper is available at
+// https://link.springer.com/article/10.1007/s00450-010-0108-2
+//
+// The original code and the constants below are from the author's
+// implementation available at http://freshmeat.net/projects/sleef.
+// The README file says, "The software is in public domain.
+// You can use the software without any obligation."
+//
+// This code is a simplified version of the original.
+
+#define LN2 0.6931471805599453094172321214581766 // log_e(2)
+#define LOG2E 1.4426950408889634073599246810018920 // 1/LN2
+#define LN2U 0.69314718055966295651160180568695068359375 // upper half LN2
+#define LN2L 0.28235290563031577122588448175013436025525412068e-12 // lower half LN2
+#define PosInf 0x7FF0000000000000
+#define NegInf 0xFFF0000000000000
+#define Overflow 7.09782712893384e+02
+
+DATA exprodata<>+0(SB)/8, $0.5
+DATA exprodata<>+8(SB)/8, $1.0
+DATA exprodata<>+16(SB)/8, $2.0
+DATA exprodata<>+24(SB)/8, $1.6666666666666666667e-1
+DATA exprodata<>+32(SB)/8, $4.1666666666666666667e-2
+DATA exprodata<>+40(SB)/8, $8.3333333333333333333e-3
+DATA exprodata<>+48(SB)/8, $1.3888888888888888889e-3
+DATA exprodata<>+56(SB)/8, $1.9841269841269841270e-4
+DATA exprodata<>+64(SB)/8, $2.4801587301587301587e-5
+GLOBL exprodata<>+0(SB), RODATA, $72
+
+// func Exp(x float64) float64
+TEXT ·Exp(SB),NOSPLIT,$0
+ // test bits for not-finite
+ MOVQ x+0(FP), BX
+ MOVQ $~(1<<63), AX // sign bit mask
+ MOVQ BX, DX
+ ANDQ AX, DX
+ MOVQ $PosInf, AX
+ CMPQ AX, DX
+ JLE notFinite
+ // check if argument will overflow
+ MOVQ BX, X0
+ MOVSD $Overflow, X1
+ COMISD X1, X0
+ JA overflow
+ MOVSD $LOG2E, X1
+ MULSD X0, X1
+ CVTSD2SL X1, BX // BX = exponent
+ CVTSL2SD BX, X1
+ CMPB ·useFMA(SB), $1
+ JE avxfma
+ MOVSD $LN2U, X2
+ MULSD X1, X2
+ SUBSD X2, X0
+ MOVSD $LN2L, X2
+ MULSD X1, X2
+ SUBSD X2, X0
+ // reduce argument
+ MULSD $0.0625, X0
+ // Taylor series evaluation
+ MOVSD exprodata<>+64(SB), X1
+ MULSD X0, X1
+ ADDSD exprodata<>+56(SB), X1
+ MULSD X0, X1
+ ADDSD exprodata<>+48(SB), X1
+ MULSD X0, X1
+ ADDSD exprodata<>+40(SB), X1
+ MULSD X0, X1
+ ADDSD exprodata<>+32(SB), X1
+ MULSD X0, X1
+ ADDSD exprodata<>+24(SB), X1
+ MULSD X0, X1
+ ADDSD exprodata<>+0(SB), X1
+ MULSD X0, X1
+ ADDSD exprodata<>+8(SB), X1
+ MULSD X1, X0
+ MOVSD exprodata<>+16(SB), X1
+ ADDSD X0, X1
+ MULSD X1, X0
+ MOVSD exprodata<>+16(SB), X1
+ ADDSD X0, X1
+ MULSD X1, X0
+ MOVSD exprodata<>+16(SB), X1
+ ADDSD X0, X1
+ MULSD X1, X0
+ MOVSD exprodata<>+16(SB), X1
+ ADDSD X0, X1
+ MULSD X1, X0
+ ADDSD exprodata<>+8(SB), X0
+ // return fr * 2**exponent
+ldexp:
+ ADDL $0x3FF, BX // add bias
+ JLE denormal
+ CMPL BX, $0x7FF
+ JGE overflow
+lastStep:
+ SHLQ $52, BX
+ MOVQ BX, X1
+ MULSD X1, X0
+ MOVSD X0, ret+8(FP)
+ RET
+notFinite:
+ // test bits for -Inf
+ MOVQ $NegInf, AX
+ CMPQ AX, BX
+ JNE notNegInf
+ // -Inf, return 0
+underflow: // return 0
+ MOVQ $0, ret+8(FP)
+ RET
+overflow: // return +Inf
+ MOVQ $PosInf, BX
+notNegInf: // NaN or +Inf, return x
+ MOVQ BX, ret+8(FP)
+ RET
+denormal:
+ CMPL BX, $-52
+ JL underflow
+ ADDL $0x3FE, BX // add bias - 1
+ SHLQ $52, BX
+ MOVQ BX, X1
+ MULSD X1, X0
+ MOVQ $1, BX
+ JMP lastStep
+
+avxfma:
+ MOVSD $LN2U, X2
+ VFNMADD231SD X2, X1, X0
+ MOVSD $LN2L, X2
+ VFNMADD231SD X2, X1, X0
+ // reduce argument
+ MULSD $0.0625, X0
+ // Taylor series evaluation
+ MOVSD exprodata<>+64(SB), X1
+ VFMADD213SD exprodata<>+56(SB), X0, X1
+ VFMADD213SD exprodata<>+48(SB), X0, X1
+ VFMADD213SD exprodata<>+40(SB), X0, X1
+ VFMADD213SD exprodata<>+32(SB), X0, X1
+ VFMADD213SD exprodata<>+24(SB), X0, X1
+ VFMADD213SD exprodata<>+0(SB), X0, X1
+ VFMADD213SD exprodata<>+8(SB), X0, X1
+ MULSD X1, X0
+ VADDSD exprodata<>+16(SB), X0, X1
+ MULSD X1, X0
+ VADDSD exprodata<>+16(SB), X0, X1
+ MULSD X1, X0
+ VADDSD exprodata<>+16(SB), X0, X1
+ MULSD X1, X0
+ VADDSD exprodata<>+16(SB), X0, X1
+ VFMADD213SD exprodata<>+8(SB), X1, X0
+ JMP ldexp
diff --git a/src/math/exp_arm64.s b/src/math/exp_arm64.s
new file mode 100644
index 0000000..19736cb
--- /dev/null
+++ b/src/math/exp_arm64.s
@@ -0,0 +1,182 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#define Ln2Hi 6.93147180369123816490e-01
+#define Ln2Lo 1.90821492927058770002e-10
+#define Log2e 1.44269504088896338700e+00
+#define Overflow 7.09782712893383973096e+02
+#define Underflow -7.45133219101941108420e+02
+#define Overflow2 1.0239999999999999e+03
+#define Underflow2 -1.0740e+03
+#define NearZero 0x3e30000000000000 // 2**-28
+#define PosInf 0x7ff0000000000000
+#define FracMask 0x000fffffffffffff
+#define C1 0x3cb0000000000000 // 2**-52
+#define P1 1.66666666666666657415e-01 // 0x3FC55555; 0x55555555
+#define P2 -2.77777777770155933842e-03 // 0xBF66C16C; 0x16BEBD93
+#define P3 6.61375632143793436117e-05 // 0x3F11566A; 0xAF25DE2C
+#define P4 -1.65339022054652515390e-06 // 0xBEBBBD41; 0xC5D26BF1
+#define P5 4.13813679705723846039e-08 // 0x3E663769; 0x72BEA4D0
+
+// Exp returns e**x, the base-e exponential of x.
+// This is an assembly implementation of the method used for function Exp in file exp.go.
+//
+// func Exp(x float64) float64
+TEXT ·Exp(SB),$0-16
+ FMOVD x+0(FP), F0 // F0 = x
+ FCMPD F0, F0
+ BNE isNaN // x = NaN, return NaN
+ FMOVD $Overflow, F1
+ FCMPD F1, F0
+ BGT overflow // x > Overflow, return PosInf
+ FMOVD $Underflow, F1
+ FCMPD F1, F0
+ BLT underflow // x < Underflow, return 0
+ MOVD $NearZero, R0
+ FMOVD R0, F2
+ FABSD F0, F3
+ FMOVD $1.0, F1 // F1 = 1.0
+ FCMPD F2, F3
+ BLT nearzero // fabs(x) < NearZero, return 1 + x
+ // argument reduction, x = k*ln2 + r, |r| <= 0.5*ln2
+ // computed as r = hi - lo for extra precision.
+ FMOVD $Log2e, F2
+ FMOVD $0.5, F3
+ FNMSUBD F0, F3, F2, F4 // Log2e*x - 0.5
+ FMADDD F0, F3, F2, F3 // Log2e*x + 0.5
+ FCMPD $0.0, F0
+ FCSELD LT, F4, F3, F3 // F3 = k
+ FCVTZSD F3, R1 // R1 = int(k)
+ SCVTFD R1, F3 // F3 = float64(int(k))
+ FMOVD $Ln2Hi, F4 // F4 = Ln2Hi
+ FMOVD $Ln2Lo, F5 // F5 = Ln2Lo
+ FMSUBD F3, F0, F4, F4 // F4 = hi = x - float64(int(k))*Ln2Hi
+ FMULD F3, F5 // F5 = lo = float64(int(k)) * Ln2Lo
+ FSUBD F5, F4, F6 // F6 = r = hi - lo
+ FMULD F6, F6, F7 // F7 = t = r * r
+ // compute y
+ FMOVD $P5, F8 // F8 = P5
+ FMOVD $P4, F9 // F9 = P4
+ FMADDD F7, F9, F8, F13 // P4+t*P5
+ FMOVD $P3, F10 // F10 = P3
+ FMADDD F7, F10, F13, F13 // P3+t*(P4+t*P5)
+ FMOVD $P2, F11 // F11 = P2
+ FMADDD F7, F11, F13, F13 // P2+t*(P3+t*(P4+t*P5))
+ FMOVD $P1, F12 // F12 = P1
+ FMADDD F7, F12, F13, F13 // P1+t*(P2+t*(P3+t*(P4+t*P5)))
+ FMSUBD F7, F6, F13, F13 // F13 = c = r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))))
+ FMOVD $2.0, F14
+ FSUBD F13, F14
+ FMULD F6, F13, F15
+ FDIVD F14, F15 // F15 = (r*c)/(2-c)
+ FSUBD F15, F5, F15 // lo-(r*c)/(2-c)
+ FSUBD F4, F15, F15 // (lo-(r*c)/(2-c))-hi
+ FSUBD F15, F1, F16 // F16 = y = 1-((lo-(r*c)/(2-c))-hi)
+ // inline Ldexp(y, k), benefit:
+ // 1, no parameter pass overhead.
+ // 2, skip unnecessary checks for Inf/NaN/Zero
+ FMOVD F16, R0
+ AND $FracMask, R0, R2 // fraction
+ LSR $52, R0, R5 // exponent
+ ADD R1, R5 // R1 = int(k)
+ CMP $1, R5
+ BGE normal
+ ADD $52, R5 // denormal
+ MOVD $C1, R8
+ FMOVD R8, F1 // m = 2**-52
+normal:
+ ORR R5<<52, R2, R0
+ FMOVD R0, F0
+ FMULD F1, F0 // return m * x
+ FMOVD F0, ret+8(FP)
+ RET
+nearzero:
+ FADDD F1, F0
+isNaN:
+ FMOVD F0, ret+8(FP)
+ RET
+underflow:
+ MOVD ZR, ret+8(FP)
+ RET
+overflow:
+ MOVD $PosInf, R0
+ MOVD R0, ret+8(FP)
+ RET
+
+
+// Exp2 returns 2**x, the base-2 exponential of x.
+// This is an assembly implementation of the method used for function Exp2 in file exp.go.
+//
+// func Exp2(x float64) float64
+TEXT ·Exp2(SB),$0-16
+ FMOVD x+0(FP), F0 // F0 = x
+ FCMPD F0, F0
+ BNE isNaN // x = NaN, return NaN
+ FMOVD $Overflow2, F1
+ FCMPD F1, F0
+ BGT overflow // x > Overflow, return PosInf
+ FMOVD $Underflow2, F1
+ FCMPD F1, F0
+ BLT underflow // x < Underflow, return 0
+ // argument reduction; x = r*lg(e) + k with |r| <= ln(2)/2
+ // computed as r = hi - lo for extra precision.
+ FMOVD $0.5, F2
+ FSUBD F2, F0, F3 // x + 0.5
+ FADDD F2, F0, F4 // x - 0.5
+ FCMPD $0.0, F0
+ FCSELD LT, F3, F4, F3 // F3 = k
+ FCVTZSD F3, R1 // R1 = int(k)
+ SCVTFD R1, F3 // F3 = float64(int(k))
+ FSUBD F3, F0, F3 // t = x - float64(int(k))
+ FMOVD $Ln2Hi, F4 // F4 = Ln2Hi
+ FMOVD $Ln2Lo, F5 // F5 = Ln2Lo
+ FMULD F3, F4 // F4 = hi = t * Ln2Hi
+ FNMULD F3, F5 // F5 = lo = -t * Ln2Lo
+ FSUBD F5, F4, F6 // F6 = r = hi - lo
+ FMULD F6, F6, F7 // F7 = t = r * r
+ // compute y
+ FMOVD $P5, F8 // F8 = P5
+ FMOVD $P4, F9 // F9 = P4
+ FMADDD F7, F9, F8, F13 // P4+t*P5
+ FMOVD $P3, F10 // F10 = P3
+ FMADDD F7, F10, F13, F13 // P3+t*(P4+t*P5)
+ FMOVD $P2, F11 // F11 = P2
+ FMADDD F7, F11, F13, F13 // P2+t*(P3+t*(P4+t*P5))
+ FMOVD $P1, F12 // F12 = P1
+ FMADDD F7, F12, F13, F13 // P1+t*(P2+t*(P3+t*(P4+t*P5)))
+ FMSUBD F7, F6, F13, F13 // F13 = c = r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))))
+ FMOVD $2.0, F14
+ FSUBD F13, F14
+ FMULD F6, F13, F15
+ FDIVD F14, F15 // F15 = (r*c)/(2-c)
+ FMOVD $1.0, F1 // F1 = 1.0
+ FSUBD F15, F5, F15 // lo-(r*c)/(2-c)
+ FSUBD F4, F15, F15 // (lo-(r*c)/(2-c))-hi
+ FSUBD F15, F1, F16 // F16 = y = 1-((lo-(r*c)/(2-c))-hi)
+ // inline Ldexp(y, k), benefit:
+ // 1, no parameter pass overhead.
+ // 2, skip unnecessary checks for Inf/NaN/Zero
+ FMOVD F16, R0
+ AND $FracMask, R0, R2 // fraction
+ LSR $52, R0, R5 // exponent
+ ADD R1, R5 // R1 = int(k)
+ CMP $1, R5
+ BGE normal
+ ADD $52, R5 // denormal
+ MOVD $C1, R8
+ FMOVD R8, F1 // m = 2**-52
+normal:
+ ORR R5<<52, R2, R0
+ FMOVD R0, F0
+ FMULD F1, F0 // return m * x
+isNaN:
+ FMOVD F0, ret+8(FP)
+ RET
+underflow:
+ MOVD ZR, ret+8(FP)
+ RET
+overflow:
+ MOVD $PosInf, R0
+ MOVD R0, ret+8(FP)
+ RET
diff --git a/src/math/exp_asm.go b/src/math/exp_asm.go
new file mode 100644
index 0000000..8dad3c8
--- /dev/null
+++ b/src/math/exp_asm.go
@@ -0,0 +1,11 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build amd64
+
+package math
+
+import "internal/cpu"
+
+var useFMA = cpu.X86.HasAVX && cpu.X86.HasFMA
diff --git a/src/math/exp_s390x.s b/src/math/exp_s390x.s
new file mode 100644
index 0000000..e0ec823
--- /dev/null
+++ b/src/math/exp_s390x.s
@@ -0,0 +1,177 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial approximation and other constants
+DATA ·exprodataL22<> + 0(SB)/8, $800.0E+00
+DATA ·exprodataL22<> + 8(SB)/8, $1.0000000000000022e+00
+DATA ·exprodataL22<> + 16(SB)/8, $0.500000000000004237e+00
+DATA ·exprodataL22<> + 24(SB)/8, $0.166666666630345592e+00
+DATA ·exprodataL22<> + 32(SB)/8, $0.138926439368309441e-02
+DATA ·exprodataL22<> + 40(SB)/8, $0.833349307718286047e-02
+DATA ·exprodataL22<> + 48(SB)/8, $0.416666664838056960e-01
+DATA ·exprodataL22<> + 56(SB)/8, $-.231904681384629956E-16
+DATA ·exprodataL22<> + 64(SB)/8, $-.693147180559945286E+00
+DATA ·exprodataL22<> + 72(SB)/8, $0.144269504088896339E+01
+DATA ·exprodataL22<> + 80(SB)/8, $704.0E+00
+GLOBL ·exprodataL22<> + 0(SB), RODATA, $88
+
+DATA ·expxinf<> + 0(SB)/8, $0x7ff0000000000000
+GLOBL ·expxinf<> + 0(SB), RODATA, $8
+DATA ·expx4ff<> + 0(SB)/8, $0x4ff0000000000000
+GLOBL ·expx4ff<> + 0(SB), RODATA, $8
+DATA ·expx2ff<> + 0(SB)/8, $0x2ff0000000000000
+GLOBL ·expx2ff<> + 0(SB), RODATA, $8
+DATA ·expxaddexp<> + 0(SB)/8, $0xc2f0000100003fef
+GLOBL ·expxaddexp<> + 0(SB), RODATA, $8
+
+// Log multipliers table
+DATA ·exptexp<> + 0(SB)/8, $0.442737824274138381E-01
+DATA ·exptexp<> + 8(SB)/8, $0.263602189790660309E-01
+DATA ·exptexp<> + 16(SB)/8, $0.122565642281703586E-01
+DATA ·exptexp<> + 24(SB)/8, $0.143757052860721398E-02
+DATA ·exptexp<> + 32(SB)/8, $-.651375034121276075E-02
+DATA ·exptexp<> + 40(SB)/8, $-.119317678849450159E-01
+DATA ·exptexp<> + 48(SB)/8, $-.150868749549871069E-01
+DATA ·exptexp<> + 56(SB)/8, $-.161992609578469234E-01
+DATA ·exptexp<> + 64(SB)/8, $-.154492360403337917E-01
+DATA ·exptexp<> + 72(SB)/8, $-.129850717389178721E-01
+DATA ·exptexp<> + 80(SB)/8, $-.892902649276657891E-02
+DATA ·exptexp<> + 88(SB)/8, $-.338202636596794887E-02
+DATA ·exptexp<> + 96(SB)/8, $0.357266307045684762E-02
+DATA ·exptexp<> + 104(SB)/8, $0.118665304327406698E-01
+DATA ·exptexp<> + 112(SB)/8, $0.214434994118118914E-01
+DATA ·exptexp<> + 120(SB)/8, $0.322580645161290314E-01
+GLOBL ·exptexp<> + 0(SB), RODATA, $128
+
+// Exp returns e**x, the base-e exponential of x.
+//
+// Special cases are:
+// Exp(+Inf) = +Inf
+// Exp(NaN) = NaN
+// Very large values overflow to 0 or +Inf.
+// Very small values underflow to 1.
+// The algorithm used is minimax polynomial approximation using a table of
+// polynomial coefficients determined with a Remez exchange algorithm.
+
+TEXT ·expAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·exprodataL22<>+0(SB), R5
+ LTDBR F0, F0
+ BLTU L20
+ FMOVD F0, F2
+L2:
+ WORD $0xED205050 //cdb %f2,.L23-.L22(%r5)
+ BYTE $0x00
+ BYTE $0x19
+ BGE L16
+ BVS L16
+ WFCEDBS V2, V2, V2
+ BVS LEXITTAGexp
+ MOVD $·expxaddexp<>+0(SB), R1
+ FMOVD 72(R5), F6
+ FMOVD 0(R1), F2
+ WFMSDB V0, V6, V2, V6
+ FMOVD 64(R5), F4
+ FADD F6, F2
+ FMOVD 56(R5), F1
+ FMADD F4, F2, F0
+ FMOVD 48(R5), F3
+ WFMADB V2, V1, V0, V2
+ FMOVD 40(R5), F1
+ FMOVD 32(R5), F4
+ FMUL F0, F0
+ WFMADB V2, V4, V1, V4
+ LGDR F6, R1
+ FMOVD 24(R5), F1
+ WFMADB V2, V3, V1, V3
+ FMOVD 16(R5), F1
+ WFMADB V0, V4, V3, V4
+ FMOVD 8(R5), F3
+ WFMADB V2, V1, V3, V1
+ RISBGZ $57, $60, $3, R1, R3
+ WFMADB V0, V4, V1, V0
+ MOVD $·exptexp<>+0(SB), R2
+ WORD $0x68432000 //ld %f4,0(%r3,%r2)
+ FMADD F4, F2, F2
+ SLD $48, R1, R2
+ WFMADB V2, V0, V4, V2
+ LDGR R2, F0
+ FMADD F0, F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+L16:
+ WFCEDBS V2, V2, V4
+ BVS LEXITTAGexp
+ WORD $0xED205000 //cdb %f2,.L33-.L22(%r5)
+ BYTE $0x00
+ BYTE $0x19
+ BLT L6
+ WFCEDBS V2, V0, V0
+ BVS L13
+ MOVD $·expxinf<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+L20:
+ WORD $0xB3130020 //lcdbr %f2,%f0
+ BR L2
+L6:
+ MOVD $·expxaddexp<>+0(SB), R1
+ FMOVD 72(R5), F3
+ FMOVD 0(R1), F4
+ WFMSDB V0, V3, V4, V3
+ FMOVD 64(R5), F6
+ FADD F3, F4
+ FMOVD 56(R5), F5
+ WFMADB V4, V6, V0, V6
+ FMOVD 32(R5), F1
+ WFMADB V4, V5, V6, V4
+ FMOVD 40(R5), F5
+ FMUL F6, F6
+ WFMADB V4, V1, V5, V1
+ FMOVD 48(R5), F7
+ LGDR F3, R1
+ FMOVD 24(R5), F5
+ WFMADB V4, V7, V5, V7
+ FMOVD 16(R5), F5
+ WFMADB V6, V1, V7, V1
+ FMOVD 8(R5), F7
+ WFMADB V4, V5, V7, V5
+ RISBGZ $57, $60, $3, R1, R3
+ WFMADB V6, V1, V5, V6
+ MOVD $·exptexp<>+0(SB), R2
+ WFCHDBS V2, V0, V0
+ WORD $0x68132000 //ld %f1,0(%r3,%r2)
+ FMADD F1, F4, F4
+ MOVD $0x4086000000000000, R2
+ WFMADB V4, V6, V1, V4
+ BEQ L21
+ ADDW $0xF000, R1
+ RISBGN $0, $15, $48, R1, R2
+ LDGR R2, F0
+ FMADD F0, F4, F0
+ MOVD $·expx4ff<>+0(SB), R3
+ FMOVD 0(R3), F2
+ FMUL F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+L13:
+ FMOVD $0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+L21:
+ ADDW $0x1000, R1
+ RISBGN $0, $15, $48, R1, R2
+ LDGR R2, F0
+ FMADD F0, F4, F0
+ MOVD $·expx2ff<>+0(SB), R3
+ FMOVD 0(R3), F2
+ FMUL F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+LEXITTAGexp:
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/expm1.go b/src/math/expm1.go
new file mode 100644
index 0000000..8e77398
--- /dev/null
+++ b/src/math/expm1.go
@@ -0,0 +1,237 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/s_expm1.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// expm1(x)
+// Returns exp(x)-1, the exponential of x minus 1.
+//
+// Method
+// 1. Argument reduction:
+// Given x, find r and integer k such that
+//
+// x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
+//
+// Here a correction term c will be computed to compensate
+// the error in r when rounded to a floating-point number.
+//
+// 2. Approximating expm1(r) by a special rational function on
+// the interval [0,0.34658]:
+// Since
+// r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 - r**4/360 + ...
+// we define R1(r*r) by
+// r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 * R1(r*r)
+// That is,
+// R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
+// = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
+// = 1 - r**2/60 + r**4/2520 - r**6/100800 + ...
+// We use a special Reme algorithm on [0,0.347] to generate
+// a polynomial of degree 5 in r*r to approximate R1. The
+// maximum error of this polynomial approximation is bounded
+// by 2**-61. In other words,
+// R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
+// where Q1 = -1.6666666666666567384E-2,
+// Q2 = 3.9682539681370365873E-4,
+// Q3 = -9.9206344733435987357E-6,
+// Q4 = 2.5051361420808517002E-7,
+// Q5 = -6.2843505682382617102E-9;
+// (where z=r*r, and the values of Q1 to Q5 are listed below)
+// with error bounded by
+// | 5 | -61
+// | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
+// | |
+//
+// expm1(r) = exp(r)-1 is then computed by the following
+// specific way which minimize the accumulation rounding error:
+// 2 3
+// r r [ 3 - (R1 + R1*r/2) ]
+// expm1(r) = r + --- + --- * [--------------------]
+// 2 2 [ 6 - r*(3 - R1*r/2) ]
+//
+// To compensate the error in the argument reduction, we use
+// expm1(r+c) = expm1(r) + c + expm1(r)*c
+// ~ expm1(r) + c + r*c
+// Thus c+r*c will be added in as the correction terms for
+// expm1(r+c). Now rearrange the term to avoid optimization
+// screw up:
+// ( 2 2 )
+// ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
+// expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
+// ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
+// ( )
+//
+// = r - E
+// 3. Scale back to obtain expm1(x):
+// From step 1, we have
+// expm1(x) = either 2**k*[expm1(r)+1] - 1
+// = or 2**k*[expm1(r) + (1-2**-k)]
+// 4. Implementation notes:
+// (A). To save one multiplication, we scale the coefficient Qi
+// to Qi*2**i, and replace z by (x**2)/2.
+// (B). To achieve maximum accuracy, we compute expm1(x) by
+// (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
+// (ii) if k=0, return r-E
+// (iii) if k=-1, return 0.5*(r-E)-0.5
+// (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
+// else return 1.0+2.0*(r-E);
+// (v) if (k<-2||k>56) return 2**k(1-(E-r)) - 1 (or exp(x)-1)
+// (vi) if k <= 20, return 2**k((1-2**-k)-(E-r)), else
+// (vii) return 2**k(1-((E+2**-k)-r))
+//
+// Special cases:
+// expm1(INF) is INF, expm1(NaN) is NaN;
+// expm1(-INF) is -1, and
+// for finite argument, only expm1(0)=0 is exact.
+//
+// Accuracy:
+// according to an error analysis, the error is always less than
+// 1 ulp (unit in the last place).
+//
+// Misc. info.
+// For IEEE double
+// if x > 7.09782712893383973096e+02 then expm1(x) overflow
+//
+// Constants:
+// The hexadecimal values are the intended ones for the following
+// constants. The decimal values may be used, provided that the
+// compiler will convert from decimal to binary accurately enough
+// to produce the hexadecimal values shown.
+//
+
+// Expm1 returns e**x - 1, the base-e exponential of x minus 1.
+// It is more accurate than Exp(x) - 1 when x is near zero.
+//
+// Special cases are:
+// Expm1(+Inf) = +Inf
+// Expm1(-Inf) = -1
+// Expm1(NaN) = NaN
+// Very large values overflow to -1 or +Inf.
+func Expm1(x float64) float64
+
+func expm1(x float64) float64 {
+ const (
+ Othreshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF
+ Ln2X56 = 3.88162421113569373274e+01 // 0x4043687a9f1af2b1
+ Ln2HalfX3 = 1.03972077083991796413e+00 // 0x3ff0a2b23f3bab73
+ Ln2Half = 3.46573590279972654709e-01 // 0x3fd62e42fefa39ef
+ Ln2Hi = 6.93147180369123816490e-01 // 0x3fe62e42fee00000
+ Ln2Lo = 1.90821492927058770002e-10 // 0x3dea39ef35793c76
+ InvLn2 = 1.44269504088896338700e+00 // 0x3ff71547652b82fe
+ Tiny = 1.0 / (1 << 54) // 2**-54 = 0x3c90000000000000
+ // scaled coefficients related to expm1
+ Q1 = -3.33333333333331316428e-02 // 0xBFA11111111110F4
+ Q2 = 1.58730158725481460165e-03 // 0x3F5A01A019FE5585
+ Q3 = -7.93650757867487942473e-05 // 0xBF14CE199EAADBB7
+ Q4 = 4.00821782732936239552e-06 // 0x3ED0CFCA86E65239
+ Q5 = -2.01099218183624371326e-07 // 0xBE8AFDB76E09C32D
+ )
+
+ // special cases
+ switch {
+ case IsInf(x, 1) || IsNaN(x):
+ return x
+ case IsInf(x, -1):
+ return -1
+ }
+
+ absx := x
+ sign := false
+ if x < 0 {
+ absx = -absx
+ sign = true
+ }
+
+ // filter out huge argument
+ if absx >= Ln2X56 { // if |x| >= 56 * ln2
+ if sign {
+ return -1 // x < -56*ln2, return -1
+ }
+ if absx >= Othreshold { // if |x| >= 709.78...
+ return Inf(1)
+ }
+ }
+
+ // argument reduction
+ var c float64
+ var k int
+ if absx > Ln2Half { // if |x| > 0.5 * ln2
+ var hi, lo float64
+ if absx < Ln2HalfX3 { // and |x| < 1.5 * ln2
+ if !sign {
+ hi = x - Ln2Hi
+ lo = Ln2Lo
+ k = 1
+ } else {
+ hi = x + Ln2Hi
+ lo = -Ln2Lo
+ k = -1
+ }
+ } else {
+ if !sign {
+ k = int(InvLn2*x + 0.5)
+ } else {
+ k = int(InvLn2*x - 0.5)
+ }
+ t := float64(k)
+ hi = x - t*Ln2Hi // t * Ln2Hi is exact here
+ lo = t * Ln2Lo
+ }
+ x = hi - lo
+ c = (hi - x) - lo
+ } else if absx < Tiny { // when |x| < 2**-54, return x
+ return x
+ } else {
+ k = 0
+ }
+
+ // x is now in primary range
+ hfx := 0.5 * x
+ hxs := x * hfx
+ r1 := 1 + hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))))
+ t := 3 - r1*hfx
+ e := hxs * ((r1 - t) / (6.0 - x*t))
+ if k == 0 {
+ return x - (x*e - hxs) // c is 0
+ }
+ e = (x*(e-c) - c)
+ e -= hxs
+ switch {
+ case k == -1:
+ return 0.5*(x-e) - 0.5
+ case k == 1:
+ if x < -0.25 {
+ return -2 * (e - (x + 0.5))
+ }
+ return 1 + 2*(x-e)
+ case k <= -2 || k > 56: // suffice to return exp(x)-1
+ y := 1 - (e - x)
+ y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
+ return y - 1
+ }
+ if k < 20 {
+ t := Float64frombits(0x3ff0000000000000 - (0x20000000000000 >> uint(k))) // t=1-2**-k
+ y := t - (e - x)
+ y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
+ return y
+ }
+ t = Float64frombits(uint64(0x3ff-k) << 52) // 2**-k
+ y := x - (e + t)
+ y++
+ y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
+ return y
+}
diff --git a/src/math/expm1_386.s b/src/math/expm1_386.s
new file mode 100644
index 0000000..d020296
--- /dev/null
+++ b/src/math/expm1_386.s
@@ -0,0 +1,57 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Expm1(x float64) float64
+TEXT ·Expm1(SB),NOSPLIT,$0
+ FLDLN2 // F0=log(2) = 1/log2(e) ~ 0.693147
+ FMOVD x+0(FP), F0 // F0=x, F1=1/log2(e)
+ FABS // F0=|x|, F1=1/log2(e)
+ FUCOMPP F0, F1 // compare F0 to F1
+ FSTSW AX
+ SAHF
+ JCC use_exp // jump if F0 >= F1
+ FLDL2E // F0=log2(e)
+ FMULD x+0(FP), F0 // F0=x*log2(e) (-1<F0<1)
+ F2XM1 // F0=e**x-1 = 2**(x*log2(e))-1
+ FMOVDP F0, ret+8(FP)
+ RET
+use_exp:
+// test bits for not-finite
+ MOVL x_hi+4(FP), AX
+ ANDL $0x7ff00000, AX
+ CMPL AX, $0x7ff00000
+ JEQ not_finite
+ FLDL2E // F0=log2(e)
+ FMULD x+0(FP), F0 // F0=x*log2(e)
+ FMOVD F0, F1 // F0=x*log2(e), F1=x*log2(e)
+ FRNDINT // F0=int(x*log2(e)), F1=x*log2(e)
+ FSUBD F0, F1 // F0=int(x*log2(e)), F1=x*log2(e)-int(x*log2(e))
+ FXCHD F0, F1 // F0=x*log2(e)-int(x*log2(e)), F1=int(x*log2(e))
+ F2XM1 // F0=2**(x*log2(e)-int(x*log2(e)))-1, F1=int(x*log2(e))
+ FLD1 // F0=1, F1=2**(x*log2(e)-int(x*log2(e)))-1, F2=int(x*log2(e))
+ FADDDP F0, F1 // F0=2**(x*log2(e)-int(x*log2(e))), F1=int(x*log2(e))
+ FSCALE // F0=e**x, F1=int(x*log2(e))
+ FMOVDP F0, F1 // F0=e**x
+ FLD1 // F0=1, F1=e**x
+ FSUBDP F0, F1 // F0=e**x-1
+ FMOVDP F0, ret+8(FP)
+ RET
+not_finite:
+// test bits for -Inf
+ MOVL x_hi+4(FP), BX
+ MOVL x_lo+0(FP), CX
+ CMPL BX, $0xfff00000
+ JNE not_neginf
+ CMPL CX, $0
+ JNE not_neginf
+ FLD1 // F0=1
+ FCHS // F0=-1
+ FMOVDP F0, ret+8(FP)
+ RET
+not_neginf:
+ MOVL CX, ret_lo+8(FP)
+ MOVL BX, ret_hi+12(FP)
+ RET
diff --git a/src/math/expm1_s390x.s b/src/math/expm1_s390x.s
new file mode 100644
index 0000000..16c861b
--- /dev/null
+++ b/src/math/expm1_s390x.s
@@ -0,0 +1,194 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial approximation and other constants
+DATA ·expm1rodataL22<> + 0(SB)/8, $-1.0
+DATA ·expm1rodataL22<> + 8(SB)/8, $800.0E+00
+DATA ·expm1rodataL22<> + 16(SB)/8, $1.0
+DATA ·expm1rodataL22<> + 24(SB)/8, $-.231904681384629956E-16
+DATA ·expm1rodataL22<> + 32(SB)/8, $0.50000000000000029671E+00
+DATA ·expm1rodataL22<> + 40(SB)/8, $0.16666666666666676570E+00
+DATA ·expm1rodataL22<> + 48(SB)/8, $0.83333333323590973444E-02
+DATA ·expm1rodataL22<> + 56(SB)/8, $0.13889096526400683566E-02
+DATA ·expm1rodataL22<> + 64(SB)/8, $0.41666666661701152924E-01
+DATA ·expm1rodataL22<> + 72(SB)/8, $0.19841562053987360264E-03
+DATA ·expm1rodataL22<> + 80(SB)/8, $-.693147180559945286E+00
+DATA ·expm1rodataL22<> + 88(SB)/8, $0.144269504088896339E+01
+DATA ·expm1rodataL22<> + 96(SB)/8, $704.0E+00
+GLOBL ·expm1rodataL22<> + 0(SB), RODATA, $104
+
+DATA ·expm1xmone<> + 0(SB)/8, $0xbff0000000000000
+GLOBL ·expm1xmone<> + 0(SB), RODATA, $8
+DATA ·expm1xinf<> + 0(SB)/8, $0x7ff0000000000000
+GLOBL ·expm1xinf<> + 0(SB), RODATA, $8
+DATA ·expm1x4ff<> + 0(SB)/8, $0x4ff0000000000000
+GLOBL ·expm1x4ff<> + 0(SB), RODATA, $8
+DATA ·expm1x2ff<> + 0(SB)/8, $0x2ff0000000000000
+GLOBL ·expm1x2ff<> + 0(SB), RODATA, $8
+DATA ·expm1xaddexp<> + 0(SB)/8, $0xc2f0000100003ff0
+GLOBL ·expm1xaddexp<> + 0(SB), RODATA, $8
+
+// Log multipliers table
+DATA ·expm1tab<> + 0(SB)/8, $0.0
+DATA ·expm1tab<> + 8(SB)/8, $-.171540871271399150E-01
+DATA ·expm1tab<> + 16(SB)/8, $-.306597931864376363E-01
+DATA ·expm1tab<> + 24(SB)/8, $-.410200970469965021E-01
+DATA ·expm1tab<> + 32(SB)/8, $-.486343079978231466E-01
+DATA ·expm1tab<> + 40(SB)/8, $-.538226193725835820E-01
+DATA ·expm1tab<> + 48(SB)/8, $-.568439602538111520E-01
+DATA ·expm1tab<> + 56(SB)/8, $-.579091847395528847E-01
+DATA ·expm1tab<> + 64(SB)/8, $-.571909584179366341E-01
+DATA ·expm1tab<> + 72(SB)/8, $-.548312665987204407E-01
+DATA ·expm1tab<> + 80(SB)/8, $-.509471843643441085E-01
+DATA ·expm1tab<> + 88(SB)/8, $-.456353588448863359E-01
+DATA ·expm1tab<> + 96(SB)/8, $-.389755254243262365E-01
+DATA ·expm1tab<> + 104(SB)/8, $-.310332908285244231E-01
+DATA ·expm1tab<> + 112(SB)/8, $-.218623539150173528E-01
+DATA ·expm1tab<> + 120(SB)/8, $-.115062908917949451E-01
+GLOBL ·expm1tab<> + 0(SB), RODATA, $128
+
+// Expm1 returns e**x - 1, the base-e exponential of x minus 1.
+// It is more accurate than Exp(x) - 1 when x is near zero.
+//
+// Special cases are:
+// Expm1(+Inf) = +Inf
+// Expm1(-Inf) = -1
+// Expm1(NaN) = NaN
+// Very large values overflow to -1 or +Inf.
+// The algorithm used is minimax polynomial approximation using a table of
+// polynomial coefficients determined with a Remez exchange algorithm.
+
+TEXT ·expm1Asm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·expm1rodataL22<>+0(SB), R5
+ LTDBR F0, F0
+ BLTU L20
+ FMOVD F0, F2
+L2:
+ WORD $0xED205060 //cdb %f2,.L23-.L22(%r5)
+ BYTE $0x00
+ BYTE $0x19
+ BGE L16
+ BVS L16
+ WFCEDBS V2, V2, V2
+ BVS LEXITTAGexpm1
+ MOVD $·expm1xaddexp<>+0(SB), R1
+ FMOVD 88(R5), F1
+ FMOVD 0(R1), F2
+ WFMSDB V0, V1, V2, V1
+ FMOVD 80(R5), F6
+ WFADB V1, V2, V4
+ FMOVD 72(R5), F2
+ FMADD F6, F4, F0
+ FMOVD 64(R5), F3
+ FMOVD 56(R5), F6
+ FMOVD 48(R5), F5
+ FMADD F2, F0, F6
+ WFMADB V0, V5, V3, V5
+ WFMDB V0, V0, V2
+ LGDR F1, R1
+ WFMADB V6, V2, V5, V6
+ FMOVD 40(R5), F3
+ FMOVD 32(R5), F5
+ WFMADB V0, V3, V5, V3
+ FMOVD 24(R5), F5
+ WFMADB V2, V6, V3, V2
+ FMADD F5, F4, F0
+ FMOVD 16(R5), F6
+ WFMADB V0, V2, V6, V2
+ RISBGZ $57, $60, $3, R1, R3
+ WORD $0xB3130022 //lcdbr %f2,%f2
+ MOVD $·expm1tab<>+0(SB), R2
+ WORD $0x68432000 //ld %f4,0(%r3,%r2)
+ FMADD F4, F0, F0
+ SLD $48, R1, R2
+ WFMSDB V2, V0, V4, V0
+ LDGR R2, F4
+ WORD $0xB3130000 //lcdbr %f0,%f0
+ FSUB F4, F6
+ WFMSDB V0, V4, V6, V0
+ FMOVD F0, ret+8(FP)
+ RET
+L16:
+ WFCEDBS V2, V2, V4
+ BVS LEXITTAGexpm1
+ WORD $0xED205008 //cdb %f2,.L34-.L22(%r5)
+ BYTE $0x00
+ BYTE $0x19
+ BLT L6
+ WFCEDBS V2, V0, V0
+ BVS L7
+ MOVD $·expm1xinf<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+L20:
+ WORD $0xB3130020 //lcdbr %f2,%f0
+ BR L2
+L6:
+ MOVD $·expm1xaddexp<>+0(SB), R1
+ FMOVD 88(R5), F5
+ FMOVD 0(R1), F4
+ WFMSDB V0, V5, V4, V5
+ FMOVD 80(R5), F3
+ WFADB V5, V4, V1
+ VLEG $0, 48(R5), V16
+ WFMADB V1, V3, V0, V3
+ FMOVD 56(R5), F4
+ FMOVD 64(R5), F7
+ FMOVD 72(R5), F6
+ WFMADB V3, V16, V7, V16
+ WFMADB V3, V6, V4, V6
+ WFMDB V3, V3, V4
+ MOVD $·expm1tab<>+0(SB), R2
+ WFMADB V6, V4, V16, V6
+ VLEG $0, 32(R5), V16
+ FMOVD 40(R5), F7
+ WFMADB V3, V7, V16, V7
+ VLEG $0, 24(R5), V16
+ WFMADB V4, V6, V7, V4
+ WFMADB V1, V16, V3, V1
+ FMOVD 16(R5), F6
+ FMADD F4, F1, F6
+ LGDR F5, R1
+ WORD $0xB3130066 //lcdbr %f6,%f6
+ RISBGZ $57, $60, $3, R1, R3
+ WORD $0x68432000 //ld %f4,0(%r3,%r2)
+ FMADD F4, F1, F1
+ MOVD $0x4086000000000000, R2
+ FMSUB F1, F6, F4
+ WORD $0xB3130044 //lcdbr %f4,%f4
+ WFCHDBS V2, V0, V0
+ BEQ L21
+ ADDW $0xF000, R1
+ RISBGN $0, $15, $48, R1, R2
+ LDGR R2, F0
+ FMADD F0, F4, F0
+ MOVD $·expm1x4ff<>+0(SB), R3
+ FMOVD 0(R5), F4
+ FMOVD 0(R3), F2
+ WFMADB V2, V0, V4, V0
+ FMOVD F0, ret+8(FP)
+ RET
+L7:
+ MOVD $·expm1xmone<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+L21:
+ ADDW $0x1000, R1
+ RISBGN $0, $15, $48, R1, R2
+ LDGR R2, F0
+ FMADD F0, F4, F0
+ MOVD $·expm1x2ff<>+0(SB), R3
+ FMOVD 0(R5), F4
+ FMOVD 0(R3), F2
+ WFMADB V2, V0, V4, V0
+ FMOVD F0, ret+8(FP)
+ RET
+LEXITTAGexpm1:
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/export_s390x_test.go b/src/math/export_s390x_test.go
new file mode 100644
index 0000000..827bf1c
--- /dev/null
+++ b/src/math/export_s390x_test.go
@@ -0,0 +1,31 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Export internal functions and variable for testing.
+var Log10NoVec = log10
+var CosNoVec = cos
+var CoshNoVec = cosh
+var SinNoVec = sin
+var SinhNoVec = sinh
+var TanhNoVec = tanh
+var Log1pNovec = log1p
+var AtanhNovec = atanh
+var AcosNovec = acos
+var AcoshNovec = acosh
+var AsinNovec = asin
+var AsinhNovec = asinh
+var ErfNovec = erf
+var ErfcNovec = erfc
+var AtanNovec = atan
+var Atan2Novec = atan2
+var CbrtNovec = cbrt
+var LogNovec = log
+var TanNovec = tan
+var ExpNovec = exp
+var Expm1Novec = expm1
+var PowNovec = pow
+var HypotNovec = hypot
+var HasVX = hasVX
diff --git a/src/math/export_test.go b/src/math/export_test.go
new file mode 100644
index 0000000..53d9205
--- /dev/null
+++ b/src/math/export_test.go
@@ -0,0 +1,14 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Export internal functions for testing.
+var ExpGo = exp
+var Exp2Go = exp2
+var HypotGo = hypot
+var SqrtGo = sqrt
+var TrigReduce = trigReduce
+
+const ReduceThreshold = reduceThreshold
diff --git a/src/math/floor.go b/src/math/floor.go
new file mode 100644
index 0000000..18e89ef
--- /dev/null
+++ b/src/math/floor.go
@@ -0,0 +1,131 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Floor returns the greatest integer value less than or equal to x.
+//
+// Special cases are:
+// Floor(±0) = ±0
+// Floor(±Inf) = ±Inf
+// Floor(NaN) = NaN
+func Floor(x float64) float64
+
+func floor(x float64) float64 {
+ if x == 0 || IsNaN(x) || IsInf(x, 0) {
+ return x
+ }
+ if x < 0 {
+ d, fract := Modf(-x)
+ if fract != 0.0 {
+ d = d + 1
+ }
+ return -d
+ }
+ d, _ := Modf(x)
+ return d
+}
+
+// Ceil returns the least integer value greater than or equal to x.
+//
+// Special cases are:
+// Ceil(±0) = ±0
+// Ceil(±Inf) = ±Inf
+// Ceil(NaN) = NaN
+func Ceil(x float64) float64
+
+func ceil(x float64) float64 {
+ return -Floor(-x)
+}
+
+// Trunc returns the integer value of x.
+//
+// Special cases are:
+// Trunc(±0) = ±0
+// Trunc(±Inf) = ±Inf
+// Trunc(NaN) = NaN
+func Trunc(x float64) float64
+
+func trunc(x float64) float64 {
+ if x == 0 || IsNaN(x) || IsInf(x, 0) {
+ return x
+ }
+ d, _ := Modf(x)
+ return d
+}
+
+// Round returns the nearest integer, rounding half away from zero.
+//
+// Special cases are:
+// Round(±0) = ±0
+// Round(±Inf) = ±Inf
+// Round(NaN) = NaN
+func Round(x float64) float64 {
+ // Round is a faster implementation of:
+ //
+ // func Round(x float64) float64 {
+ // t := Trunc(x)
+ // if Abs(x-t) >= 0.5 {
+ // return t + Copysign(1, x)
+ // }
+ // return t
+ // }
+ bits := Float64bits(x)
+ e := uint(bits>>shift) & mask
+ if e < bias {
+ // Round abs(x) < 1 including denormals.
+ bits &= signMask // +-0
+ if e == bias-1 {
+ bits |= uvone // +-1
+ }
+ } else if e < bias+shift {
+ // Round any abs(x) >= 1 containing a fractional component [0,1).
+ //
+ // Numbers with larger exponents are returned unchanged since they
+ // must be either an integer, infinity, or NaN.
+ const half = 1 << (shift - 1)
+ e -= bias
+ bits += half >> e
+ bits &^= fracMask >> e
+ }
+ return Float64frombits(bits)
+}
+
+// RoundToEven returns the nearest integer, rounding ties to even.
+//
+// Special cases are:
+// RoundToEven(±0) = ±0
+// RoundToEven(±Inf) = ±Inf
+// RoundToEven(NaN) = NaN
+func RoundToEven(x float64) float64 {
+ // RoundToEven is a faster implementation of:
+ //
+ // func RoundToEven(x float64) float64 {
+ // t := math.Trunc(x)
+ // odd := math.Remainder(t, 2) != 0
+ // if d := math.Abs(x - t); d > 0.5 || (d == 0.5 && odd) {
+ // return t + math.Copysign(1, x)
+ // }
+ // return t
+ // }
+ bits := Float64bits(x)
+ e := uint(bits>>shift) & mask
+ if e >= bias {
+ // Round abs(x) >= 1.
+ // - Large numbers without fractional components, infinity, and NaN are unchanged.
+ // - Add 0.499.. or 0.5 before truncating depending on whether the truncated
+ // number is even or odd (respectively).
+ const halfMinusULP = (1 << (shift - 1)) - 1
+ e -= bias
+ bits += (halfMinusULP + (bits>>(shift-e))&1) >> e
+ bits &^= fracMask >> e
+ } else if e == bias-1 && bits&fracMask != 0 {
+ // Round 0.5 < abs(x) < 1.
+ bits = bits&signMask | uvone // +-1
+ } else {
+ // Round abs(x) <= 0.5 including denormals.
+ bits &= signMask // +-0
+ }
+ return Float64frombits(bits)
+}
diff --git a/src/math/floor_386.s b/src/math/floor_386.s
new file mode 100644
index 0000000..0960eec
--- /dev/null
+++ b/src/math/floor_386.s
@@ -0,0 +1,46 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Ceil(x float64) float64
+TEXT ·Ceil(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FSTCW -2(SP) // save old Control Word
+ MOVW -2(SP), AX
+ ANDW $0xf3ff, AX
+ ORW $0x0800, AX // Rounding Control set to +Inf
+ MOVW AX, -4(SP) // store new Control Word
+ FLDCW -4(SP) // load new Control Word
+ FRNDINT // F0=Ceil(x)
+ FLDCW -2(SP) // load old Control Word
+ FMOVDP F0, ret+8(FP)
+ RET
+
+// func Floor(x float64) float64
+TEXT ·Floor(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FSTCW -2(SP) // save old Control Word
+ MOVW -2(SP), AX
+ ANDW $0xf3ff, AX
+ ORW $0x0400, AX // Rounding Control set to -Inf
+ MOVW AX, -4(SP) // store new Control Word
+ FLDCW -4(SP) // load new Control Word
+ FRNDINT // F0=Floor(x)
+ FLDCW -2(SP) // load old Control Word
+ FMOVDP F0, ret+8(FP)
+ RET
+
+// func Trunc(x float64) float64
+TEXT ·Trunc(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FSTCW -2(SP) // save old Control Word
+ MOVW -2(SP), AX
+ ORW $0x0c00, AX // Rounding Control set to truncate
+ MOVW AX, -4(SP) // store new Control Word
+ FLDCW -4(SP) // load new Control Word
+ FRNDINT // F0=Trunc(x)
+ FLDCW -2(SP) // load old Control Word
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/floor_amd64.s b/src/math/floor_amd64.s
new file mode 100644
index 0000000..4ef02eb
--- /dev/null
+++ b/src/math/floor_amd64.s
@@ -0,0 +1,76 @@
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define Big 0x4330000000000000 // 2**52
+
+// func Floor(x float64) float64
+TEXT ·Floor(SB),NOSPLIT,$0
+ MOVQ x+0(FP), AX
+ MOVQ $~(1<<63), DX // sign bit mask
+ ANDQ AX,DX // DX = |x|
+ SUBQ $1,DX
+ MOVQ $(Big - 1), CX // if |x| >= 2**52-1 or IsNaN(x) or |x| == 0, return x
+ CMPQ DX,CX
+ JAE isBig_floor
+ MOVQ AX, X0 // X0 = x
+ CVTTSD2SQ X0, AX
+ CVTSQ2SD AX, X1 // X1 = float(int(x))
+ CMPSD X1, X0, 1 // compare LT; X0 = 0xffffffffffffffff or 0
+ MOVSD $(-1.0), X2
+ ANDPD X2, X0 // if x < float(int(x)) {X0 = -1} else {X0 = 0}
+ ADDSD X1, X0
+ MOVSD X0, ret+8(FP)
+ RET
+isBig_floor:
+ MOVQ AX, ret+8(FP) // return x
+ RET
+
+// func Ceil(x float64) float64
+TEXT ·Ceil(SB),NOSPLIT,$0
+ MOVQ x+0(FP), AX
+ MOVQ $~(1<<63), DX // sign bit mask
+ MOVQ AX, BX // BX = copy of x
+ ANDQ DX, BX // BX = |x|
+ MOVQ $Big, CX // if |x| >= 2**52 or IsNaN(x), return x
+ CMPQ BX, CX
+ JAE isBig_ceil
+ MOVQ AX, X0 // X0 = x
+ MOVQ DX, X2 // X2 = sign bit mask
+ CVTTSD2SQ X0, AX
+ ANDNPD X0, X2 // X2 = sign
+ CVTSQ2SD AX, X1 // X1 = float(int(x))
+ CMPSD X1, X0, 2 // compare LE; X0 = 0xffffffffffffffff or 0
+ ORPD X2, X1 // if X1 = 0.0, incorporate sign
+ MOVSD $1.0, X3
+ ANDNPD X3, X0
+ ORPD X2, X0 // if float(int(x)) <= x {X0 = 1} else {X0 = -0}
+ ADDSD X1, X0
+ MOVSD X0, ret+8(FP)
+ RET
+isBig_ceil:
+ MOVQ AX, ret+8(FP)
+ RET
+
+// func Trunc(x float64) float64
+TEXT ·Trunc(SB),NOSPLIT,$0
+ MOVQ x+0(FP), AX
+ MOVQ $~(1<<63), DX // sign bit mask
+ MOVQ AX, BX // BX = copy of x
+ ANDQ DX, BX // BX = |x|
+ MOVQ $Big, CX // if |x| >= 2**52 or IsNaN(x), return x
+ CMPQ BX, CX
+ JAE isBig_trunc
+ MOVQ AX, X0
+ MOVQ DX, X2 // X2 = sign bit mask
+ CVTTSD2SQ X0, AX
+ ANDNPD X0, X2 // X2 = sign
+ CVTSQ2SD AX, X0 // X0 = float(int(x))
+ ORPD X2, X0 // if X0 = 0.0, incorporate sign
+ MOVSD X0, ret+8(FP)
+ RET
+isBig_trunc:
+ MOVQ AX, ret+8(FP) // return x
+ RET
diff --git a/src/math/floor_arm64.s b/src/math/floor_arm64.s
new file mode 100644
index 0000000..6d240d4
--- /dev/null
+++ b/src/math/floor_arm64.s
@@ -0,0 +1,26 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Floor(x float64) float64
+TEXT ·Floor(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FRINTMD F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+// func Ceil(x float64) float64
+TEXT ·Ceil(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FRINTPD F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+// func Trunc(x float64) float64
+TEXT ·Trunc(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FRINTZD F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/floor_ppc64x.s b/src/math/floor_ppc64x.s
new file mode 100644
index 0000000..2ab011d
--- /dev/null
+++ b/src/math/floor_ppc64x.s
@@ -0,0 +1,25 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build ppc64 ppc64le
+
+#include "textflag.h"
+
+TEXT ·Floor(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FRIM F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+TEXT ·Ceil(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FRIP F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+TEXT ·Trunc(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FRIZ F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/floor_s390x.s b/src/math/floor_s390x.s
new file mode 100644
index 0000000..896e79b
--- /dev/null
+++ b/src/math/floor_s390x.s
@@ -0,0 +1,26 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Floor(x float64) float64
+TEXT ·Floor(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FIDBR $7, F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+// func Ceil(x float64) float64
+TEXT ·Ceil(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FIDBR $6, F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+// func Trunc(x float64) float64
+TEXT ·Trunc(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FIDBR $5, F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/floor_wasm.s b/src/math/floor_wasm.s
new file mode 100644
index 0000000..4d8a0eb
--- /dev/null
+++ b/src/math/floor_wasm.s
@@ -0,0 +1,26 @@
+// Copyright 2018 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Floor(SB),NOSPLIT,$0
+ Get SP
+ F64Load x+0(FP)
+ F64Floor
+ F64Store ret+8(FP)
+ RET
+
+TEXT ·Ceil(SB),NOSPLIT,$0
+ Get SP
+ F64Load x+0(FP)
+ F64Ceil
+ F64Store ret+8(FP)
+ RET
+
+TEXT ·Trunc(SB),NOSPLIT,$0
+ Get SP
+ F64Load x+0(FP)
+ F64Trunc
+ F64Store ret+8(FP)
+ RET
diff --git a/src/math/fma.go b/src/math/fma.go
new file mode 100644
index 0000000..db78dfa
--- /dev/null
+++ b/src/math/fma.go
@@ -0,0 +1,170 @@
+// Copyright 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+import "math/bits"
+
+func zero(x uint64) uint64 {
+ if x == 0 {
+ return 1
+ }
+ return 0
+ // branchless:
+ // return ((x>>1 | x&1) - 1) >> 63
+}
+
+func nonzero(x uint64) uint64 {
+ if x != 0 {
+ return 1
+ }
+ return 0
+ // branchless:
+ // return 1 - ((x>>1|x&1)-1)>>63
+}
+
+func shl(u1, u2 uint64, n uint) (r1, r2 uint64) {
+ r1 = u1<<n | u2>>(64-n) | u2<<(n-64)
+ r2 = u2 << n
+ return
+}
+
+func shr(u1, u2 uint64, n uint) (r1, r2 uint64) {
+ r2 = u2>>n | u1<<(64-n) | u1>>(n-64)
+ r1 = u1 >> n
+ return
+}
+
+// shrcompress compresses the bottom n+1 bits of the two-word
+// value into a single bit. the result is equal to the value
+// shifted to the right by n, except the result's 0th bit is
+// set to the bitwise OR of the bottom n+1 bits.
+func shrcompress(u1, u2 uint64, n uint) (r1, r2 uint64) {
+ // TODO: Performance here is really sensitive to the
+ // order/placement of these branches. n == 0 is common
+ // enough to be in the fast path. Perhaps more measurement
+ // needs to be done to find the optimal order/placement?
+ switch {
+ case n == 0:
+ return u1, u2
+ case n == 64:
+ return 0, u1 | nonzero(u2)
+ case n >= 128:
+ return 0, nonzero(u1 | u2)
+ case n < 64:
+ r1, r2 = shr(u1, u2, n)
+ r2 |= nonzero(u2 & (1<<n - 1))
+ case n < 128:
+ r1, r2 = shr(u1, u2, n)
+ r2 |= nonzero(u1&(1<<(n-64)-1) | u2)
+ }
+ return
+}
+
+func lz(u1, u2 uint64) (l int32) {
+ l = int32(bits.LeadingZeros64(u1))
+ if l == 64 {
+ l += int32(bits.LeadingZeros64(u2))
+ }
+ return l
+}
+
+// split splits b into sign, biased exponent, and mantissa.
+// It adds the implicit 1 bit to the mantissa for normal values,
+// and normalizes subnormal values.
+func split(b uint64) (sign uint32, exp int32, mantissa uint64) {
+ sign = uint32(b >> 63)
+ exp = int32(b>>52) & mask
+ mantissa = b & fracMask
+
+ if exp == 0 {
+ // Normalize value if subnormal.
+ shift := uint(bits.LeadingZeros64(mantissa) - 11)
+ mantissa <<= shift
+ exp = 1 - int32(shift)
+ } else {
+ // Add implicit 1 bit
+ mantissa |= 1 << 52
+ }
+ return
+}
+
+// FMA returns x * y + z, computed with only one rounding.
+// (That is, FMA returns the fused multiply-add of x, y, and z.)
+func FMA(x, y, z float64) float64 {
+ bx, by, bz := Float64bits(x), Float64bits(y), Float64bits(z)
+
+ // Inf or NaN or zero involved. At most one rounding will occur.
+ if x == 0.0 || y == 0.0 || z == 0.0 || bx&uvinf == uvinf || by&uvinf == uvinf {
+ return x*y + z
+ }
+ // Handle non-finite z separately. Evaluating x*y+z where
+ // x and y are finite, but z is infinite, should always result in z.
+ if bz&uvinf == uvinf {
+ return z
+ }
+
+ // Inputs are (sub)normal.
+ // Split x, y, z into sign, exponent, mantissa.
+ xs, xe, xm := split(bx)
+ ys, ye, ym := split(by)
+ zs, ze, zm := split(bz)
+
+ // Compute product p = x*y as sign, exponent, two-word mantissa.
+ // Start with exponent. "is normal" bit isn't subtracted yet.
+ pe := xe + ye - bias + 1
+
+ // pm1:pm2 is the double-word mantissa for the product p.
+ // Shift left to leave top bit in product. Effectively
+ // shifts the 106-bit product to the left by 21.
+ pm1, pm2 := bits.Mul64(xm<<10, ym<<11)
+ zm1, zm2 := zm<<10, uint64(0)
+ ps := xs ^ ys // product sign
+
+ // normalize to 62nd bit
+ is62zero := uint((^pm1 >> 62) & 1)
+ pm1, pm2 = shl(pm1, pm2, is62zero)
+ pe -= int32(is62zero)
+
+ // Swap addition operands so |p| >= |z|
+ if pe < ze || (pe == ze && (pm1 < zm1 || (pm1 == zm1 && pm2 < zm2))) {
+ ps, pe, pm1, pm2, zs, ze, zm1, zm2 = zs, ze, zm1, zm2, ps, pe, pm1, pm2
+ }
+
+ // Align significands
+ zm1, zm2 = shrcompress(zm1, zm2, uint(pe-ze))
+
+ // Compute resulting significands, normalizing if necessary.
+ var m, c uint64
+ if ps == zs {
+ // Adding (pm1:pm2) + (zm1:zm2)
+ pm2, c = bits.Add64(pm2, zm2, 0)
+ pm1, _ = bits.Add64(pm1, zm1, c)
+ pe -= int32(^pm1 >> 63)
+ pm1, m = shrcompress(pm1, pm2, uint(64+pm1>>63))
+ } else {
+ // Subtracting (pm1:pm2) - (zm1:zm2)
+ // TODO: should we special-case cancellation?
+ pm2, c = bits.Sub64(pm2, zm2, 0)
+ pm1, _ = bits.Sub64(pm1, zm1, c)
+ nz := lz(pm1, pm2)
+ pe -= nz
+ m, pm2 = shl(pm1, pm2, uint(nz-1))
+ m |= nonzero(pm2)
+ }
+
+ // Round and break ties to even
+ if pe > 1022+bias || pe == 1022+bias && (m+1<<9)>>63 == 1 {
+ // rounded value overflows exponent range
+ return Float64frombits(uint64(ps)<<63 | uvinf)
+ }
+ if pe < 0 {
+ n := uint(-pe)
+ m = m>>n | nonzero(m&(1<<n-1))
+ pe = 0
+ }
+ m = ((m + 1<<9) >> 10) & ^zero((m&(1<<10-1))^1<<9)
+ pe &= -int32(nonzero(m))
+ return Float64frombits(uint64(ps)<<63 + uint64(pe)<<52 + m)
+}
diff --git a/src/math/frexp.go b/src/math/frexp.go
new file mode 100644
index 0000000..0e26feb
--- /dev/null
+++ b/src/math/frexp.go
@@ -0,0 +1,33 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Frexp breaks f into a normalized fraction
+// and an integral power of two.
+// It returns frac and exp satisfying f == frac × 2**exp,
+// with the absolute value of frac in the interval [½, 1).
+//
+// Special cases are:
+// Frexp(±0) = ±0, 0
+// Frexp(±Inf) = ±Inf, 0
+// Frexp(NaN) = NaN, 0
+func Frexp(f float64) (frac float64, exp int)
+
+func frexp(f float64) (frac float64, exp int) {
+ // special cases
+ switch {
+ case f == 0:
+ return f, 0 // correctly return -0
+ case IsInf(f, 0) || IsNaN(f):
+ return f, 0
+ }
+ f, exp = normalize(f)
+ x := Float64bits(f)
+ exp += int((x>>shift)&mask) - bias + 1
+ x &^= mask << shift
+ x |= (-1 + bias) << shift
+ frac = Float64frombits(x)
+ return
+}
diff --git a/src/math/frexp_386.s b/src/math/frexp_386.s
new file mode 100644
index 0000000..832b435
--- /dev/null
+++ b/src/math/frexp_386.s
@@ -0,0 +1,25 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Frexp(f float64) (frac float64, exp int)
+TEXT ·Frexp(SB),NOSPLIT,$0
+ FMOVD f+0(FP), F0 // F0=f
+ FXAM
+ FSTSW AX
+ SAHF
+ JNP nan_zero_inf
+ JCS nan_zero_inf
+ FXTRACT // F0=f (0<=f<1), F1=e
+ FMULD $(0.5), F0 // F0=f (0.5<=f<1), F1=e
+ FMOVDP F0, frac+8(FP) // F0=e
+ FLD1 // F0=1, F1=e
+ FADDDP F0, F1 // F0=e+1
+ FMOVLP F0, exp+16(FP) // (int=int32)
+ RET
+nan_zero_inf:
+ FMOVDP F0, frac+8(FP) // F0=e
+ MOVL $0, exp+16(FP) // exp=0
+ RET
diff --git a/src/math/gamma.go b/src/math/gamma.go
new file mode 100644
index 0000000..cc9e869
--- /dev/null
+++ b/src/math/gamma.go
@@ -0,0 +1,221 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/cprob/gamma.c.
+// The go code is a simplified version of the original C.
+//
+// tgamma.c
+//
+// Gamma function
+//
+// SYNOPSIS:
+//
+// double x, y, tgamma();
+// extern int signgam;
+//
+// y = tgamma( x );
+//
+// DESCRIPTION:
+//
+// Returns gamma function of the argument. The result is
+// correctly signed, and the sign (+1 or -1) is also
+// returned in a global (extern) variable named signgam.
+// This variable is also filled in by the logarithmic gamma
+// function lgamma().
+//
+// Arguments |x| <= 34 are reduced by recurrence and the function
+// approximated by a rational function of degree 6/7 in the
+// interval (2,3). Large arguments are handled by Stirling's
+// formula. Large negative arguments are made positive using
+// a reflection formula.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -34, 34 10000 1.3e-16 2.5e-17
+// IEEE -170,-33 20000 2.3e-15 3.3e-16
+// IEEE -33, 33 20000 9.4e-16 2.2e-16
+// IEEE 33, 171.6 20000 2.3e-15 3.2e-16
+//
+// Error for arguments outside the test range will be larger
+// owing to error amplification by the exponential function.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+var _gamP = [...]float64{
+ 1.60119522476751861407e-04,
+ 1.19135147006586384913e-03,
+ 1.04213797561761569935e-02,
+ 4.76367800457137231464e-02,
+ 2.07448227648435975150e-01,
+ 4.94214826801497100753e-01,
+ 9.99999999999999996796e-01,
+}
+var _gamQ = [...]float64{
+ -2.31581873324120129819e-05,
+ 5.39605580493303397842e-04,
+ -4.45641913851797240494e-03,
+ 1.18139785222060435552e-02,
+ 3.58236398605498653373e-02,
+ -2.34591795718243348568e-01,
+ 7.14304917030273074085e-02,
+ 1.00000000000000000320e+00,
+}
+var _gamS = [...]float64{
+ 7.87311395793093628397e-04,
+ -2.29549961613378126380e-04,
+ -2.68132617805781232825e-03,
+ 3.47222221605458667310e-03,
+ 8.33333333333482257126e-02,
+}
+
+// Gamma function computed by Stirling's formula.
+// The pair of results must be multiplied together to get the actual answer.
+// The multiplication is left to the caller so that, if careful, the caller can avoid
+// infinity for 172 <= x <= 180.
+// The polynomial is valid for 33 <= x <= 172; larger values are only used
+// in reciprocal and produce denormalized floats. The lower precision there
+// masks any imprecision in the polynomial.
+func stirling(x float64) (float64, float64) {
+ if x > 200 {
+ return Inf(1), 1
+ }
+ const (
+ SqrtTwoPi = 2.506628274631000502417
+ MaxStirling = 143.01608
+ )
+ w := 1 / x
+ w = 1 + w*((((_gamS[0]*w+_gamS[1])*w+_gamS[2])*w+_gamS[3])*w+_gamS[4])
+ y1 := Exp(x)
+ y2 := 1.0
+ if x > MaxStirling { // avoid Pow() overflow
+ v := Pow(x, 0.5*x-0.25)
+ y1, y2 = v, v/y1
+ } else {
+ y1 = Pow(x, x-0.5) / y1
+ }
+ return y1, SqrtTwoPi * w * y2
+}
+
+// Gamma returns the Gamma function of x.
+//
+// Special cases are:
+// Gamma(+Inf) = +Inf
+// Gamma(+0) = +Inf
+// Gamma(-0) = -Inf
+// Gamma(x) = NaN for integer x < 0
+// Gamma(-Inf) = NaN
+// Gamma(NaN) = NaN
+func Gamma(x float64) float64 {
+ const Euler = 0.57721566490153286060651209008240243104215933593992 // A001620
+ // special cases
+ switch {
+ case isNegInt(x) || IsInf(x, -1) || IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return Inf(1)
+ case x == 0:
+ if Signbit(x) {
+ return Inf(-1)
+ }
+ return Inf(1)
+ }
+ q := Abs(x)
+ p := Floor(q)
+ if q > 33 {
+ if x >= 0 {
+ y1, y2 := stirling(x)
+ return y1 * y2
+ }
+ // Note: x is negative but (checked above) not a negative integer,
+ // so x must be small enough to be in range for conversion to int64.
+ // If |x| were >= 2⁶³ it would have to be an integer.
+ signgam := 1
+ if ip := int64(p); ip&1 == 0 {
+ signgam = -1
+ }
+ z := q - p
+ if z > 0.5 {
+ p = p + 1
+ z = q - p
+ }
+ z = q * Sin(Pi*z)
+ if z == 0 {
+ return Inf(signgam)
+ }
+ sq1, sq2 := stirling(q)
+ absz := Abs(z)
+ d := absz * sq1 * sq2
+ if IsInf(d, 0) {
+ z = Pi / absz / sq1 / sq2
+ } else {
+ z = Pi / d
+ }
+ return float64(signgam) * z
+ }
+
+ // Reduce argument
+ z := 1.0
+ for x >= 3 {
+ x = x - 1
+ z = z * x
+ }
+ for x < 0 {
+ if x > -1e-09 {
+ goto small
+ }
+ z = z / x
+ x = x + 1
+ }
+ for x < 2 {
+ if x < 1e-09 {
+ goto small
+ }
+ z = z / x
+ x = x + 1
+ }
+
+ if x == 2 {
+ return z
+ }
+
+ x = x - 2
+ p = (((((x*_gamP[0]+_gamP[1])*x+_gamP[2])*x+_gamP[3])*x+_gamP[4])*x+_gamP[5])*x + _gamP[6]
+ q = ((((((x*_gamQ[0]+_gamQ[1])*x+_gamQ[2])*x+_gamQ[3])*x+_gamQ[4])*x+_gamQ[5])*x+_gamQ[6])*x + _gamQ[7]
+ return z * p / q
+
+small:
+ if x == 0 {
+ return Inf(1)
+ }
+ return z / ((1 + Euler*x) * x)
+}
+
+func isNegInt(x float64) bool {
+ if x < 0 {
+ _, xf := Modf(x)
+ return xf == 0
+ }
+ return false
+}
diff --git a/src/math/huge_test.go b/src/math/huge_test.go
new file mode 100644
index 0000000..9448edc
--- /dev/null
+++ b/src/math/huge_test.go
@@ -0,0 +1,115 @@
+// Copyright 2018 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Disabled for s390x because it uses assembly routines that are not
+// accurate for huge arguments.
+
+// +build !s390x
+
+package math_test
+
+import (
+ . "math"
+ "testing"
+)
+
+// Inputs to test trig_reduce
+var trigHuge = []float64{
+ 1 << 28,
+ 1 << 29,
+ 1 << 30,
+ 1 << 35,
+ 1 << 120,
+ 1 << 240,
+ 1 << 480,
+ 1234567891234567 << 180,
+ 1234567891234567 << 300,
+ MaxFloat64,
+}
+
+// Results for trigHuge[i] calculated with https://github.com/robpike/ivy
+// using 4096 bits of working precision. Values requiring less than
+// 102 decimal digits (1 << 120, 1 << 240, 1 << 480, 1234567891234567 << 180)
+// were confirmed via https://keisan.casio.com/
+var cosHuge = []float64{
+ -0.16556897949057876,
+ -0.94517382606089662,
+ 0.78670712294118812,
+ -0.76466301249635305,
+ -0.92587902285483787,
+ 0.93601042593353793,
+ -0.28282777640193788,
+ -0.14616431394103619,
+ -0.79456058210671406,
+ -0.99998768942655994,
+}
+
+var sinHuge = []float64{
+ -0.98619821183697566,
+ 0.32656766301856334,
+ -0.61732641504604217,
+ -0.64443035102329113,
+ 0.37782010936075202,
+ -0.35197227524865778,
+ 0.95917070894368716,
+ 0.98926032637023618,
+ -0.60718488235646949,
+ 0.00496195478918406,
+}
+
+var tanHuge = []float64{
+ 5.95641897939639421,
+ -0.34551069233430392,
+ -0.78469661331920043,
+ 0.84276385870875983,
+ -0.40806638884180424,
+ -0.37603456702698076,
+ -3.39135965054779932,
+ -6.76813854009065030,
+ 0.76417695016604922,
+ -0.00496201587444489,
+}
+
+// Check that trig values of huge angles return accurate results.
+// This confirms that argument reduction works for very large values
+// up to MaxFloat64.
+func TestHugeCos(t *testing.T) {
+ for i := 0; i < len(trigHuge); i++ {
+ f1 := cosHuge[i]
+ f2 := Cos(trigHuge[i])
+ if !close(f1, f2) {
+ t.Errorf("Cos(%g) = %g, want %g", trigHuge[i], f2, f1)
+ }
+ }
+}
+
+func TestHugeSin(t *testing.T) {
+ for i := 0; i < len(trigHuge); i++ {
+ f1 := sinHuge[i]
+ f2 := Sin(trigHuge[i])
+ if !close(f1, f2) {
+ t.Errorf("Sin(%g) = %g, want %g", trigHuge[i], f2, f1)
+ }
+ }
+}
+
+func TestHugeSinCos(t *testing.T) {
+ for i := 0; i < len(trigHuge); i++ {
+ f1, g1 := sinHuge[i], cosHuge[i]
+ f2, g2 := Sincos(trigHuge[i])
+ if !close(f1, f2) || !close(g1, g2) {
+ t.Errorf("Sincos(%g) = %g, %g, want %g, %g", trigHuge[i], f2, g2, f1, g1)
+ }
+ }
+}
+
+func TestHugeTan(t *testing.T) {
+ for i := 0; i < len(trigHuge); i++ {
+ f1 := tanHuge[i]
+ f2 := Tan(trigHuge[i])
+ if !close(f1, f2) {
+ t.Errorf("Tan(%g) = %g, want %g", trigHuge[i], f2, f1)
+ }
+ }
+}
diff --git a/src/math/hypot.go b/src/math/hypot.go
new file mode 100644
index 0000000..c7f19d4
--- /dev/null
+++ b/src/math/hypot.go
@@ -0,0 +1,38 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Hypot -- sqrt(p*p + q*q), but overflows only if the result does.
+*/
+
+// Hypot returns Sqrt(p*p + q*q), taking care to avoid
+// unnecessary overflow and underflow.
+//
+// Special cases are:
+// Hypot(±Inf, q) = +Inf
+// Hypot(p, ±Inf) = +Inf
+// Hypot(NaN, q) = NaN
+// Hypot(p, NaN) = NaN
+func Hypot(p, q float64) float64
+
+func hypot(p, q float64) float64 {
+ // special cases
+ switch {
+ case IsInf(p, 0) || IsInf(q, 0):
+ return Inf(1)
+ case IsNaN(p) || IsNaN(q):
+ return NaN()
+ }
+ p, q = Abs(p), Abs(q)
+ if p < q {
+ p, q = q, p
+ }
+ if p == 0 {
+ return 0
+ }
+ q = q / p
+ return p * Sqrt(1+q*q)
+}
diff --git a/src/math/hypot_386.s b/src/math/hypot_386.s
new file mode 100644
index 0000000..a89cdf7
--- /dev/null
+++ b/src/math/hypot_386.s
@@ -0,0 +1,59 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Hypot(p, q float64) float64
+TEXT ·Hypot(SB),NOSPLIT,$0
+// test bits for not-finite
+ MOVL p_hi+4(FP), AX // high word p
+ ANDL $0x7ff00000, AX
+ CMPL AX, $0x7ff00000
+ JEQ not_finite
+ MOVL q_hi+12(FP), AX // high word q
+ ANDL $0x7ff00000, AX
+ CMPL AX, $0x7ff00000
+ JEQ not_finite
+ FMOVD p+0(FP), F0 // F0=p
+ FABS // F0=|p|
+ FMOVD q+8(FP), F0 // F0=q, F1=|p|
+ FABS // F0=|q|, F1=|p|
+ FUCOMI F0, F1 // compare F0 to F1
+ JCC 2(PC) // jump if F0 >= F1
+ FXCHD F0, F1 // F0=|p| (larger), F1=|q| (smaller)
+ FTST // compare F0 to 0
+ FSTSW AX
+ ANDW $0x4000, AX
+ JNE 10(PC) // jump if F0 = 0
+ FXCHD F0, F1 // F0=q (smaller), F1=p (larger)
+ FDIVD F1, F0 // F0=q(=q/p), F1=p
+ FMULD F0, F0 // F0=q*q, F1=p
+ FLD1 // F0=1, F1=q*q, F2=p
+ FADDDP F0, F1 // F0=1+q*q, F1=p
+ FSQRT // F0=sqrt(1+q*q), F1=p
+ FMULDP F0, F1 // F0=p*sqrt(1+q*q)
+ FMOVDP F0, ret+16(FP)
+ RET
+ FMOVDP F0, F1 // F0=0
+ FMOVDP F0, ret+16(FP)
+ RET
+not_finite:
+// test bits for -Inf or +Inf
+ MOVL p_hi+4(FP), AX // high word p
+ ORL p_lo+0(FP), AX // low word p
+ ANDL $0x7fffffff, AX
+ CMPL AX, $0x7ff00000
+ JEQ is_inf
+ MOVL q_hi+12(FP), AX // high word q
+ ORL q_lo+8(FP), AX // low word q
+ ANDL $0x7fffffff, AX
+ CMPL AX, $0x7ff00000
+ JEQ is_inf
+ MOVL $0x7ff80000, ret_hi+20(FP) // return NaN = 0x7FF8000000000001
+ MOVL $0x00000001, ret_lo+16(FP)
+ RET
+is_inf:
+ MOVL AX, ret_hi+20(FP) // return +Inf = 0x7FF0000000000000
+ MOVL $0x00000000, ret_lo+16(FP)
+ RET
diff --git a/src/math/hypot_amd64.s b/src/math/hypot_amd64.s
new file mode 100644
index 0000000..d7983a7
--- /dev/null
+++ b/src/math/hypot_amd64.s
@@ -0,0 +1,52 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define PosInf 0x7FF0000000000000
+#define NaN 0x7FF8000000000001
+
+// func Hypot(p, q float64) float64
+TEXT ·Hypot(SB),NOSPLIT,$0
+ // test bits for special cases
+ MOVQ p+0(FP), BX
+ MOVQ $~(1<<63), AX
+ ANDQ AX, BX // p = |p|
+ MOVQ q+8(FP), CX
+ ANDQ AX, CX // q = |q|
+ MOVQ $PosInf, AX
+ CMPQ AX, BX
+ JLE isInfOrNaN
+ CMPQ AX, CX
+ JLE isInfOrNaN
+ // hypot = max * sqrt(1 + (min/max)**2)
+ MOVQ BX, X0
+ MOVQ CX, X1
+ ORQ CX, BX
+ JEQ isZero
+ MOVAPD X0, X2
+ MAXSD X1, X0
+ MINSD X2, X1
+ DIVSD X0, X1
+ MULSD X1, X1
+ ADDSD $1.0, X1
+ SQRTSD X1, X1
+ MULSD X1, X0
+ MOVSD X0, ret+16(FP)
+ RET
+isInfOrNaN:
+ CMPQ AX, BX
+ JEQ isInf
+ CMPQ AX, CX
+ JEQ isInf
+ MOVQ $NaN, AX
+ MOVQ AX, ret+16(FP) // return NaN
+ RET
+isInf:
+ MOVQ AX, ret+16(FP) // return +Inf
+ RET
+isZero:
+ MOVQ $0, AX
+ MOVQ AX, ret+16(FP) // return 0
+ RET
diff --git a/src/math/j0.go b/src/math/j0.go
new file mode 100644
index 0000000..cb5f07b
--- /dev/null
+++ b/src/math/j0.go
@@ -0,0 +1,427 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Bessel function of the first and second kinds of order zero.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_j0.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_j0(x), __ieee754_y0(x)
+// Bessel function of the first and second kinds of order zero.
+// Method -- j0(x):
+// 1. For tiny x, we use j0(x) = 1 - x**2/4 + x**4/64 - ...
+// 2. Reduce x to |x| since j0(x)=j0(-x), and
+// for x in (0,2)
+// j0(x) = 1-z/4+ z**2*R0/S0, where z = x*x;
+// (precision: |j0-1+z/4-z**2R0/S0 |<2**-63.67 )
+// for x in (2,inf)
+// j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
+// where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
+// as follow:
+// cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
+// = 1/sqrt(2) * (cos(x) + sin(x))
+// sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
+// = 1/sqrt(2) * (sin(x) - cos(x))
+// (To avoid cancellation, use
+// sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+// to compute the worse one.)
+//
+// 3 Special cases
+// j0(nan)= nan
+// j0(0) = 1
+// j0(inf) = 0
+//
+// Method -- y0(x):
+// 1. For x<2.
+// Since
+// y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x**2/4 - ...)
+// therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
+// We use the following function to approximate y0,
+// y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x**2
+// where
+// U(z) = u00 + u01*z + ... + u06*z**6
+// V(z) = 1 + v01*z + ... + v04*z**4
+// with absolute approximation error bounded by 2**-72.
+// Note: For tiny x, U/V = u0 and j0(x)~1, hence
+// y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
+// 2. For x>=2.
+// y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
+// where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
+// by the method mentioned above.
+// 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
+//
+
+// J0 returns the order-zero Bessel function of the first kind.
+//
+// Special cases are:
+// J0(±Inf) = 0
+// J0(0) = 1
+// J0(NaN) = NaN
+func J0(x float64) float64 {
+ const (
+ Huge = 1e300
+ TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
+ TwoM13 = 1.0 / (1 << 13) // 2**-13 0x3f20000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ // R0/S0 on [0, 2]
+ R02 = 1.56249999999999947958e-02 // 0x3F8FFFFFFFFFFFFD
+ R03 = -1.89979294238854721751e-04 // 0xBF28E6A5B61AC6E9
+ R04 = 1.82954049532700665670e-06 // 0x3EBEB1D10C503919
+ R05 = -4.61832688532103189199e-09 // 0xBE33D5E773D63FCE
+ S01 = 1.56191029464890010492e-02 // 0x3F8FFCE882C8C2A4
+ S02 = 1.16926784663337450260e-04 // 0x3F1EA6D2DD57DBF4
+ S03 = 5.13546550207318111446e-07 // 0x3EA13B54CE84D5A9
+ S04 = 1.16614003333790000205e-09 // 0x3E1408BCF4745D8F
+ )
+ // special cases
+ switch {
+ case IsNaN(x):
+ return x
+ case IsInf(x, 0):
+ return 0
+ case x == 0:
+ return 1
+ }
+
+ x = Abs(x)
+ if x >= 2 {
+ s, c := Sincos(x)
+ ss := s - c
+ cc := s + c
+
+ // make sure x+x does not overflow
+ if x < MaxFloat64/2 {
+ z := -Cos(x + x)
+ if s*c < 0 {
+ cc = z / ss
+ } else {
+ ss = z / cc
+ }
+ }
+
+ // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
+ // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
+
+ var z float64
+ if x > Two129 { // |x| > ~6.8056e+38
+ z = (1 / SqrtPi) * cc / Sqrt(x)
+ } else {
+ u := pzero(x)
+ v := qzero(x)
+ z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
+ }
+ return z // |x| >= 2.0
+ }
+ if x < TwoM13 { // |x| < ~1.2207e-4
+ if x < TwoM27 {
+ return 1 // |x| < ~7.4506e-9
+ }
+ return 1 - 0.25*x*x // ~7.4506e-9 < |x| < ~1.2207e-4
+ }
+ z := x * x
+ r := z * (R02 + z*(R03+z*(R04+z*R05)))
+ s := 1 + z*(S01+z*(S02+z*(S03+z*S04)))
+ if x < 1 {
+ return 1 + z*(-0.25+(r/s)) // |x| < 1.00
+ }
+ u := 0.5 * x
+ return (1+u)*(1-u) + z*(r/s) // 1.0 < |x| < 2.0
+}
+
+// Y0 returns the order-zero Bessel function of the second kind.
+//
+// Special cases are:
+// Y0(+Inf) = 0
+// Y0(0) = -Inf
+// Y0(x < 0) = NaN
+// Y0(NaN) = NaN
+func Y0(x float64) float64 {
+ const (
+ TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ U00 = -7.38042951086872317523e-02 // 0xBFB2E4D699CBD01F
+ U01 = 1.76666452509181115538e-01 // 0x3FC69D019DE9E3FC
+ U02 = -1.38185671945596898896e-02 // 0xBF8C4CE8B16CFA97
+ U03 = 3.47453432093683650238e-04 // 0x3F36C54D20B29B6B
+ U04 = -3.81407053724364161125e-06 // 0xBECFFEA773D25CAD
+ U05 = 1.95590137035022920206e-08 // 0x3E5500573B4EABD4
+ U06 = -3.98205194132103398453e-11 // 0xBDC5E43D693FB3C8
+ V01 = 1.27304834834123699328e-02 // 0x3F8A127091C9C71A
+ V02 = 7.60068627350353253702e-05 // 0x3F13ECBBF578C6C1
+ V03 = 2.59150851840457805467e-07 // 0x3E91642D7FF202FD
+ V04 = 4.41110311332675467403e-10 // 0x3DFE50183BD6D9EF
+ )
+ // special cases
+ switch {
+ case x < 0 || IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return 0
+ case x == 0:
+ return Inf(-1)
+ }
+
+ if x >= 2 { // |x| >= 2.0
+
+ // y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
+ // where x0 = x-pi/4
+ // Better formula:
+ // cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
+ // = 1/sqrt(2) * (sin(x) + cos(x))
+ // sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
+ // = 1/sqrt(2) * (sin(x) - cos(x))
+ // To avoid cancellation, use
+ // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+ // to compute the worse one.
+
+ s, c := Sincos(x)
+ ss := s - c
+ cc := s + c
+
+ // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
+ // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
+
+ // make sure x+x does not overflow
+ if x < MaxFloat64/2 {
+ z := -Cos(x + x)
+ if s*c < 0 {
+ cc = z / ss
+ } else {
+ ss = z / cc
+ }
+ }
+ var z float64
+ if x > Two129 { // |x| > ~6.8056e+38
+ z = (1 / SqrtPi) * ss / Sqrt(x)
+ } else {
+ u := pzero(x)
+ v := qzero(x)
+ z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
+ }
+ return z // |x| >= 2.0
+ }
+ if x <= TwoM27 {
+ return U00 + (2/Pi)*Log(x) // |x| < ~7.4506e-9
+ }
+ z := x * x
+ u := U00 + z*(U01+z*(U02+z*(U03+z*(U04+z*(U05+z*U06)))))
+ v := 1 + z*(V01+z*(V02+z*(V03+z*V04)))
+ return u/v + (2/Pi)*J0(x)*Log(x) // ~7.4506e-9 < |x| < 2.0
+}
+
+// The asymptotic expansions of pzero is
+// 1 - 9/128 s**2 + 11025/98304 s**4 - ..., where s = 1/x.
+// For x >= 2, We approximate pzero by
+// pzero(x) = 1 + (R/S)
+// where R = pR0 + pR1*s**2 + pR2*s**4 + ... + pR5*s**10
+// S = 1 + pS0*s**2 + ... + pS4*s**10
+// and
+// | pzero(x)-1-R/S | <= 2 ** ( -60.26)
+
+// for x in [inf, 8]=1/[0,0.125]
+var p0R8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ -7.03124999999900357484e-02, // 0xBFB1FFFFFFFFFD32
+ -8.08167041275349795626e+00, // 0xC02029D0B44FA779
+ -2.57063105679704847262e+02, // 0xC07011027B19E863
+ -2.48521641009428822144e+03, // 0xC0A36A6ECD4DCAFC
+ -5.25304380490729545272e+03, // 0xC0B4850B36CC643D
+}
+var p0S8 = [5]float64{
+ 1.16534364619668181717e+02, // 0x405D223307A96751
+ 3.83374475364121826715e+03, // 0x40ADF37D50596938
+ 4.05978572648472545552e+04, // 0x40E3D2BB6EB6B05F
+ 1.16752972564375915681e+05, // 0x40FC810F8F9FA9BD
+ 4.76277284146730962675e+04, // 0x40E741774F2C49DC
+}
+
+// for x in [8,4.5454]=1/[0.125,0.22001]
+var p0R5 = [6]float64{
+ -1.14125464691894502584e-11, // 0xBDA918B147E495CC
+ -7.03124940873599280078e-02, // 0xBFB1FFFFE69AFBC6
+ -4.15961064470587782438e+00, // 0xC010A370F90C6BBF
+ -6.76747652265167261021e+01, // 0xC050EB2F5A7D1783
+ -3.31231299649172967747e+02, // 0xC074B3B36742CC63
+ -3.46433388365604912451e+02, // 0xC075A6EF28A38BD7
+}
+var p0S5 = [5]float64{
+ 6.07539382692300335975e+01, // 0x404E60810C98C5DE
+ 1.05125230595704579173e+03, // 0x40906D025C7E2864
+ 5.97897094333855784498e+03, // 0x40B75AF88FBE1D60
+ 9.62544514357774460223e+03, // 0x40C2CCB8FA76FA38
+ 2.40605815922939109441e+03, // 0x40A2CC1DC70BE864
+}
+
+// for x in [4.547,2.8571]=1/[0.2199,0.35001]
+var p0R3 = [6]float64{
+ -2.54704601771951915620e-09, // 0xBE25E1036FE1AA86
+ -7.03119616381481654654e-02, // 0xBFB1FFF6F7C0E24B
+ -2.40903221549529611423e+00, // 0xC00345B2AEA48074
+ -2.19659774734883086467e+01, // 0xC035F74A4CB94E14
+ -5.80791704701737572236e+01, // 0xC04D0A22420A1A45
+ -3.14479470594888503854e+01, // 0xC03F72ACA892D80F
+}
+var p0S3 = [5]float64{
+ 3.58560338055209726349e+01, // 0x4041ED9284077DD3
+ 3.61513983050303863820e+02, // 0x40769839464A7C0E
+ 1.19360783792111533330e+03, // 0x4092A66E6D1061D6
+ 1.12799679856907414432e+03, // 0x40919FFCB8C39B7E
+ 1.73580930813335754692e+02, // 0x4065B296FC379081
+}
+
+// for x in [2.8570,2]=1/[0.3499,0.5]
+var p0R2 = [6]float64{
+ -8.87534333032526411254e-08, // 0xBE77D316E927026D
+ -7.03030995483624743247e-02, // 0xBFB1FF62495E1E42
+ -1.45073846780952986357e+00, // 0xBFF736398A24A843
+ -7.63569613823527770791e+00, // 0xC01E8AF3EDAFA7F3
+ -1.11931668860356747786e+01, // 0xC02662E6C5246303
+ -3.23364579351335335033e+00, // 0xC009DE81AF8FE70F
+}
+var p0S2 = [5]float64{
+ 2.22202997532088808441e+01, // 0x40363865908B5959
+ 1.36206794218215208048e+02, // 0x4061069E0EE8878F
+ 2.70470278658083486789e+02, // 0x4070E78642EA079B
+ 1.53875394208320329881e+02, // 0x40633C033AB6FAFF
+ 1.46576176948256193810e+01, // 0x402D50B344391809
+}
+
+func pzero(x float64) float64 {
+ var p *[6]float64
+ var q *[5]float64
+ if x >= 8 {
+ p = &p0R8
+ q = &p0S8
+ } else if x >= 4.5454 {
+ p = &p0R5
+ q = &p0S5
+ } else if x >= 2.8571 {
+ p = &p0R3
+ q = &p0S3
+ } else if x >= 2 {
+ p = &p0R2
+ q = &p0S2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
+ return 1 + r/s
+}
+
+// For x >= 8, the asymptotic expansions of qzero is
+// -1/8 s + 75/1024 s**3 - ..., where s = 1/x.
+// We approximate pzero by
+// qzero(x) = s*(-1.25 + (R/S))
+// where R = qR0 + qR1*s**2 + qR2*s**4 + ... + qR5*s**10
+// S = 1 + qS0*s**2 + ... + qS5*s**12
+// and
+// | qzero(x)/s +1.25-R/S | <= 2**(-61.22)
+
+// for x in [inf, 8]=1/[0,0.125]
+var q0R8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ 7.32421874999935051953e-02, // 0x3FB2BFFFFFFFFE2C
+ 1.17682064682252693899e+01, // 0x402789525BB334D6
+ 5.57673380256401856059e+02, // 0x40816D6315301825
+ 8.85919720756468632317e+03, // 0x40C14D993E18F46D
+ 3.70146267776887834771e+04, // 0x40E212D40E901566
+}
+var q0S8 = [6]float64{
+ 1.63776026895689824414e+02, // 0x406478D5365B39BC
+ 8.09834494656449805916e+03, // 0x40BFA2584E6B0563
+ 1.42538291419120476348e+05, // 0x4101665254D38C3F
+ 8.03309257119514397345e+05, // 0x412883DA83A52B43
+ 8.40501579819060512818e+05, // 0x4129A66B28DE0B3D
+ -3.43899293537866615225e+05, // 0xC114FD6D2C9530C5
+}
+
+// for x in [8,4.5454]=1/[0.125,0.22001]
+var q0R5 = [6]float64{
+ 1.84085963594515531381e-11, // 0x3DB43D8F29CC8CD9
+ 7.32421766612684765896e-02, // 0x3FB2BFFFD172B04C
+ 5.83563508962056953777e+00, // 0x401757B0B9953DD3
+ 1.35111577286449829671e+02, // 0x4060E3920A8788E9
+ 1.02724376596164097464e+03, // 0x40900CF99DC8C481
+ 1.98997785864605384631e+03, // 0x409F17E953C6E3A6
+}
+var q0S5 = [6]float64{
+ 8.27766102236537761883e+01, // 0x4054B1B3FB5E1543
+ 2.07781416421392987104e+03, // 0x40A03BA0DA21C0CE
+ 1.88472887785718085070e+04, // 0x40D267D27B591E6D
+ 5.67511122894947329769e+04, // 0x40EBB5E397E02372
+ 3.59767538425114471465e+04, // 0x40E191181F7A54A0
+ -5.35434275601944773371e+03, // 0xC0B4EA57BEDBC609
+}
+
+// for x in [4.547,2.8571]=1/[0.2199,0.35001]
+var q0R3 = [6]float64{
+ 4.37741014089738620906e-09, // 0x3E32CD036ADECB82
+ 7.32411180042911447163e-02, // 0x3FB2BFEE0E8D0842
+ 3.34423137516170720929e+00, // 0x400AC0FC61149CF5
+ 4.26218440745412650017e+01, // 0x40454F98962DAEDD
+ 1.70808091340565596283e+02, // 0x406559DBE25EFD1F
+ 1.66733948696651168575e+02, // 0x4064D77C81FA21E0
+}
+var q0S3 = [6]float64{
+ 4.87588729724587182091e+01, // 0x40486122BFE343A6
+ 7.09689221056606015736e+02, // 0x40862D8386544EB3
+ 3.70414822620111362994e+03, // 0x40ACF04BE44DFC63
+ 6.46042516752568917582e+03, // 0x40B93C6CD7C76A28
+ 2.51633368920368957333e+03, // 0x40A3A8AAD94FB1C0
+ -1.49247451836156386662e+02, // 0xC062A7EB201CF40F
+}
+
+// for x in [2.8570,2]=1/[0.3499,0.5]
+var q0R2 = [6]float64{
+ 1.50444444886983272379e-07, // 0x3E84313B54F76BDB
+ 7.32234265963079278272e-02, // 0x3FB2BEC53E883E34
+ 1.99819174093815998816e+00, // 0x3FFFF897E727779C
+ 1.44956029347885735348e+01, // 0x402CFDBFAAF96FE5
+ 3.16662317504781540833e+01, // 0x403FAA8E29FBDC4A
+ 1.62527075710929267416e+01, // 0x403040B171814BB4
+}
+var q0S2 = [6]float64{
+ 3.03655848355219184498e+01, // 0x403E5D96F7C07AED
+ 2.69348118608049844624e+02, // 0x4070D591E4D14B40
+ 8.44783757595320139444e+02, // 0x408A664522B3BF22
+ 8.82935845112488550512e+02, // 0x408B977C9C5CC214
+ 2.12666388511798828631e+02, // 0x406A95530E001365
+ -5.31095493882666946917e+00, // 0xC0153E6AF8B32931
+}
+
+func qzero(x float64) float64 {
+ var p, q *[6]float64
+ if x >= 8 {
+ p = &q0R8
+ q = &q0S8
+ } else if x >= 4.5454 {
+ p = &q0R5
+ q = &q0S5
+ } else if x >= 2.8571 {
+ p = &q0R3
+ q = &q0S3
+ } else if x >= 2 {
+ p = &q0R2
+ q = &q0S2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
+ return (-0.125 + r/s) / x
+}
diff --git a/src/math/j1.go b/src/math/j1.go
new file mode 100644
index 0000000..7c7d279
--- /dev/null
+++ b/src/math/j1.go
@@ -0,0 +1,422 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Bessel function of the first and second kinds of order one.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_j1.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_j1(x), __ieee754_y1(x)
+// Bessel function of the first and second kinds of order one.
+// Method -- j1(x):
+// 1. For tiny x, we use j1(x) = x/2 - x**3/16 + x**5/384 - ...
+// 2. Reduce x to |x| since j1(x)=-j1(-x), and
+// for x in (0,2)
+// j1(x) = x/2 + x*z*R0/S0, where z = x*x;
+// (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
+// for x in (2,inf)
+// j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
+// y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
+// where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
+// as follow:
+// cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
+// = 1/sqrt(2) * (sin(x) - cos(x))
+// sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
+// = -1/sqrt(2) * (sin(x) + cos(x))
+// (To avoid cancellation, use
+// sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+// to compute the worse one.)
+//
+// 3 Special cases
+// j1(nan)= nan
+// j1(0) = 0
+// j1(inf) = 0
+//
+// Method -- y1(x):
+// 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
+// 2. For x<2.
+// Since
+// y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x**3-...)
+// therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
+// We use the following function to approximate y1,
+// y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x**2
+// where for x in [0,2] (abs err less than 2**-65.89)
+// U(z) = U0[0] + U0[1]*z + ... + U0[4]*z**4
+// V(z) = 1 + v0[0]*z + ... + v0[4]*z**5
+// Note: For tiny x, 1/x dominate y1 and hence
+// y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
+// 3. For x>=2.
+// y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
+// where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
+// by method mentioned above.
+
+// J1 returns the order-one Bessel function of the first kind.
+//
+// Special cases are:
+// J1(±Inf) = 0
+// J1(NaN) = NaN
+func J1(x float64) float64 {
+ const (
+ TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ // R0/S0 on [0, 2]
+ R00 = -6.25000000000000000000e-02 // 0xBFB0000000000000
+ R01 = 1.40705666955189706048e-03 // 0x3F570D9F98472C61
+ R02 = -1.59955631084035597520e-05 // 0xBEF0C5C6BA169668
+ R03 = 4.96727999609584448412e-08 // 0x3E6AAAFA46CA0BD9
+ S01 = 1.91537599538363460805e-02 // 0x3F939D0B12637E53
+ S02 = 1.85946785588630915560e-04 // 0x3F285F56B9CDF664
+ S03 = 1.17718464042623683263e-06 // 0x3EB3BFF8333F8498
+ S04 = 5.04636257076217042715e-09 // 0x3E35AC88C97DFF2C
+ S05 = 1.23542274426137913908e-11 // 0x3DAB2ACFCFB97ED8
+ )
+ // special cases
+ switch {
+ case IsNaN(x):
+ return x
+ case IsInf(x, 0) || x == 0:
+ return 0
+ }
+
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ if x >= 2 {
+ s, c := Sincos(x)
+ ss := -s - c
+ cc := s - c
+
+ // make sure x+x does not overflow
+ if x < MaxFloat64/2 {
+ z := Cos(x + x)
+ if s*c > 0 {
+ cc = z / ss
+ } else {
+ ss = z / cc
+ }
+ }
+
+ // j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
+ // y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
+
+ var z float64
+ if x > Two129 {
+ z = (1 / SqrtPi) * cc / Sqrt(x)
+ } else {
+ u := pone(x)
+ v := qone(x)
+ z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
+ }
+ if sign {
+ return -z
+ }
+ return z
+ }
+ if x < TwoM27 { // |x|<2**-27
+ return 0.5 * x // inexact if x!=0 necessary
+ }
+ z := x * x
+ r := z * (R00 + z*(R01+z*(R02+z*R03)))
+ s := 1.0 + z*(S01+z*(S02+z*(S03+z*(S04+z*S05))))
+ r *= x
+ z = 0.5*x + r/s
+ if sign {
+ return -z
+ }
+ return z
+}
+
+// Y1 returns the order-one Bessel function of the second kind.
+//
+// Special cases are:
+// Y1(+Inf) = 0
+// Y1(0) = -Inf
+// Y1(x < 0) = NaN
+// Y1(NaN) = NaN
+func Y1(x float64) float64 {
+ const (
+ TwoM54 = 1.0 / (1 << 54) // 2**-54 0x3c90000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ U00 = -1.96057090646238940668e-01 // 0xBFC91866143CBC8A
+ U01 = 5.04438716639811282616e-02 // 0x3FA9D3C776292CD1
+ U02 = -1.91256895875763547298e-03 // 0xBF5F55E54844F50F
+ U03 = 2.35252600561610495928e-05 // 0x3EF8AB038FA6B88E
+ U04 = -9.19099158039878874504e-08 // 0xBE78AC00569105B8
+ V00 = 1.99167318236649903973e-02 // 0x3F94650D3F4DA9F0
+ V01 = 2.02552581025135171496e-04 // 0x3F2A8C896C257764
+ V02 = 1.35608801097516229404e-06 // 0x3EB6C05A894E8CA6
+ V03 = 6.22741452364621501295e-09 // 0x3E3ABF1D5BA69A86
+ V04 = 1.66559246207992079114e-11 // 0x3DB25039DACA772A
+ )
+ // special cases
+ switch {
+ case x < 0 || IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return 0
+ case x == 0:
+ return Inf(-1)
+ }
+
+ if x >= 2 {
+ s, c := Sincos(x)
+ ss := -s - c
+ cc := s - c
+
+ // make sure x+x does not overflow
+ if x < MaxFloat64/2 {
+ z := Cos(x + x)
+ if s*c > 0 {
+ cc = z / ss
+ } else {
+ ss = z / cc
+ }
+ }
+ // y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
+ // where x0 = x-3pi/4
+ // Better formula:
+ // cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
+ // = 1/sqrt(2) * (sin(x) - cos(x))
+ // sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
+ // = -1/sqrt(2) * (cos(x) + sin(x))
+ // To avoid cancellation, use
+ // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+ // to compute the worse one.
+
+ var z float64
+ if x > Two129 {
+ z = (1 / SqrtPi) * ss / Sqrt(x)
+ } else {
+ u := pone(x)
+ v := qone(x)
+ z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
+ }
+ return z
+ }
+ if x <= TwoM54 { // x < 2**-54
+ return -(2 / Pi) / x
+ }
+ z := x * x
+ u := U00 + z*(U01+z*(U02+z*(U03+z*U04)))
+ v := 1 + z*(V00+z*(V01+z*(V02+z*(V03+z*V04))))
+ return x*(u/v) + (2/Pi)*(J1(x)*Log(x)-1/x)
+}
+
+// For x >= 8, the asymptotic expansions of pone is
+// 1 + 15/128 s**2 - 4725/2**15 s**4 - ..., where s = 1/x.
+// We approximate pone by
+// pone(x) = 1 + (R/S)
+// where R = pr0 + pr1*s**2 + pr2*s**4 + ... + pr5*s**10
+// S = 1 + ps0*s**2 + ... + ps4*s**10
+// and
+// | pone(x)-1-R/S | <= 2**(-60.06)
+
+// for x in [inf, 8]=1/[0,0.125]
+var p1R8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ 1.17187499999988647970e-01, // 0x3FBDFFFFFFFFFCCE
+ 1.32394806593073575129e+01, // 0x402A7A9D357F7FCE
+ 4.12051854307378562225e+02, // 0x4079C0D4652EA590
+ 3.87474538913960532227e+03, // 0x40AE457DA3A532CC
+ 7.91447954031891731574e+03, // 0x40BEEA7AC32782DD
+}
+var p1S8 = [5]float64{
+ 1.14207370375678408436e+02, // 0x405C8D458E656CAC
+ 3.65093083420853463394e+03, // 0x40AC85DC964D274F
+ 3.69562060269033463555e+04, // 0x40E20B8697C5BB7F
+ 9.76027935934950801311e+04, // 0x40F7D42CB28F17BB
+ 3.08042720627888811578e+04, // 0x40DE1511697A0B2D
+}
+
+// for x in [8,4.5454] = 1/[0.125,0.22001]
+var p1R5 = [6]float64{
+ 1.31990519556243522749e-11, // 0x3DAD0667DAE1CA7D
+ 1.17187493190614097638e-01, // 0x3FBDFFFFE2C10043
+ 6.80275127868432871736e+00, // 0x401B36046E6315E3
+ 1.08308182990189109773e+02, // 0x405B13B9452602ED
+ 5.17636139533199752805e+02, // 0x40802D16D052D649
+ 5.28715201363337541807e+02, // 0x408085B8BB7E0CB7
+}
+var p1S5 = [5]float64{
+ 5.92805987221131331921e+01, // 0x404DA3EAA8AF633D
+ 9.91401418733614377743e+02, // 0x408EFB361B066701
+ 5.35326695291487976647e+03, // 0x40B4E9445706B6FB
+ 7.84469031749551231769e+03, // 0x40BEA4B0B8A5BB15
+ 1.50404688810361062679e+03, // 0x40978030036F5E51
+}
+
+// for x in[4.5453,2.8571] = 1/[0.2199,0.35001]
+var p1R3 = [6]float64{
+ 3.02503916137373618024e-09, // 0x3E29FC21A7AD9EDD
+ 1.17186865567253592491e-01, // 0x3FBDFFF55B21D17B
+ 3.93297750033315640650e+00, // 0x400F76BCE85EAD8A
+ 3.51194035591636932736e+01, // 0x40418F489DA6D129
+ 9.10550110750781271918e+01, // 0x4056C3854D2C1837
+ 4.85590685197364919645e+01, // 0x4048478F8EA83EE5
+}
+var p1S3 = [5]float64{
+ 3.47913095001251519989e+01, // 0x40416549A134069C
+ 3.36762458747825746741e+02, // 0x40750C3307F1A75F
+ 1.04687139975775130551e+03, // 0x40905B7C5037D523
+ 8.90811346398256432622e+02, // 0x408BD67DA32E31E9
+ 1.03787932439639277504e+02, // 0x4059F26D7C2EED53
+}
+
+// for x in [2.8570,2] = 1/[0.3499,0.5]
+var p1R2 = [6]float64{
+ 1.07710830106873743082e-07, // 0x3E7CE9D4F65544F4
+ 1.17176219462683348094e-01, // 0x3FBDFF42BE760D83
+ 2.36851496667608785174e+00, // 0x4002F2B7F98FAEC0
+ 1.22426109148261232917e+01, // 0x40287C377F71A964
+ 1.76939711271687727390e+01, // 0x4031B1A8177F8EE2
+ 5.07352312588818499250e+00, // 0x40144B49A574C1FE
+}
+var p1S2 = [5]float64{
+ 2.14364859363821409488e+01, // 0x40356FBD8AD5ECDC
+ 1.25290227168402751090e+02, // 0x405F529314F92CD5
+ 2.32276469057162813669e+02, // 0x406D08D8D5A2DBD9
+ 1.17679373287147100768e+02, // 0x405D6B7ADA1884A9
+ 8.36463893371618283368e+00, // 0x4020BAB1F44E5192
+}
+
+func pone(x float64) float64 {
+ var p *[6]float64
+ var q *[5]float64
+ if x >= 8 {
+ p = &p1R8
+ q = &p1S8
+ } else if x >= 4.5454 {
+ p = &p1R5
+ q = &p1S5
+ } else if x >= 2.8571 {
+ p = &p1R3
+ q = &p1S3
+ } else if x >= 2 {
+ p = &p1R2
+ q = &p1S2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1.0 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
+ return 1 + r/s
+}
+
+// For x >= 8, the asymptotic expansions of qone is
+// 3/8 s - 105/1024 s**3 - ..., where s = 1/x.
+// We approximate qone by
+// qone(x) = s*(0.375 + (R/S))
+// where R = qr1*s**2 + qr2*s**4 + ... + qr5*s**10
+// S = 1 + qs1*s**2 + ... + qs6*s**12
+// and
+// | qone(x)/s -0.375-R/S | <= 2**(-61.13)
+
+// for x in [inf, 8] = 1/[0,0.125]
+var q1R8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ -1.02539062499992714161e-01, // 0xBFBA3FFFFFFFFDF3
+ -1.62717534544589987888e+01, // 0xC0304591A26779F7
+ -7.59601722513950107896e+02, // 0xC087BCD053E4B576
+ -1.18498066702429587167e+04, // 0xC0C724E740F87415
+ -4.84385124285750353010e+04, // 0xC0E7A6D065D09C6A
+}
+var q1S8 = [6]float64{
+ 1.61395369700722909556e+02, // 0x40642CA6DE5BCDE5
+ 7.82538599923348465381e+03, // 0x40BE9162D0D88419
+ 1.33875336287249578163e+05, // 0x4100579AB0B75E98
+ 7.19657723683240939863e+05, // 0x4125F65372869C19
+ 6.66601232617776375264e+05, // 0x412457D27719AD5C
+ -2.94490264303834643215e+05, // 0xC111F9690EA5AA18
+}
+
+// for x in [8,4.5454] = 1/[0.125,0.22001]
+var q1R5 = [6]float64{
+ -2.08979931141764104297e-11, // 0xBDB6FA431AA1A098
+ -1.02539050241375426231e-01, // 0xBFBA3FFFCB597FEF
+ -8.05644828123936029840e+00, // 0xC0201CE6CA03AD4B
+ -1.83669607474888380239e+02, // 0xC066F56D6CA7B9B0
+ -1.37319376065508163265e+03, // 0xC09574C66931734F
+ -2.61244440453215656817e+03, // 0xC0A468E388FDA79D
+}
+var q1S5 = [6]float64{
+ 8.12765501384335777857e+01, // 0x405451B2FF5A11B2
+ 1.99179873460485964642e+03, // 0x409F1F31E77BF839
+ 1.74684851924908907677e+04, // 0x40D10F1F0D64CE29
+ 4.98514270910352279316e+04, // 0x40E8576DAABAD197
+ 2.79480751638918118260e+04, // 0x40DB4B04CF7C364B
+ -4.71918354795128470869e+03, // 0xC0B26F2EFCFFA004
+}
+
+// for x in [4.5454,2.8571] = 1/[0.2199,0.35001] ???
+var q1R3 = [6]float64{
+ -5.07831226461766561369e-09, // 0xBE35CFA9D38FC84F
+ -1.02537829820837089745e-01, // 0xBFBA3FEB51AEED54
+ -4.61011581139473403113e+00, // 0xC01270C23302D9FF
+ -5.78472216562783643212e+01, // 0xC04CEC71C25D16DA
+ -2.28244540737631695038e+02, // 0xC06C87D34718D55F
+ -2.19210128478909325622e+02, // 0xC06B66B95F5C1BF6
+}
+var q1S3 = [6]float64{
+ 4.76651550323729509273e+01, // 0x4047D523CCD367E4
+ 6.73865112676699709482e+02, // 0x40850EEBC031EE3E
+ 3.38015286679526343505e+03, // 0x40AA684E448E7C9A
+ 5.54772909720722782367e+03, // 0x40B5ABBAA61D54A6
+ 1.90311919338810798763e+03, // 0x409DBC7A0DD4DF4B
+ -1.35201191444307340817e+02, // 0xC060E670290A311F
+}
+
+// for x in [2.8570,2] = 1/[0.3499,0.5]
+var q1R2 = [6]float64{
+ -1.78381727510958865572e-07, // 0xBE87F12644C626D2
+ -1.02517042607985553460e-01, // 0xBFBA3E8E9148B010
+ -2.75220568278187460720e+00, // 0xC006048469BB4EDA
+ -1.96636162643703720221e+01, // 0xC033A9E2C168907F
+ -4.23253133372830490089e+01, // 0xC04529A3DE104AAA
+ -2.13719211703704061733e+01, // 0xC0355F3639CF6E52
+}
+var q1S2 = [6]float64{
+ 2.95333629060523854548e+01, // 0x403D888A78AE64FF
+ 2.52981549982190529136e+02, // 0x406F9F68DB821CBA
+ 7.57502834868645436472e+02, // 0x4087AC05CE49A0F7
+ 7.39393205320467245656e+02, // 0x40871B2548D4C029
+ 1.55949003336666123687e+02, // 0x40637E5E3C3ED8D4
+ -4.95949898822628210127e+00, // 0xC013D686E71BE86B
+}
+
+func qone(x float64) float64 {
+ var p, q *[6]float64
+ if x >= 8 {
+ p = &q1R8
+ q = &q1S8
+ } else if x >= 4.5454 {
+ p = &q1R5
+ q = &q1S5
+ } else if x >= 2.8571 {
+ p = &q1R3
+ q = &q1S3
+ } else if x >= 2 {
+ p = &q1R2
+ q = &q1S2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
+ return (0.375 + r/s) / x
+}
diff --git a/src/math/jn.go b/src/math/jn.go
new file mode 100644
index 0000000..b1aca8f
--- /dev/null
+++ b/src/math/jn.go
@@ -0,0 +1,304 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Bessel function of the first and second kinds of order n.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_jn.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_jn(n, x), __ieee754_yn(n, x)
+// floating point Bessel's function of the 1st and 2nd kind
+// of order n
+//
+// Special cases:
+// y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
+// y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
+// Note 2. About jn(n,x), yn(n,x)
+// For n=0, j0(x) is called,
+// for n=1, j1(x) is called,
+// for n<x, forward recursion is used starting
+// from values of j0(x) and j1(x).
+// for n>x, a continued fraction approximation to
+// j(n,x)/j(n-1,x) is evaluated and then backward
+// recursion is used starting from a supposed value
+// for j(n,x). The resulting value of j(0,x) is
+// compared with the actual value to correct the
+// supposed value of j(n,x).
+//
+// yn(n,x) is similar in all respects, except
+// that forward recursion is used for all
+// values of n>1.
+
+// Jn returns the order-n Bessel function of the first kind.
+//
+// Special cases are:
+// Jn(n, ±Inf) = 0
+// Jn(n, NaN) = NaN
+func Jn(n int, x float64) float64 {
+ const (
+ TwoM29 = 1.0 / (1 << 29) // 2**-29 0x3e10000000000000
+ Two302 = 1 << 302 // 2**302 0x52D0000000000000
+ )
+ // special cases
+ switch {
+ case IsNaN(x):
+ return x
+ case IsInf(x, 0):
+ return 0
+ }
+ // J(-n, x) = (-1)**n * J(n, x), J(n, -x) = (-1)**n * J(n, x)
+ // Thus, J(-n, x) = J(n, -x)
+
+ if n == 0 {
+ return J0(x)
+ }
+ if x == 0 {
+ return 0
+ }
+ if n < 0 {
+ n, x = -n, -x
+ }
+ if n == 1 {
+ return J1(x)
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ if n&1 == 1 {
+ sign = true // odd n and negative x
+ }
+ }
+ var b float64
+ if float64(n) <= x {
+ // Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
+ if x >= Two302 { // x > 2**302
+
+ // (x >> n**2)
+ // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Let s=sin(x), c=cos(x),
+ // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
+ //
+ // n sin(xn)*sqt2 cos(xn)*sqt2
+ // ----------------------------------
+ // 0 s-c c+s
+ // 1 -s-c -c+s
+ // 2 -s+c -c-s
+ // 3 s+c c-s
+
+ var temp float64
+ switch s, c := Sincos(x); n & 3 {
+ case 0:
+ temp = c + s
+ case 1:
+ temp = -c + s
+ case 2:
+ temp = -c - s
+ case 3:
+ temp = c - s
+ }
+ b = (1 / SqrtPi) * temp / Sqrt(x)
+ } else {
+ b = J1(x)
+ for i, a := 1, J0(x); i < n; i++ {
+ a, b = b, b*(float64(i+i)/x)-a // avoid underflow
+ }
+ }
+ } else {
+ if x < TwoM29 { // x < 2**-29
+ // x is tiny, return the first Taylor expansion of J(n,x)
+ // J(n,x) = 1/n!*(x/2)**n - ...
+
+ if n > 33 { // underflow
+ b = 0
+ } else {
+ temp := x * 0.5
+ b = temp
+ a := 1.0
+ for i := 2; i <= n; i++ {
+ a *= float64(i) // a = n!
+ b *= temp // b = (x/2)**n
+ }
+ b /= a
+ }
+ } else {
+ // use backward recurrence
+ // x x**2 x**2
+ // J(n,x)/J(n-1,x) = ---- ------ ------ .....
+ // 2n - 2(n+1) - 2(n+2)
+ //
+ // 1 1 1
+ // (for large x) = ---- ------ ------ .....
+ // 2n 2(n+1) 2(n+2)
+ // -- - ------ - ------ -
+ // x x x
+ //
+ // Let w = 2n/x and h=2/x, then the above quotient
+ // is equal to the continued fraction:
+ // 1
+ // = -----------------------
+ // 1
+ // w - -----------------
+ // 1
+ // w+h - ---------
+ // w+2h - ...
+ //
+ // To determine how many terms needed, let
+ // Q(0) = w, Q(1) = w(w+h) - 1,
+ // Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
+ // When Q(k) > 1e4 good for single
+ // When Q(k) > 1e9 good for double
+ // When Q(k) > 1e17 good for quadruple
+
+ // determine k
+ w := float64(n+n) / x
+ h := 2 / x
+ q0 := w
+ z := w + h
+ q1 := w*z - 1
+ k := 1
+ for q1 < 1e9 {
+ k++
+ z += h
+ q0, q1 = q1, z*q1-q0
+ }
+ m := n + n
+ t := 0.0
+ for i := 2 * (n + k); i >= m; i -= 2 {
+ t = 1 / (float64(i)/x - t)
+ }
+ a := t
+ b = 1
+ // estimate log((2/x)**n*n!) = n*log(2/x)+n*ln(n)
+ // Hence, if n*(log(2n/x)) > ...
+ // single 8.8722839355e+01
+ // double 7.09782712893383973096e+02
+ // long double 1.1356523406294143949491931077970765006170e+04
+ // then recurrent value may overflow and the result is
+ // likely underflow to zero
+
+ tmp := float64(n)
+ v := 2 / x
+ tmp = tmp * Log(Abs(v*tmp))
+ if tmp < 7.09782712893383973096e+02 {
+ for i := n - 1; i > 0; i-- {
+ di := float64(i + i)
+ a, b = b, b*di/x-a
+ }
+ } else {
+ for i := n - 1; i > 0; i-- {
+ di := float64(i + i)
+ a, b = b, b*di/x-a
+ // scale b to avoid spurious overflow
+ if b > 1e100 {
+ a /= b
+ t /= b
+ b = 1
+ }
+ }
+ }
+ b = t * J0(x) / b
+ }
+ }
+ if sign {
+ return -b
+ }
+ return b
+}
+
+// Yn returns the order-n Bessel function of the second kind.
+//
+// Special cases are:
+// Yn(n, +Inf) = 0
+// Yn(n ≥ 0, 0) = -Inf
+// Yn(n < 0, 0) = +Inf if n is odd, -Inf if n is even
+// Yn(n, x < 0) = NaN
+// Yn(n, NaN) = NaN
+func Yn(n int, x float64) float64 {
+ const Two302 = 1 << 302 // 2**302 0x52D0000000000000
+ // special cases
+ switch {
+ case x < 0 || IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return 0
+ }
+
+ if n == 0 {
+ return Y0(x)
+ }
+ if x == 0 {
+ if n < 0 && n&1 == 1 {
+ return Inf(1)
+ }
+ return Inf(-1)
+ }
+ sign := false
+ if n < 0 {
+ n = -n
+ if n&1 == 1 {
+ sign = true // sign true if n < 0 && |n| odd
+ }
+ }
+ if n == 1 {
+ if sign {
+ return -Y1(x)
+ }
+ return Y1(x)
+ }
+ var b float64
+ if x >= Two302 { // x > 2**302
+ // (x >> n**2)
+ // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Let s=sin(x), c=cos(x),
+ // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
+ //
+ // n sin(xn)*sqt2 cos(xn)*sqt2
+ // ----------------------------------
+ // 0 s-c c+s
+ // 1 -s-c -c+s
+ // 2 -s+c -c-s
+ // 3 s+c c-s
+
+ var temp float64
+ switch s, c := Sincos(x); n & 3 {
+ case 0:
+ temp = s - c
+ case 1:
+ temp = -s - c
+ case 2:
+ temp = -s + c
+ case 3:
+ temp = s + c
+ }
+ b = (1 / SqrtPi) * temp / Sqrt(x)
+ } else {
+ a := Y0(x)
+ b = Y1(x)
+ // quit if b is -inf
+ for i := 1; i < n && !IsInf(b, -1); i++ {
+ a, b = b, (float64(i+i)/x)*b-a
+ }
+ }
+ if sign {
+ return -b
+ }
+ return b
+}
diff --git a/src/math/ldexp.go b/src/math/ldexp.go
new file mode 100644
index 0000000..aa50a49
--- /dev/null
+++ b/src/math/ldexp.go
@@ -0,0 +1,45 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Ldexp is the inverse of Frexp.
+// It returns frac × 2**exp.
+//
+// Special cases are:
+// Ldexp(±0, exp) = ±0
+// Ldexp(±Inf, exp) = ±Inf
+// Ldexp(NaN, exp) = NaN
+func Ldexp(frac float64, exp int) float64
+
+func ldexp(frac float64, exp int) float64 {
+ // special cases
+ switch {
+ case frac == 0:
+ return frac // correctly return -0
+ case IsInf(frac, 0) || IsNaN(frac):
+ return frac
+ }
+ frac, e := normalize(frac)
+ exp += e
+ x := Float64bits(frac)
+ exp += int(x>>shift)&mask - bias
+ if exp < -1075 {
+ return Copysign(0, frac) // underflow
+ }
+ if exp > 1023 { // overflow
+ if frac < 0 {
+ return Inf(-1)
+ }
+ return Inf(1)
+ }
+ var m float64 = 1
+ if exp < -1022 { // denormal
+ exp += 53
+ m = 1.0 / (1 << 53) // 2**-53
+ }
+ x &^= mask << shift
+ x |= uint64(exp+bias) << shift
+ return m * Float64frombits(x)
+}
diff --git a/src/math/ldexp_386.s b/src/math/ldexp_386.s
new file mode 100644
index 0000000..6f67ae1
--- /dev/null
+++ b/src/math/ldexp_386.s
@@ -0,0 +1,14 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Ldexp(frac float64, exp int) float64
+TEXT ·Ldexp(SB),NOSPLIT,$0
+ FMOVL exp+8(FP), F0 // F0=exp
+ FMOVD frac+0(FP), F0 // F0=frac, F1=e
+ FSCALE // F0=x*2**e, F1=e
+ FMOVDP F0, F1 // F0=x*2**e
+ FMOVDP F0, ret+12(FP)
+ RET
diff --git a/src/math/lgamma.go b/src/math/lgamma.go
new file mode 100644
index 0000000..7af5871
--- /dev/null
+++ b/src/math/lgamma.go
@@ -0,0 +1,365 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point logarithm of the Gamma function.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_lgamma_r.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_lgamma_r(x, signgamp)
+// Reentrant version of the logarithm of the Gamma function
+// with user provided pointer for the sign of Gamma(x).
+//
+// Method:
+// 1. Argument Reduction for 0 < x <= 8
+// Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
+// reduce x to a number in [1.5,2.5] by
+// lgamma(1+s) = log(s) + lgamma(s)
+// for example,
+// lgamma(7.3) = log(6.3) + lgamma(6.3)
+// = log(6.3*5.3) + lgamma(5.3)
+// = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
+// 2. Polynomial approximation of lgamma around its
+// minimum (ymin=1.461632144968362245) to maintain monotonicity.
+// On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
+// Let z = x-ymin;
+// lgamma(x) = -1.214862905358496078218 + z**2*poly(z)
+// poly(z) is a 14 degree polynomial.
+// 2. Rational approximation in the primary interval [2,3]
+// We use the following approximation:
+// s = x-2.0;
+// lgamma(x) = 0.5*s + s*P(s)/Q(s)
+// with accuracy
+// |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
+// Our algorithms are based on the following observation
+//
+// zeta(2)-1 2 zeta(3)-1 3
+// lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
+// 2 3
+//
+// where Euler = 0.5772156649... is the Euler constant, which
+// is very close to 0.5.
+//
+// 3. For x>=8, we have
+// lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
+// (better formula:
+// lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
+// Let z = 1/x, then we approximation
+// f(z) = lgamma(x) - (x-0.5)(log(x)-1)
+// by
+// 3 5 11
+// w = w0 + w1*z + w2*z + w3*z + ... + w6*z
+// where
+// |w - f(z)| < 2**-58.74
+//
+// 4. For negative x, since (G is gamma function)
+// -x*G(-x)*G(x) = pi/sin(pi*x),
+// we have
+// G(x) = pi/(sin(pi*x)*(-x)*G(-x))
+// since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
+// Hence, for x<0, signgam = sign(sin(pi*x)) and
+// lgamma(x) = log(|Gamma(x)|)
+// = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
+// Note: one should avoid computing pi*(-x) directly in the
+// computation of sin(pi*(-x)).
+//
+// 5. Special Cases
+// lgamma(2+s) ~ s*(1-Euler) for tiny s
+// lgamma(1)=lgamma(2)=0
+// lgamma(x) ~ -log(x) for tiny x
+// lgamma(0) = lgamma(inf) = inf
+// lgamma(-integer) = +-inf
+//
+//
+
+var _lgamA = [...]float64{
+ 7.72156649015328655494e-02, // 0x3FB3C467E37DB0C8
+ 3.22467033424113591611e-01, // 0x3FD4A34CC4A60FAD
+ 6.73523010531292681824e-02, // 0x3FB13E001A5562A7
+ 2.05808084325167332806e-02, // 0x3F951322AC92547B
+ 7.38555086081402883957e-03, // 0x3F7E404FB68FEFE8
+ 2.89051383673415629091e-03, // 0x3F67ADD8CCB7926B
+ 1.19270763183362067845e-03, // 0x3F538A94116F3F5D
+ 5.10069792153511336608e-04, // 0x3F40B6C689B99C00
+ 2.20862790713908385557e-04, // 0x3F2CF2ECED10E54D
+ 1.08011567247583939954e-04, // 0x3F1C5088987DFB07
+ 2.52144565451257326939e-05, // 0x3EFA7074428CFA52
+ 4.48640949618915160150e-05, // 0x3F07858E90A45837
+}
+var _lgamR = [...]float64{
+ 1.0, // placeholder
+ 1.39200533467621045958e+00, // 0x3FF645A762C4AB74
+ 7.21935547567138069525e-01, // 0x3FE71A1893D3DCDC
+ 1.71933865632803078993e-01, // 0x3FC601EDCCFBDF27
+ 1.86459191715652901344e-02, // 0x3F9317EA742ED475
+ 7.77942496381893596434e-04, // 0x3F497DDACA41A95B
+ 7.32668430744625636189e-06, // 0x3EDEBAF7A5B38140
+}
+var _lgamS = [...]float64{
+ -7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
+ 2.14982415960608852501e-01, // 0x3FCB848B36E20878
+ 3.25778796408930981787e-01, // 0x3FD4D98F4F139F59
+ 1.46350472652464452805e-01, // 0x3FC2BB9CBEE5F2F7
+ 2.66422703033638609560e-02, // 0x3F9B481C7E939961
+ 1.84028451407337715652e-03, // 0x3F5E26B67368F239
+ 3.19475326584100867617e-05, // 0x3F00BFECDD17E945
+}
+var _lgamT = [...]float64{
+ 4.83836122723810047042e-01, // 0x3FDEF72BC8EE38A2
+ -1.47587722994593911752e-01, // 0xBFC2E4278DC6C509
+ 6.46249402391333854778e-02, // 0x3FB08B4294D5419B
+ -3.27885410759859649565e-02, // 0xBFA0C9A8DF35B713
+ 1.79706750811820387126e-02, // 0x3F9266E7970AF9EC
+ -1.03142241298341437450e-02, // 0xBF851F9FBA91EC6A
+ 6.10053870246291332635e-03, // 0x3F78FCE0E370E344
+ -3.68452016781138256760e-03, // 0xBF6E2EFFB3E914D7
+ 2.25964780900612472250e-03, // 0x3F6282D32E15C915
+ -1.40346469989232843813e-03, // 0xBF56FE8EBF2D1AF1
+ 8.81081882437654011382e-04, // 0x3F4CDF0CEF61A8E9
+ -5.38595305356740546715e-04, // 0xBF41A6109C73E0EC
+ 3.15632070903625950361e-04, // 0x3F34AF6D6C0EBBF7
+ -3.12754168375120860518e-04, // 0xBF347F24ECC38C38
+ 3.35529192635519073543e-04, // 0x3F35FD3EE8C2D3F4
+}
+var _lgamU = [...]float64{
+ -7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
+ 6.32827064025093366517e-01, // 0x3FE4401E8B005DFF
+ 1.45492250137234768737e+00, // 0x3FF7475CD119BD6F
+ 9.77717527963372745603e-01, // 0x3FEF497644EA8450
+ 2.28963728064692451092e-01, // 0x3FCD4EAEF6010924
+ 1.33810918536787660377e-02, // 0x3F8B678BBF2BAB09
+}
+var _lgamV = [...]float64{
+ 1.0,
+ 2.45597793713041134822e+00, // 0x4003A5D7C2BD619C
+ 2.12848976379893395361e+00, // 0x40010725A42B18F5
+ 7.69285150456672783825e-01, // 0x3FE89DFBE45050AF
+ 1.04222645593369134254e-01, // 0x3FBAAE55D6537C88
+ 3.21709242282423911810e-03, // 0x3F6A5ABB57D0CF61
+}
+var _lgamW = [...]float64{
+ 4.18938533204672725052e-01, // 0x3FDACFE390C97D69
+ 8.33333333333329678849e-02, // 0x3FB555555555553B
+ -2.77777777728775536470e-03, // 0xBF66C16C16B02E5C
+ 7.93650558643019558500e-04, // 0x3F4A019F98CF38B6
+ -5.95187557450339963135e-04, // 0xBF4380CB8C0FE741
+ 8.36339918996282139126e-04, // 0x3F4B67BA4CDAD5D1
+ -1.63092934096575273989e-03, // 0xBF5AB89D0B9E43E4
+}
+
+// Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x).
+//
+// Special cases are:
+// Lgamma(+Inf) = +Inf
+// Lgamma(0) = +Inf
+// Lgamma(-integer) = +Inf
+// Lgamma(-Inf) = -Inf
+// Lgamma(NaN) = NaN
+func Lgamma(x float64) (lgamma float64, sign int) {
+ const (
+ Ymin = 1.461632144968362245
+ Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
+ Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
+ Two58 = 1 << 58 // 0x4390000000000000 ~2.8823e+17
+ Tiny = 1.0 / (1 << 70) // 0x3b90000000000000 ~8.47033e-22
+ Tc = 1.46163214496836224576e+00 // 0x3FF762D86356BE3F
+ Tf = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
+ // Tt = -(tail of Tf)
+ Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
+ )
+ // special cases
+ sign = 1
+ switch {
+ case IsNaN(x):
+ lgamma = x
+ return
+ case IsInf(x, 0):
+ lgamma = x
+ return
+ case x == 0:
+ lgamma = Inf(1)
+ return
+ }
+
+ neg := false
+ if x < 0 {
+ x = -x
+ neg = true
+ }
+
+ if x < Tiny { // if |x| < 2**-70, return -log(|x|)
+ if neg {
+ sign = -1
+ }
+ lgamma = -Log(x)
+ return
+ }
+ var nadj float64
+ if neg {
+ if x >= Two52 { // |x| >= 2**52, must be -integer
+ lgamma = Inf(1)
+ return
+ }
+ t := sinPi(x)
+ if t == 0 {
+ lgamma = Inf(1) // -integer
+ return
+ }
+ nadj = Log(Pi / Abs(t*x))
+ if t < 0 {
+ sign = -1
+ }
+ }
+
+ switch {
+ case x == 1 || x == 2: // purge off 1 and 2
+ lgamma = 0
+ return
+ case x < 2: // use lgamma(x) = lgamma(x+1) - log(x)
+ var y float64
+ var i int
+ if x <= 0.9 {
+ lgamma = -Log(x)
+ switch {
+ case x >= (Ymin - 1 + 0.27): // 0.7316 <= x <= 0.9
+ y = 1 - x
+ i = 0
+ case x >= (Ymin - 1 - 0.27): // 0.2316 <= x < 0.7316
+ y = x - (Tc - 1)
+ i = 1
+ default: // 0 < x < 0.2316
+ y = x
+ i = 2
+ }
+ } else {
+ lgamma = 0
+ switch {
+ case x >= (Ymin + 0.27): // 1.7316 <= x < 2
+ y = 2 - x
+ i = 0
+ case x >= (Ymin - 0.27): // 1.2316 <= x < 1.7316
+ y = x - Tc
+ i = 1
+ default: // 0.9 < x < 1.2316
+ y = x - 1
+ i = 2
+ }
+ }
+ switch i {
+ case 0:
+ z := y * y
+ p1 := _lgamA[0] + z*(_lgamA[2]+z*(_lgamA[4]+z*(_lgamA[6]+z*(_lgamA[8]+z*_lgamA[10]))))
+ p2 := z * (_lgamA[1] + z*(+_lgamA[3]+z*(_lgamA[5]+z*(_lgamA[7]+z*(_lgamA[9]+z*_lgamA[11])))))
+ p := y*p1 + p2
+ lgamma += (p - 0.5*y)
+ case 1:
+ z := y * y
+ w := z * y
+ p1 := _lgamT[0] + w*(_lgamT[3]+w*(_lgamT[6]+w*(_lgamT[9]+w*_lgamT[12]))) // parallel comp
+ p2 := _lgamT[1] + w*(_lgamT[4]+w*(_lgamT[7]+w*(_lgamT[10]+w*_lgamT[13])))
+ p3 := _lgamT[2] + w*(_lgamT[5]+w*(_lgamT[8]+w*(_lgamT[11]+w*_lgamT[14])))
+ p := z*p1 - (Tt - w*(p2+y*p3))
+ lgamma += (Tf + p)
+ case 2:
+ p1 := y * (_lgamU[0] + y*(_lgamU[1]+y*(_lgamU[2]+y*(_lgamU[3]+y*(_lgamU[4]+y*_lgamU[5])))))
+ p2 := 1 + y*(_lgamV[1]+y*(_lgamV[2]+y*(_lgamV[3]+y*(_lgamV[4]+y*_lgamV[5]))))
+ lgamma += (-0.5*y + p1/p2)
+ }
+ case x < 8: // 2 <= x < 8
+ i := int(x)
+ y := x - float64(i)
+ p := y * (_lgamS[0] + y*(_lgamS[1]+y*(_lgamS[2]+y*(_lgamS[3]+y*(_lgamS[4]+y*(_lgamS[5]+y*_lgamS[6]))))))
+ q := 1 + y*(_lgamR[1]+y*(_lgamR[2]+y*(_lgamR[3]+y*(_lgamR[4]+y*(_lgamR[5]+y*_lgamR[6])))))
+ lgamma = 0.5*y + p/q
+ z := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s)
+ switch i {
+ case 7:
+ z *= (y + 6)
+ fallthrough
+ case 6:
+ z *= (y + 5)
+ fallthrough
+ case 5:
+ z *= (y + 4)
+ fallthrough
+ case 4:
+ z *= (y + 3)
+ fallthrough
+ case 3:
+ z *= (y + 2)
+ lgamma += Log(z)
+ }
+ case x < Two58: // 8 <= x < 2**58
+ t := Log(x)
+ z := 1 / x
+ y := z * z
+ w := _lgamW[0] + z*(_lgamW[1]+y*(_lgamW[2]+y*(_lgamW[3]+y*(_lgamW[4]+y*(_lgamW[5]+y*_lgamW[6])))))
+ lgamma = (x-0.5)*(t-1) + w
+ default: // 2**58 <= x <= Inf
+ lgamma = x * (Log(x) - 1)
+ }
+ if neg {
+ lgamma = nadj - lgamma
+ }
+ return
+}
+
+// sinPi(x) is a helper function for negative x
+func sinPi(x float64) float64 {
+ const (
+ Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
+ Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
+ )
+ if x < 0.25 {
+ return -Sin(Pi * x)
+ }
+
+ // argument reduction
+ z := Floor(x)
+ var n int
+ if z != x { // inexact
+ x = Mod(x, 2)
+ n = int(x * 4)
+ } else {
+ if x >= Two53 { // x must be even
+ x = 0
+ n = 0
+ } else {
+ if x < Two52 {
+ z = x + Two52 // exact
+ }
+ n = int(1 & Float64bits(z))
+ x = float64(n)
+ n <<= 2
+ }
+ }
+ switch n {
+ case 0:
+ x = Sin(Pi * x)
+ case 1, 2:
+ x = Cos(Pi * (0.5 - x))
+ case 3, 4:
+ x = Sin(Pi * (1 - x))
+ case 5, 6:
+ x = -Cos(Pi * (x - 1.5))
+ default:
+ x = Sin(Pi * (x - 2))
+ }
+ return -x
+}
diff --git a/src/math/log.go b/src/math/log.go
new file mode 100644
index 0000000..e328348
--- /dev/null
+++ b/src/math/log.go
@@ -0,0 +1,123 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point logarithm.
+*/
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
+// and came with this notice. The go code is a simpler
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_log(x)
+// Return the logarithm of x
+//
+// Method :
+// 1. Argument Reduction: find k and f such that
+// x = 2**k * (1+f),
+// where sqrt(2)/2 < 1+f < sqrt(2) .
+//
+// 2. Approximation of log(1+f).
+// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
+// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
+// = 2s + s*R
+// We use a special Reme algorithm on [0,0.1716] to generate
+// a polynomial of degree 14 to approximate R. The maximum error
+// of this polynomial approximation is bounded by 2**-58.45. In
+// other words,
+// 2 4 6 8 10 12 14
+// R(z) ~ L1*s +L2*s +L3*s +L4*s +L5*s +L6*s +L7*s
+// (the values of L1 to L7 are listed in the program) and
+// | 2 14 | -58.45
+// | L1*s +...+L7*s - R(z) | <= 2
+// | |
+// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
+// In order to guarantee error in log below 1ulp, we compute log by
+// log(1+f) = f - s*(f - R) (if f is not too large)
+// log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
+//
+// 3. Finally, log(x) = k*Ln2 + log(1+f).
+// = k*Ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*Ln2_lo)))
+// Here Ln2 is split into two floating point number:
+// Ln2_hi + Ln2_lo,
+// where n*Ln2_hi is always exact for |n| < 2000.
+//
+// Special cases:
+// log(x) is NaN with signal if x < 0 (including -INF) ;
+// log(+INF) is +INF; log(0) is -INF with signal;
+// log(NaN) is that NaN with no signal.
+//
+// Accuracy:
+// according to an error analysis, the error is always less than
+// 1 ulp (unit in the last place).
+//
+// Constants:
+// The hexadecimal values are the intended ones for the following
+// constants. The decimal values may be used, provided that the
+// compiler will convert from decimal to binary accurately enough
+// to produce the hexadecimal values shown.
+
+// Log returns the natural logarithm of x.
+//
+// Special cases are:
+// Log(+Inf) = +Inf
+// Log(0) = -Inf
+// Log(x < 0) = NaN
+// Log(NaN) = NaN
+func Log(x float64) float64
+
+func log(x float64) float64 {
+ const (
+ Ln2Hi = 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
+ Ln2Lo = 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
+ L1 = 6.666666666666735130e-01 /* 3FE55555 55555593 */
+ L2 = 3.999999999940941908e-01 /* 3FD99999 9997FA04 */
+ L3 = 2.857142874366239149e-01 /* 3FD24924 94229359 */
+ L4 = 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
+ L5 = 1.818357216161805012e-01 /* 3FC74664 96CB03DE */
+ L6 = 1.531383769920937332e-01 /* 3FC39A09 D078C69F */
+ L7 = 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */
+ )
+
+ // special cases
+ switch {
+ case IsNaN(x) || IsInf(x, 1):
+ return x
+ case x < 0:
+ return NaN()
+ case x == 0:
+ return Inf(-1)
+ }
+
+ // reduce
+ f1, ki := Frexp(x)
+ if f1 < Sqrt2/2 {
+ f1 *= 2
+ ki--
+ }
+ f := f1 - 1
+ k := float64(ki)
+
+ // compute
+ s := f / (2 + f)
+ s2 := s * s
+ s4 := s2 * s2
+ t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
+ t2 := s4 * (L2 + s4*(L4+s4*L6))
+ R := t1 + t2
+ hfsq := 0.5 * f * f
+ return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f)
+}
diff --git a/src/math/log10.go b/src/math/log10.go
new file mode 100644
index 0000000..ccd079d
--- /dev/null
+++ b/src/math/log10.go
@@ -0,0 +1,27 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Log10 returns the decimal logarithm of x.
+// The special cases are the same as for Log.
+func Log10(x float64) float64
+
+func log10(x float64) float64 {
+ return Log(x) * (1 / Ln10)
+}
+
+// Log2 returns the binary logarithm of x.
+// The special cases are the same as for Log.
+func Log2(x float64) float64
+
+func log2(x float64) float64 {
+ frac, exp := Frexp(x)
+ // Make sure exact powers of two give an exact answer.
+ // Don't depend on Log(0.5)*(1/Ln2)+exp being exactly exp-1.
+ if frac == 0.5 {
+ return float64(exp - 1)
+ }
+ return Log(frac)*(1/Ln2) + float64(exp)
+}
diff --git a/src/math/log10_386.s b/src/math/log10_386.s
new file mode 100644
index 0000000..10eecd9
--- /dev/null
+++ b/src/math/log10_386.s
@@ -0,0 +1,21 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Log10(x float64) float64
+TEXT ·Log10(SB),NOSPLIT,$0
+ FLDLG2 // F0=log10(2)
+ FMOVD x+0(FP), F0 // F0=x, F1=log10(2)
+ FYL2X // F0=log10(x)=log2(x)*log10(2)
+ FMOVDP F0, ret+8(FP)
+ RET
+
+// func Log2(x float64) float64
+TEXT ·Log2(SB),NOSPLIT,$0
+ FLD1 // F0=1
+ FMOVD x+0(FP), F0 // F0=x, F1=1
+ FYL2X // F0=log2(x)
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/log10_s390x.s b/src/math/log10_s390x.s
new file mode 100644
index 0000000..3638afe
--- /dev/null
+++ b/src/math/log10_s390x.s
@@ -0,0 +1,156 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial coefficients and other constants
+DATA log10rodataL19<>+0(SB)/8, $0.000000000000000000E+00
+DATA log10rodataL19<>+8(SB)/8, $-1.0
+DATA log10rodataL19<>+16(SB)/8, $0x7FF8000000000000 //+NanN
+DATA log10rodataL19<>+24(SB)/8, $.15375570329280596749
+DATA log10rodataL19<>+32(SB)/8, $.60171950900703668594E+04
+DATA log10rodataL19<>+40(SB)/8, $-1.9578460454940795898
+DATA log10rodataL19<>+48(SB)/8, $0.78962633073318517310E-01
+DATA log10rodataL19<>+56(SB)/8, $-.71784211884836937993E-02
+DATA log10rodataL19<>+64(SB)/8, $0.87011165920689940661E-03
+DATA log10rodataL19<>+72(SB)/8, $-.11865158981621437541E-03
+DATA log10rodataL19<>+80(SB)/8, $0.17258413403018680410E-04
+DATA log10rodataL19<>+88(SB)/8, $0.40752932047883484315E-06
+DATA log10rodataL19<>+96(SB)/8, $-.26149194688832680410E-05
+DATA log10rodataL19<>+104(SB)/8, $0.92453396963875026759E-08
+DATA log10rodataL19<>+112(SB)/8, $-.64572084905921579630E-07
+DATA log10rodataL19<>+120(SB)/8, $-5.5
+DATA log10rodataL19<>+128(SB)/8, $18446744073709551616.
+GLOBL log10rodataL19<>+0(SB), RODATA, $136
+
+// Table of log10 correction terms
+DATA log10tab2074<>+0(SB)/8, $0.254164497922885069E-01
+DATA log10tab2074<>+8(SB)/8, $0.179018857989381839E-01
+DATA log10tab2074<>+16(SB)/8, $0.118926768029048674E-01
+DATA log10tab2074<>+24(SB)/8, $0.722595568238080033E-02
+DATA log10tab2074<>+32(SB)/8, $0.376393570022739135E-02
+DATA log10tab2074<>+40(SB)/8, $0.138901135928814326E-02
+DATA log10tab2074<>+48(SB)/8, $0
+DATA log10tab2074<>+56(SB)/8, $-0.490780466387818203E-03
+DATA log10tab2074<>+64(SB)/8, $-0.159811431402137571E-03
+DATA log10tab2074<>+72(SB)/8, $0.925796337165100494E-03
+DATA log10tab2074<>+80(SB)/8, $0.270683176738357035E-02
+DATA log10tab2074<>+88(SB)/8, $0.513079030821304758E-02
+DATA log10tab2074<>+96(SB)/8, $0.815089785397996303E-02
+DATA log10tab2074<>+104(SB)/8, $0.117253060262419215E-01
+DATA log10tab2074<>+112(SB)/8, $0.158164239345343963E-01
+DATA log10tab2074<>+120(SB)/8, $0.203903595489229786E-01
+GLOBL log10tab2074<>+0(SB), RODATA, $128
+
+// Log10 returns the decimal logarithm of the argument.
+//
+// Special cases are:
+// Log(+Inf) = +Inf
+// Log(0) = -Inf
+// Log(x < 0) = NaN
+// Log(NaN) = NaN
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·log10Asm(SB),NOSPLIT,$8-16
+ FMOVD x+0(FP), F0
+ MOVD $log10rodataL19<>+0(SB), R9
+ FMOVD F0, x-8(SP)
+ WORD $0xC0298006 //iilf %r2,2147909631
+ BYTE $0x7F
+ BYTE $0xFF
+ WORD $0x5840F008 //l %r4, 8(%r15)
+ SUBW R4, R2, R3
+ RISBGZ $32, $47, $0, R3, R5
+ MOVH $0x0, R1
+ RISBGN $0, $31, $32, R5, R1
+ WORD $0xC0590016 //iilf %r5,1507327
+ BYTE $0xFF
+ BYTE $0xFF
+ MOVW R4, R10
+ MOVW R5, R11
+ CMPBLE R10, R11, L2
+ WORD $0xC0297FEF //iilf %r2,2146435071
+ BYTE $0xFF
+ BYTE $0xFF
+ MOVW R4, R10
+ MOVW R2, R11
+ CMPBLE R10, R11, L16
+L3:
+L1:
+ FMOVD F0, ret+8(FP)
+ RET
+
+L2:
+ LTDBR F0, F0
+ BLEU L13
+ WORD $0xED009080 //mdb %f0,.L20-.L19(%r9)
+ BYTE $0x00
+ BYTE $0x1C
+ FMOVD F0, x-8(SP)
+ WORD $0x5B20F008 //s %r2, 8(%r15)
+ RISBGZ $57, $60, $51, R2, R3
+ ANDW $0xFFFF0000, R2
+ RISBGN $0, $31, $32, R2, R1
+ ADDW $0x4000000, R2
+ BLEU L17
+L8:
+ SRW $8, R2, R2
+ ORW $0x45000000, R2
+L4:
+ FMOVD log10rodataL19<>+120(SB), F2
+ LDGR R1, F4
+ WFMADB V4, V0, V2, V0
+ FMOVD log10rodataL19<>+112(SB), F4
+ FMOVD log10rodataL19<>+104(SB), F6
+ WFMADB V0, V6, V4, V6
+ FMOVD log10rodataL19<>+96(SB), F4
+ FMOVD log10rodataL19<>+88(SB), F1
+ WFMADB V0, V1, V4, V1
+ WFMDB V0, V0, V4
+ FMOVD log10rodataL19<>+80(SB), F2
+ WFMADB V6, V4, V1, V6
+ FMOVD log10rodataL19<>+72(SB), F1
+ WFMADB V0, V2, V1, V2
+ FMOVD log10rodataL19<>+64(SB), F1
+ RISBGZ $57, $60, $0, R3, R3
+ WFMADB V4, V6, V2, V6
+ FMOVD log10rodataL19<>+56(SB), F2
+ WFMADB V0, V1, V2, V1
+ VLVGF $0, R2, V2
+ WFMADB V4, V6, V1, V4
+ LDEBR F2, F2
+ FMOVD log10rodataL19<>+48(SB), F6
+ WFMADB V0, V4, V6, V4
+ FMOVD log10rodataL19<>+40(SB), F1
+ FMOVD log10rodataL19<>+32(SB), F6
+ MOVD $log10tab2074<>+0(SB), R1
+ WFMADB V2, V1, V6, V2
+ WORD $0x68331000 //ld %f3,0(%r3,%r1)
+ WFMADB V0, V4, V3, V0
+ FMOVD log10rodataL19<>+24(SB), F4
+ FMADD F4, F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L16:
+ RISBGZ $40, $55, $56, R3, R2
+ RISBGZ $57, $60, $51, R3, R3
+ ORW $0x45000000, R2
+ BR L4
+L13:
+ BGE L18 //jnl .L18
+ BVS L18
+ FMOVD log10rodataL19<>+16(SB), F0
+ BR L1
+L17:
+ SRAW $1, R2, R2
+ SUBW $0x40000000, R2
+ BR L8
+L18:
+ FMOVD log10rodataL19<>+8(SB), F0
+ WORD $0xED009000 //ddb %f0,.L36-.L19(%r9)
+ BYTE $0x00
+ BYTE $0x1D
+ BR L1
diff --git a/src/math/log1p.go b/src/math/log1p.go
new file mode 100644
index 0000000..e34e1ff
--- /dev/null
+++ b/src/math/log1p.go
@@ -0,0 +1,197 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/s_log1p.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// double log1p(double x)
+//
+// Method :
+// 1. Argument Reduction: find k and f such that
+// 1+x = 2**k * (1+f),
+// where sqrt(2)/2 < 1+f < sqrt(2) .
+//
+// Note. If k=0, then f=x is exact. However, if k!=0, then f
+// may not be representable exactly. In that case, a correction
+// term is need. Let u=1+x rounded. Let c = (1+x)-u, then
+// log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
+// and add back the correction term c/u.
+// (Note: when x > 2**53, one can simply return log(x))
+//
+// 2. Approximation of log1p(f).
+// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
+// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
+// = 2s + s*R
+// We use a special Reme algorithm on [0,0.1716] to generate
+// a polynomial of degree 14 to approximate R The maximum error
+// of this polynomial approximation is bounded by 2**-58.45. In
+// other words,
+// 2 4 6 8 10 12 14
+// R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
+// (the values of Lp1 to Lp7 are listed in the program)
+// and
+// | 2 14 | -58.45
+// | Lp1*s +...+Lp7*s - R(z) | <= 2
+// | |
+// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
+// In order to guarantee error in log below 1ulp, we compute log
+// by
+// log1p(f) = f - (hfsq - s*(hfsq+R)).
+//
+// 3. Finally, log1p(x) = k*ln2 + log1p(f).
+// = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
+// Here ln2 is split into two floating point number:
+// ln2_hi + ln2_lo,
+// where n*ln2_hi is always exact for |n| < 2000.
+//
+// Special cases:
+// log1p(x) is NaN with signal if x < -1 (including -INF) ;
+// log1p(+INF) is +INF; log1p(-1) is -INF with signal;
+// log1p(NaN) is that NaN with no signal.
+//
+// Accuracy:
+// according to an error analysis, the error is always less than
+// 1 ulp (unit in the last place).
+//
+// Constants:
+// The hexadecimal values are the intended ones for the following
+// constants. The decimal values may be used, provided that the
+// compiler will convert from decimal to binary accurately enough
+// to produce the hexadecimal values shown.
+//
+// Note: Assuming log() return accurate answer, the following
+// algorithm can be used to compute log1p(x) to within a few ULP:
+//
+// u = 1+x;
+// if(u==1.0) return x ; else
+// return log(u)*(x/(u-1.0));
+//
+// See HP-15C Advanced Functions Handbook, p.193.
+
+// Log1p returns the natural logarithm of 1 plus its argument x.
+// It is more accurate than Log(1 + x) when x is near zero.
+//
+// Special cases are:
+// Log1p(+Inf) = +Inf
+// Log1p(±0) = ±0
+// Log1p(-1) = -Inf
+// Log1p(x < -1) = NaN
+// Log1p(NaN) = NaN
+func Log1p(x float64) float64
+
+func log1p(x float64) float64 {
+ const (
+ Sqrt2M1 = 4.142135623730950488017e-01 // Sqrt(2)-1 = 0x3fda827999fcef34
+ Sqrt2HalfM1 = -2.928932188134524755992e-01 // Sqrt(2)/2-1 = 0xbfd2bec333018866
+ Small = 1.0 / (1 << 29) // 2**-29 = 0x3e20000000000000
+ Tiny = 1.0 / (1 << 54) // 2**-54
+ Two53 = 1 << 53 // 2**53
+ Ln2Hi = 6.93147180369123816490e-01 // 3fe62e42fee00000
+ Ln2Lo = 1.90821492927058770002e-10 // 3dea39ef35793c76
+ Lp1 = 6.666666666666735130e-01 // 3FE5555555555593
+ Lp2 = 3.999999999940941908e-01 // 3FD999999997FA04
+ Lp3 = 2.857142874366239149e-01 // 3FD2492494229359
+ Lp4 = 2.222219843214978396e-01 // 3FCC71C51D8E78AF
+ Lp5 = 1.818357216161805012e-01 // 3FC7466496CB03DE
+ Lp6 = 1.531383769920937332e-01 // 3FC39A09D078C69F
+ Lp7 = 1.479819860511658591e-01 // 3FC2F112DF3E5244
+ )
+
+ // special cases
+ switch {
+ case x < -1 || IsNaN(x): // includes -Inf
+ return NaN()
+ case x == -1:
+ return Inf(-1)
+ case IsInf(x, 1):
+ return Inf(1)
+ }
+
+ absx := Abs(x)
+
+ var f float64
+ var iu uint64
+ k := 1
+ if absx < Sqrt2M1 { // |x| < Sqrt(2)-1
+ if absx < Small { // |x| < 2**-29
+ if absx < Tiny { // |x| < 2**-54
+ return x
+ }
+ return x - x*x*0.5
+ }
+ if x > Sqrt2HalfM1 { // Sqrt(2)/2-1 < x
+ // (Sqrt(2)/2-1) < x < (Sqrt(2)-1)
+ k = 0
+ f = x
+ iu = 1
+ }
+ }
+ var c float64
+ if k != 0 {
+ var u float64
+ if absx < Two53 { // 1<<53
+ u = 1.0 + x
+ iu = Float64bits(u)
+ k = int((iu >> 52) - 1023)
+ // correction term
+ if k > 0 {
+ c = 1.0 - (u - x)
+ } else {
+ c = x - (u - 1.0)
+ }
+ c /= u
+ } else {
+ u = x
+ iu = Float64bits(u)
+ k = int((iu >> 52) - 1023)
+ c = 0
+ }
+ iu &= 0x000fffffffffffff
+ if iu < 0x0006a09e667f3bcd { // mantissa of Sqrt(2)
+ u = Float64frombits(iu | 0x3ff0000000000000) // normalize u
+ } else {
+ k++
+ u = Float64frombits(iu | 0x3fe0000000000000) // normalize u/2
+ iu = (0x0010000000000000 - iu) >> 2
+ }
+ f = u - 1.0 // Sqrt(2)/2 < u < Sqrt(2)
+ }
+ hfsq := 0.5 * f * f
+ var s, R, z float64
+ if iu == 0 { // |f| < 2**-20
+ if f == 0 {
+ if k == 0 {
+ return 0
+ }
+ c += float64(k) * Ln2Lo
+ return float64(k)*Ln2Hi + c
+ }
+ R = hfsq * (1.0 - 0.66666666666666666*f) // avoid division
+ if k == 0 {
+ return f - R
+ }
+ return float64(k)*Ln2Hi - ((R - (float64(k)*Ln2Lo + c)) - f)
+ }
+ s = f / (2.0 + f)
+ z = s * s
+ R = z * (Lp1 + z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))))
+ if k == 0 {
+ return f - (hfsq - s*(hfsq+R))
+ }
+ return float64(k)*Ln2Hi - ((hfsq - (s*(hfsq+R) + (float64(k)*Ln2Lo + c))) - f)
+}
diff --git a/src/math/log1p_386.s b/src/math/log1p_386.s
new file mode 100644
index 0000000..63b7e62
--- /dev/null
+++ b/src/math/log1p_386.s
@@ -0,0 +1,27 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Log1p(x float64) float64
+TEXT ·Log1p(SB),NOSPLIT,$0
+ FMOVD $(2.928932188134524e-01), F0
+ FMOVD x+0(FP), F0 // F0=x, F1=1-sqrt(2)/2 = 0.29289321881345247559915564
+ FABS // F0=|x|, F1=1-sqrt(2)/2
+ FUCOMPP F0, F1 // compare F0 to F1
+ FSTSW AX
+ FLDLN2 // F0=log(2)
+ ANDW $0x0100, AX
+ JEQ use_fyl2x // jump if F0 >= F1
+ FMOVD x+0(FP), F0 // F0=x, F1=log(2)
+ FYL2XP1 // F0=log(1+x)=log2(1+x)*log(2)
+ FMOVDP F0, ret+8(FP)
+ RET
+use_fyl2x:
+ FLD1 // F0=1, F2=log(2)
+ FADDD x+0(FP), F0 // F0=1+x, F1=log(2)
+ FYL2X // F0=log(1+x)=log2(1+x)*log(2)
+ FMOVDP F0, ret+8(FP)
+ RET
+
diff --git a/src/math/log1p_s390x.s b/src/math/log1p_s390x.s
new file mode 100644
index 0000000..00eb374
--- /dev/null
+++ b/src/math/log1p_s390x.s
@@ -0,0 +1,180 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Constants
+DATA ·log1pxlim<> + 0(SB)/4, $0xfff00000
+GLOBL ·log1pxlim<> + 0(SB), RODATA, $4
+DATA ·log1pxzero<> + 0(SB)/8, $0.0
+GLOBL ·log1pxzero<> + 0(SB), RODATA, $8
+DATA ·log1pxminf<> + 0(SB)/8, $0xfff0000000000000
+GLOBL ·log1pxminf<> + 0(SB), RODATA, $8
+DATA ·log1pxnan<> + 0(SB)/8, $0x7ff8000000000000
+GLOBL ·log1pxnan<> + 0(SB), RODATA, $8
+DATA ·log1pyout<> + 0(SB)/8, $0x40fce621e71da000
+GLOBL ·log1pyout<> + 0(SB), RODATA, $8
+DATA ·log1pxout<> + 0(SB)/8, $0x40f1000000000000
+GLOBL ·log1pxout<> + 0(SB), RODATA, $8
+DATA ·log1pxl2<> + 0(SB)/8, $0xbfda7aecbeba4e46
+GLOBL ·log1pxl2<> + 0(SB), RODATA, $8
+DATA ·log1pxl1<> + 0(SB)/8, $0x3ffacde700000000
+GLOBL ·log1pxl1<> + 0(SB), RODATA, $8
+DATA ·log1pxa<> + 0(SB)/8, $5.5
+GLOBL ·log1pxa<> + 0(SB), RODATA, $8
+DATA ·log1pxmone<> + 0(SB)/8, $-1.0
+GLOBL ·log1pxmone<> + 0(SB), RODATA, $8
+
+// Minimax polynomial approximations
+DATA ·log1pc8<> + 0(SB)/8, $0.212881813645679599E-07
+GLOBL ·log1pc8<> + 0(SB), RODATA, $8
+DATA ·log1pc7<> + 0(SB)/8, $-.148682720127920854E-06
+GLOBL ·log1pc7<> + 0(SB), RODATA, $8
+DATA ·log1pc6<> + 0(SB)/8, $0.938370938292558173E-06
+GLOBL ·log1pc6<> + 0(SB), RODATA, $8
+DATA ·log1pc5<> + 0(SB)/8, $-.602107458843052029E-05
+GLOBL ·log1pc5<> + 0(SB), RODATA, $8
+DATA ·log1pc4<> + 0(SB)/8, $0.397389654305194527E-04
+GLOBL ·log1pc4<> + 0(SB), RODATA, $8
+DATA ·log1pc3<> + 0(SB)/8, $-.273205381970859341E-03
+GLOBL ·log1pc3<> + 0(SB), RODATA, $8
+DATA ·log1pc2<> + 0(SB)/8, $0.200350613573012186E-02
+GLOBL ·log1pc2<> + 0(SB), RODATA, $8
+DATA ·log1pc1<> + 0(SB)/8, $-.165289256198351540E-01
+GLOBL ·log1pc1<> + 0(SB), RODATA, $8
+DATA ·log1pc0<> + 0(SB)/8, $0.181818181818181826E+00
+GLOBL ·log1pc0<> + 0(SB), RODATA, $8
+
+
+// Table of log10 correction terms
+DATA ·log1ptab<> + 0(SB)/8, $0.585235384085551248E-01
+DATA ·log1ptab<> + 8(SB)/8, $0.412206153771168640E-01
+DATA ·log1ptab<> + 16(SB)/8, $0.273839003221648339E-01
+DATA ·log1ptab<> + 24(SB)/8, $0.166383778368856480E-01
+DATA ·log1ptab<> + 32(SB)/8, $0.866678223433169637E-02
+DATA ·log1ptab<> + 40(SB)/8, $0.319831684989627514E-02
+DATA ·log1ptab<> + 48(SB)/8, $-.000000000000000000E+00
+DATA ·log1ptab<> + 56(SB)/8, $-.113006378583725549E-02
+DATA ·log1ptab<> + 64(SB)/8, $-.367979419636602491E-03
+DATA ·log1ptab<> + 72(SB)/8, $0.213172484510484979E-02
+DATA ·log1ptab<> + 80(SB)/8, $0.623271047682013536E-02
+DATA ·log1ptab<> + 88(SB)/8, $0.118140812789696885E-01
+DATA ·log1ptab<> + 96(SB)/8, $0.187681358930914206E-01
+DATA ·log1ptab<> + 104(SB)/8, $0.269985148668178992E-01
+DATA ·log1ptab<> + 112(SB)/8, $0.364186619761331328E-01
+DATA ·log1ptab<> + 120(SB)/8, $0.469505379381388441E-01
+GLOBL ·log1ptab<> + 0(SB), RODATA, $128
+
+// Log1p returns the natural logarithm of 1 plus its argument x.
+// It is more accurate than Log(1 + x) when x is near zero.
+//
+// Special cases are:
+// Log1p(+Inf) = +Inf
+// Log1p(±0) = ±0
+// Log1p(-1) = -Inf
+// Log1p(x < -1) = NaN
+// Log1p(NaN) = NaN
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·log1pAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·log1pxmone<>+0(SB), R1
+ MOVD ·log1pxout<>+0(SB), R2
+ FMOVD 0(R1), F3
+ MOVD $·log1pxa<>+0(SB), R1
+ MOVWZ ·log1pxlim<>+0(SB), R0
+ FMOVD 0(R1), F1
+ MOVD $·log1pc8<>+0(SB), R1
+ FMOVD 0(R1), F5
+ MOVD $·log1pc7<>+0(SB), R1
+ VLEG $0, 0(R1), V20
+ MOVD $·log1pc6<>+0(SB), R1
+ WFSDB V0, V3, V4
+ VLEG $0, 0(R1), V18
+ MOVD $·log1pc5<>+0(SB), R1
+ VLEG $0, 0(R1), V16
+ MOVD R2, R5
+ LGDR F4, R3
+ WORD $0xC0190006 //iilf %r1,425983
+ BYTE $0x7F
+ BYTE $0xFF
+ SRAD $32, R3, R3
+ SUBW R3, R1
+ SRW $16, R1, R1
+ BYTE $0x18 //lr %r4,%r1
+ BYTE $0x41
+ RISBGN $0, $15, $48, R4, R2
+ RISBGN $16, $31, $32, R4, R5
+ MOVW R0, R6
+ MOVW R3, R7
+ CMPBGT R6, R7, L8
+ WFCEDBS V4, V4, V6
+ MOVD $·log1pxzero<>+0(SB), R1
+ FMOVD 0(R1), F2
+ BVS LEXITTAGlog1p
+ WORD $0xB3130044 // lcdbr %f4,%f4
+ WFCEDBS V2, V4, V6
+ BEQ L9
+ WFCHDBS V4, V2, V2
+ BEQ LEXITTAGlog1p
+ MOVD $·log1pxnan<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L8:
+ LDGR R2, F2
+ FSUB F4, F3
+ FMADD F2, F4, F1
+ MOVD $·log1pc4<>+0(SB), R2
+ WORD $0xB3130041 // lcdbr %f4,%f1
+ FMOVD 0(R2), F7
+ FSUB F3, F0
+ MOVD $·log1pc3<>+0(SB), R2
+ FMOVD 0(R2), F3
+ MOVD $·log1pc2<>+0(SB), R2
+ WFMDB V1, V1, V6
+ FMADD F7, F4, F3
+ WFMSDB V0, V2, V1, V0
+ FMOVD 0(R2), F7
+ WFMADB V4, V5, V20, V5
+ MOVD $·log1pc1<>+0(SB), R2
+ FMOVD 0(R2), F2
+ FMADD F7, F4, F2
+ WFMADB V4, V18, V16, V4
+ FMADD F3, F6, F2
+ WFMADB V5, V6, V4, V5
+ FMUL F6, F6
+ MOVD $·log1pc0<>+0(SB), R2
+ WFMADB V6, V5, V2, V6
+ FMOVD 0(R2), F4
+ WFMADB V0, V6, V4, V6
+ RISBGZ $57, $60, $3, R1, R1
+ MOVD $·log1ptab<>+0(SB), R2
+ MOVD $·log1pxl1<>+0(SB), R3
+ WORD $0x68112000 //ld %f1,0(%r1,%r2)
+ FMOVD 0(R3), F2
+ WFMADB V0, V6, V1, V0
+ MOVD $·log1pyout<>+0(SB), R1
+ LDGR R5, F6
+ FMOVD 0(R1), F4
+ WFMSDB V2, V6, V4, V2
+ MOVD $·log1pxl2<>+0(SB), R1
+ FMOVD 0(R1), F4
+ FMADD F4, F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L9:
+ MOVD $·log1pxminf<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+
+LEXITTAGlog1p:
+ FMOVD F0, ret+8(FP)
+ RET
+
diff --git a/src/math/log_386.s b/src/math/log_386.s
new file mode 100644
index 0000000..0b64b50
--- /dev/null
+++ b/src/math/log_386.s
@@ -0,0 +1,13 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Log(x float64) float64
+TEXT ·Log(SB),NOSPLIT,$0
+ FLDLN2 // F0=log(2)
+ FMOVD x+0(FP), F0 // F0=x, F1=log(2)
+ FYL2X // F0=log(x)=log2(x)*log(2)
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/log_amd64.s b/src/math/log_amd64.s
new file mode 100644
index 0000000..3d76389
--- /dev/null
+++ b/src/math/log_amd64.s
@@ -0,0 +1,112 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define HSqrt2 7.07106781186547524401e-01 // sqrt(2)/2
+#define Ln2Hi 6.93147180369123816490e-01 // 0x3fe62e42fee00000
+#define Ln2Lo 1.90821492927058770002e-10 // 0x3dea39ef35793c76
+#define L1 6.666666666666735130e-01 // 0x3FE5555555555593
+#define L2 3.999999999940941908e-01 // 0x3FD999999997FA04
+#define L3 2.857142874366239149e-01 // 0x3FD2492494229359
+#define L4 2.222219843214978396e-01 // 0x3FCC71C51D8E78AF
+#define L5 1.818357216161805012e-01 // 0x3FC7466496CB03DE
+#define L6 1.531383769920937332e-01 // 0x3FC39A09D078C69F
+#define L7 1.479819860511658591e-01 // 0x3FC2F112DF3E5244
+#define NaN 0x7FF8000000000001
+#define NegInf 0xFFF0000000000000
+#define PosInf 0x7FF0000000000000
+
+// func Log(x float64) float64
+TEXT ·Log(SB),NOSPLIT,$0
+ // test bits for special cases
+ MOVQ x+0(FP), BX
+ MOVQ $~(1<<63), AX // sign bit mask
+ ANDQ BX, AX
+ JEQ isZero
+ MOVQ $0, AX
+ CMPQ AX, BX
+ JGT isNegative
+ MOVQ $PosInf, AX
+ CMPQ AX, BX
+ JLE isInfOrNaN
+ // f1, ki := math.Frexp(x); k := float64(ki)
+ MOVQ BX, X0
+ MOVQ $0x000FFFFFFFFFFFFF, AX
+ MOVQ AX, X2
+ ANDPD X0, X2
+ MOVSD $0.5, X0 // 0x3FE0000000000000
+ ORPD X0, X2 // X2= f1
+ SHRQ $52, BX
+ ANDL $0x7FF, BX
+ SUBL $0x3FE, BX
+ XORPS X1, X1 // break dependency for CVTSL2SD
+ CVTSL2SD BX, X1 // x1= k, x2= f1
+ // if f1 < math.Sqrt2/2 { k -= 1; f1 *= 2 }
+ MOVSD $HSqrt2, X0 // x0= 0.7071, x1= k, x2= f1
+ CMPSD X2, X0, 5 // cmpnlt; x0= 0 or ^0, x1= k, x2 = f1
+ MOVSD $1.0, X3 // x0= 0 or ^0, x1= k, x2 = f1, x3= 1
+ ANDPD X0, X3 // x0= 0 or ^0, x1= k, x2 = f1, x3= 0 or 1
+ SUBSD X3, X1 // x0= 0 or ^0, x1= k, x2 = f1, x3= 0 or 1
+ MOVSD $1.0, X0 // x0= 1, x1= k, x2= f1, x3= 0 or 1
+ ADDSD X0, X3 // x0= 1, x1= k, x2= f1, x3= 1 or 2
+ MULSD X3, X2 // x0= 1, x1= k, x2= f1
+ // f := f1 - 1
+ SUBSD X0, X2 // x1= k, x2= f
+ // s := f / (2 + f)
+ MOVSD $2.0, X0
+ ADDSD X2, X0
+ MOVAPD X2, X3
+ DIVSD X0, X3 // x1=k, x2= f, x3= s
+ // s2 := s * s
+ MOVAPD X3, X4 // x1= k, x2= f, x3= s
+ MULSD X4, X4 // x1= k, x2= f, x3= s, x4= s2
+ // s4 := s2 * s2
+ MOVAPD X4, X5 // x1= k, x2= f, x3= s, x4= s2
+ MULSD X5, X5 // x1= k, x2= f, x3= s, x4= s2, x5= s4
+ // t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
+ MOVSD $L7, X6
+ MULSD X5, X6
+ ADDSD $L5, X6
+ MULSD X5, X6
+ ADDSD $L3, X6
+ MULSD X5, X6
+ ADDSD $L1, X6
+ MULSD X6, X4 // x1= k, x2= f, x3= s, x4= t1, x5= s4
+ // t2 := s4 * (L2 + s4*(L4+s4*L6))
+ MOVSD $L6, X6
+ MULSD X5, X6
+ ADDSD $L4, X6
+ MULSD X5, X6
+ ADDSD $L2, X6
+ MULSD X6, X5 // x1= k, x2= f, x3= s, x4= t1, x5= t2
+ // R := t1 + t2
+ ADDSD X5, X4 // x1= k, x2= f, x3= s, x4= R
+ // hfsq := 0.5 * f * f
+ MOVSD $0.5, X0
+ MULSD X2, X0
+ MULSD X2, X0 // x0= hfsq, x1= k, x2= f, x3= s, x4= R
+ // return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f)
+ ADDSD X0, X4 // x0= hfsq, x1= k, x2= f, x3= s, x4= hfsq+R
+ MULSD X4, X3 // x0= hfsq, x1= k, x2= f, x3= s*(hfsq+R)
+ MOVSD $Ln2Lo, X4
+ MULSD X1, X4 // x4= k*Ln2Lo
+ ADDSD X4, X3 // x0= hfsq, x1= k, x2= f, x3= s*(hfsq+R)+k*Ln2Lo
+ SUBSD X3, X0 // x0= hfsq-(s*(hfsq+R)+k*Ln2Lo), x1= k, x2= f
+ SUBSD X2, X0 // x0= (hfsq-(s*(hfsq+R)+k*Ln2Lo))-f, x1= k
+ MULSD $Ln2Hi, X1 // x0= (hfsq-(s*(hfsq+R)+k*Ln2Lo))-f, x1= k*Ln2Hi
+ SUBSD X0, X1 // x1= k*Ln2Hi-((hfsq-(s*(hfsq+R)+k*Ln2Lo))-f)
+ MOVSD X1, ret+8(FP)
+ RET
+isInfOrNaN:
+ MOVQ BX, ret+8(FP) // +Inf or NaN, return x
+ RET
+isNegative:
+ MOVQ $NaN, AX
+ MOVQ AX, ret+8(FP) // return NaN
+ RET
+isZero:
+ MOVQ $NegInf, AX
+ MOVQ AX, ret+8(FP) // return -Inf
+ RET
diff --git a/src/math/log_s390x.s b/src/math/log_s390x.s
new file mode 100644
index 0000000..4b514f3
--- /dev/null
+++ b/src/math/log_s390x.s
@@ -0,0 +1,168 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial approximations
+DATA ·logrodataL21<> + 0(SB)/8, $-.499999999999999778E+00
+DATA ·logrodataL21<> + 8(SB)/8, $0.333333333333343751E+00
+DATA ·logrodataL21<> + 16(SB)/8, $-.250000000001606881E+00
+DATA ·logrodataL21<> + 24(SB)/8, $0.199999999971603032E+00
+DATA ·logrodataL21<> + 32(SB)/8, $-.166666663114122038E+00
+DATA ·logrodataL21<> + 40(SB)/8, $-.125002923782692399E+00
+DATA ·logrodataL21<> + 48(SB)/8, $0.111142014580396256E+00
+DATA ·logrodataL21<> + 56(SB)/8, $0.759438932618934220E-01
+DATA ·logrodataL21<> + 64(SB)/8, $0.142857144267212549E+00
+DATA ·logrodataL21<> + 72(SB)/8, $-.993038938793590759E-01
+DATA ·logrodataL21<> + 80(SB)/8, $-1.0
+GLOBL ·logrodataL21<> + 0(SB), RODATA, $88
+
+// Constants
+DATA ·logxminf<> + 0(SB)/8, $0xfff0000000000000
+GLOBL ·logxminf<> + 0(SB), RODATA, $8
+DATA ·logxnan<> + 0(SB)/8, $0x7ff8000000000000
+GLOBL ·logxnan<> + 0(SB), RODATA, $8
+DATA ·logx43f<> + 0(SB)/8, $0x43f0000000000000
+GLOBL ·logx43f<> + 0(SB), RODATA, $8
+DATA ·logxl2<> + 0(SB)/8, $0x3fda7aecbeba4e46
+GLOBL ·logxl2<> + 0(SB), RODATA, $8
+DATA ·logxl1<> + 0(SB)/8, $0x3ffacde700000000
+GLOBL ·logxl1<> + 0(SB), RODATA, $8
+
+/* Input transform scale and add constants */
+DATA ·logxm<> + 0(SB)/8, $0x3fc77604e63c84b1
+DATA ·logxm<> + 8(SB)/8, $0x40fb39456ab53250
+DATA ·logxm<> + 16(SB)/8, $0x3fc9ee358b945f3f
+DATA ·logxm<> + 24(SB)/8, $0x40fb39418bf3b137
+DATA ·logxm<> + 32(SB)/8, $0x3fccfb2e1304f4b6
+DATA ·logxm<> + 40(SB)/8, $0x40fb393d3eda3022
+DATA ·logxm<> + 48(SB)/8, $0x3fd0000000000000
+DATA ·logxm<> + 56(SB)/8, $0x40fb393969e70000
+DATA ·logxm<> + 64(SB)/8, $0x3fd11117aafbfe04
+DATA ·logxm<> + 72(SB)/8, $0x40fb3936eaefafcf
+DATA ·logxm<> + 80(SB)/8, $0x3fd2492af5e658b2
+DATA ·logxm<> + 88(SB)/8, $0x40fb39343ff01715
+DATA ·logxm<> + 96(SB)/8, $0x3fd3b50c622a43dd
+DATA ·logxm<> + 104(SB)/8, $0x40fb39315adae2f3
+DATA ·logxm<> + 112(SB)/8, $0x3fd56bbeea918777
+DATA ·logxm<> + 120(SB)/8, $0x40fb392e21698552
+GLOBL ·logxm<> + 0(SB), RODATA, $128
+
+// Log returns the natural logarithm of the argument.
+//
+// Special cases are:
+// Log(+Inf) = +Inf
+// Log(0) = -Inf
+// Log(x < 0) = NaN
+// Log(NaN) = NaN
+// The algorithm used is minimax polynomial approximation using a table of
+// polynomial coefficients determined with a Remez exchange algorithm.
+
+TEXT ·logAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ MOVD $·logrodataL21<>+0(SB), R9
+ MOVH $0x8006, R4
+ LGDR F0, R1
+ MOVD $0x3FF0000000000000, R6
+ SRAD $48, R1, R1
+ MOVD $0x40F03E8000000000, R8
+ SUBW R1, R4
+ RISBGZ $32, $59, $0, R4, R2
+ RISBGN $0, $15, $48, R2, R6
+ RISBGN $16, $31, $32, R2, R8
+ MOVW R1, R7
+ CMPBGT R7, $22, L17
+ LTDBR F0, F0
+ MOVD $·logx43f<>+0(SB), R1
+ FMOVD 0(R1), F2
+ BLEU L3
+ MOVH $0x8005, R12
+ MOVH $0x8405, R0
+ BR L15
+L7:
+ LTDBR F0, F0
+ BLEU L3
+L15:
+ FMUL F2, F0
+ LGDR F0, R1
+ SRAD $48, R1, R1
+ SUBW R1, R0, R2
+ SUBW R1, R12, R3
+ BYTE $0x18 //lr %r4,%r2
+ BYTE $0x42
+ ANDW $0xFFFFFFF0, R3
+ ANDW $0xFFFFFFF0, R2
+ BYTE $0x18 //lr %r5,%r1
+ BYTE $0x51
+ MOVW R1, R7
+ CMPBLE R7, $22, L7
+ RISBGN $0, $15, $48, R3, R6
+ RISBGN $16, $31, $32, R2, R8
+L2:
+ MOVH R5, R5
+ MOVH $0x7FEF, R1
+ CMPW R5, R1
+ BGT L1
+ LDGR R6, F2
+ FMUL F2, F0
+ RISBGZ $57, $59, $3, R4, R4
+ FMOVD 80(R9), F2
+ MOVD $·logxm<>+0(SB), R7
+ ADD R7, R4
+ FMOVD 72(R9), F4
+ WORD $0xED004000 //madb %f2,%f0,0(%r4)
+ BYTE $0x20
+ BYTE $0x1E
+ FMOVD 64(R9), F1
+ FMOVD F2, F0
+ FMOVD 56(R9), F2
+ WFMADB V0, V2, V4, V2
+ WFMDB V0, V0, V6
+ FMOVD 48(R9), F4
+ WFMADB V0, V2, V4, V2
+ FMOVD 40(R9), F4
+ WFMADB V2, V6, V1, V2
+ FMOVD 32(R9), F1
+ WFMADB V6, V4, V1, V4
+ FMOVD 24(R9), F1
+ WFMADB V6, V2, V1, V2
+ FMOVD 16(R9), F1
+ WFMADB V6, V4, V1, V4
+ MOVD $·logxl1<>+0(SB), R1
+ FMOVD 8(R9), F1
+ WFMADB V6, V2, V1, V2
+ FMOVD 0(R9), F1
+ WFMADB V6, V4, V1, V4
+ FMOVD 8(R4), F1
+ WFMADB V0, V2, V4, V2
+ LDGR R8, F4
+ WFMADB V6, V2, V0, V2
+ WORD $0xED401000 //msdb %f1,%f4,0(%r1)
+ BYTE $0x10
+ BYTE $0x1F
+ MOVD ·logxl2<>+0(SB), R1
+ WORD $0xB3130001 //lcdbr %f0,%f1
+ LDGR R1, F4
+ WFMADB V0, V4, V2, V0
+L1:
+ FMOVD F0, ret+8(FP)
+ RET
+L3:
+ LTDBR F0, F0
+ BEQ L20
+ BGE L1
+ BVS L1
+
+ MOVD $·logxnan<>+0(SB), R1
+ FMOVD 0(R1), F0
+ BR L1
+L20:
+ MOVD $·logxminf<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+L17:
+ BYTE $0x18 //lr %r5,%r1
+ BYTE $0x51
+ BR L2
diff --git a/src/math/logb.go b/src/math/logb.go
new file mode 100644
index 0000000..f2769d4
--- /dev/null
+++ b/src/math/logb.go
@@ -0,0 +1,50 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Logb returns the binary exponent of x.
+//
+// Special cases are:
+// Logb(±Inf) = +Inf
+// Logb(0) = -Inf
+// Logb(NaN) = NaN
+func Logb(x float64) float64 {
+ // special cases
+ switch {
+ case x == 0:
+ return Inf(-1)
+ case IsInf(x, 0):
+ return Inf(1)
+ case IsNaN(x):
+ return x
+ }
+ return float64(ilogb(x))
+}
+
+// Ilogb returns the binary exponent of x as an integer.
+//
+// Special cases are:
+// Ilogb(±Inf) = MaxInt32
+// Ilogb(0) = MinInt32
+// Ilogb(NaN) = MaxInt32
+func Ilogb(x float64) int {
+ // special cases
+ switch {
+ case x == 0:
+ return MinInt32
+ case IsNaN(x):
+ return MaxInt32
+ case IsInf(x, 0):
+ return MaxInt32
+ }
+ return ilogb(x)
+}
+
+// logb returns the binary exponent of x. It assumes x is finite and
+// non-zero.
+func ilogb(x float64) int {
+ x, exp := normalize(x)
+ return int((Float64bits(x)>>shift)&mask) - bias + exp
+}
diff --git a/src/math/mod.go b/src/math/mod.go
new file mode 100644
index 0000000..7efc018
--- /dev/null
+++ b/src/math/mod.go
@@ -0,0 +1,46 @@
+// Copyright 2009-2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point mod function.
+*/
+
+// Mod returns the floating-point remainder of x/y.
+// The magnitude of the result is less than y and its
+// sign agrees with that of x.
+//
+// Special cases are:
+// Mod(±Inf, y) = NaN
+// Mod(NaN, y) = NaN
+// Mod(x, 0) = NaN
+// Mod(x, ±Inf) = x
+// Mod(x, NaN) = NaN
+func Mod(x, y float64) float64
+
+func mod(x, y float64) float64 {
+ if y == 0 || IsInf(x, 0) || IsNaN(x) || IsNaN(y) {
+ return NaN()
+ }
+ y = Abs(y)
+
+ yfr, yexp := Frexp(y)
+ r := x
+ if x < 0 {
+ r = -x
+ }
+
+ for r >= y {
+ rfr, rexp := Frexp(r)
+ if rfr < yfr {
+ rexp = rexp - 1
+ }
+ r = r - Ldexp(y, rexp-yexp)
+ }
+ if x < 0 {
+ r = -r
+ }
+ return r
+}
diff --git a/src/math/mod_386.s b/src/math/mod_386.s
new file mode 100644
index 0000000..10ad98b
--- /dev/null
+++ b/src/math/mod_386.s
@@ -0,0 +1,17 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Mod(x, y float64) float64
+TEXT ·Mod(SB),NOSPLIT,$0
+ FMOVD y+8(FP), F0 // F0=y
+ FMOVD x+0(FP), F0 // F0=x, F1=y
+ FPREM // F0=reduced_x, F1=y
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE -3(PC) // jump if reduction incomplete
+ FMOVDP F0, F1 // F0=x-q*y
+ FMOVDP F0, ret+16(FP)
+ RET
diff --git a/src/math/modf.go b/src/math/modf.go
new file mode 100644
index 0000000..c5bb894
--- /dev/null
+++ b/src/math/modf.go
@@ -0,0 +1,37 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Modf returns integer and fractional floating-point numbers
+// that sum to f. Both values have the same sign as f.
+//
+// Special cases are:
+// Modf(±Inf) = ±Inf, NaN
+// Modf(NaN) = NaN, NaN
+func Modf(f float64) (int float64, frac float64)
+
+func modf(f float64) (int float64, frac float64) {
+ if f < 1 {
+ switch {
+ case f < 0:
+ int, frac = Modf(-f)
+ return -int, -frac
+ case f == 0:
+ return f, f // Return -0, -0 when f == -0
+ }
+ return 0, f
+ }
+
+ x := Float64bits(f)
+ e := uint(x>>shift)&mask - bias
+
+ // Keep the top 12+e bits, the integer part; clear the rest.
+ if e < 64-12 {
+ x &^= 1<<(64-12-e) - 1
+ }
+ int = Float64frombits(x)
+ frac = f - int
+ return
+}
diff --git a/src/math/modf_386.s b/src/math/modf_386.s
new file mode 100644
index 0000000..e916073
--- /dev/null
+++ b/src/math/modf_386.s
@@ -0,0 +1,34 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Modf(f float64) (int float64, frac float64)
+TEXT ·Modf(SB),NOSPLIT,$0
+ // special case for f == -0.0
+ MOVL f_hi+4(FP), DX // high word
+ MOVL f_lo+0(FP), AX // low word
+ CMPL DX, $(1<<31) // beginning of -0.0
+ JNE notNegativeZero
+ CMPL AX, $0 // could be denormalized
+ JNE notNegativeZero
+ MOVL AX, int_lo+8(FP)
+ MOVL DX, int_hi+12(FP)
+ MOVL AX, frac_lo+16(FP)
+ MOVL DX, frac_hi+20(FP)
+ RET
+notNegativeZero:
+ FMOVD f+0(FP), F0 // F0=f
+ FMOVD F0, F1 // F0=f, F1=f
+ FSTCW -2(SP) // save old Control Word
+ MOVW -2(SP), AX
+ ORW $0x0c00, AX // Rounding Control set to truncate
+ MOVW AX, -4(SP) // store new Control Word
+ FLDCW -4(SP) // load new Control Word
+ FRNDINT // F0=trunc(f), F1=f
+ FLDCW -2(SP) // load old Control Word
+ FSUBD F0, F1 // F0=trunc(f), F1=f-trunc(f)
+ FMOVDP F0, int+8(FP) // F0=f-trunc(f)
+ FMOVDP F0, frac+16(FP)
+ RET
diff --git a/src/math/modf_arm64.s b/src/math/modf_arm64.s
new file mode 100644
index 0000000..7c70ef3
--- /dev/null
+++ b/src/math/modf_arm64.s
@@ -0,0 +1,18 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Modf(f float64) (int float64, frac float64)
+TEXT ·Modf(SB),NOSPLIT,$0
+ MOVD f+0(FP), R0
+ FMOVD R0, F0
+ FRINTZD F0, F1
+ FMOVD F1, int+8(FP)
+ FSUBD F1, F0
+ FMOVD F0, R1
+ AND $(1<<63), R0
+ ORR R0, R1 // must have same sign
+ MOVD R1, frac+16(FP)
+ RET
diff --git a/src/math/modf_ppc64x.s b/src/math/modf_ppc64x.s
new file mode 100644
index 0000000..da58653
--- /dev/null
+++ b/src/math/modf_ppc64x.s
@@ -0,0 +1,17 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build ppc64 ppc64le
+
+#include "textflag.h"
+
+// func Modf(f float64) (int float64, frac float64)
+TEXT ·Modf(SB),NOSPLIT,$0
+ FMOVD f+0(FP), F0
+ FRIZ F0, F1
+ FMOVD F1, int+8(FP)
+ FSUB F1, F0, F2
+ FCPSGN F2, F0, F2
+ FMOVD F2, frac+16(FP)
+ RET
diff --git a/src/math/nextafter.go b/src/math/nextafter.go
new file mode 100644
index 0000000..9088e4d
--- /dev/null
+++ b/src/math/nextafter.go
@@ -0,0 +1,49 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Nextafter32 returns the next representable float32 value after x towards y.
+//
+// Special cases are:
+// Nextafter32(x, x) = x
+// Nextafter32(NaN, y) = NaN
+// Nextafter32(x, NaN) = NaN
+func Nextafter32(x, y float32) (r float32) {
+ switch {
+ case IsNaN(float64(x)) || IsNaN(float64(y)): // special case
+ r = float32(NaN())
+ case x == y:
+ r = x
+ case x == 0:
+ r = float32(Copysign(float64(Float32frombits(1)), float64(y)))
+ case (y > x) == (x > 0):
+ r = Float32frombits(Float32bits(x) + 1)
+ default:
+ r = Float32frombits(Float32bits(x) - 1)
+ }
+ return
+}
+
+// Nextafter returns the next representable float64 value after x towards y.
+//
+// Special cases are:
+// Nextafter(x, x) = x
+// Nextafter(NaN, y) = NaN
+// Nextafter(x, NaN) = NaN
+func Nextafter(x, y float64) (r float64) {
+ switch {
+ case IsNaN(x) || IsNaN(y): // special case
+ r = NaN()
+ case x == y:
+ r = x
+ case x == 0:
+ r = Copysign(Float64frombits(1), y)
+ case (y > x) == (x > 0):
+ r = Float64frombits(Float64bits(x) + 1)
+ default:
+ r = Float64frombits(Float64bits(x) - 1)
+ }
+ return
+}
diff --git a/src/math/pow.go b/src/math/pow.go
new file mode 100644
index 0000000..2219a90
--- /dev/null
+++ b/src/math/pow.go
@@ -0,0 +1,151 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+func isOddInt(x float64) bool {
+ xi, xf := Modf(x)
+ return xf == 0 && int64(xi)&1 == 1
+}
+
+// Special cases taken from FreeBSD's /usr/src/lib/msun/src/e_pow.c
+// updated by IEEE Std. 754-2008 "Section 9.2.1 Special values".
+
+// Pow returns x**y, the base-x exponential of y.
+//
+// Special cases are (in order):
+// Pow(x, ±0) = 1 for any x
+// Pow(1, y) = 1 for any y
+// Pow(x, 1) = x for any x
+// Pow(NaN, y) = NaN
+// Pow(x, NaN) = NaN
+// Pow(±0, y) = ±Inf for y an odd integer < 0
+// Pow(±0, -Inf) = +Inf
+// Pow(±0, +Inf) = +0
+// Pow(±0, y) = +Inf for finite y < 0 and not an odd integer
+// Pow(±0, y) = ±0 for y an odd integer > 0
+// Pow(±0, y) = +0 for finite y > 0 and not an odd integer
+// Pow(-1, ±Inf) = 1
+// Pow(x, +Inf) = +Inf for |x| > 1
+// Pow(x, -Inf) = +0 for |x| > 1
+// Pow(x, +Inf) = +0 for |x| < 1
+// Pow(x, -Inf) = +Inf for |x| < 1
+// Pow(+Inf, y) = +Inf for y > 0
+// Pow(+Inf, y) = +0 for y < 0
+// Pow(-Inf, y) = Pow(-0, -y)
+// Pow(x, y) = NaN for finite x < 0 and finite non-integer y
+func Pow(x, y float64) float64
+
+func pow(x, y float64) float64 {
+ switch {
+ case y == 0 || x == 1:
+ return 1
+ case y == 1:
+ return x
+ case IsNaN(x) || IsNaN(y):
+ return NaN()
+ case x == 0:
+ switch {
+ case y < 0:
+ if isOddInt(y) {
+ return Copysign(Inf(1), x)
+ }
+ return Inf(1)
+ case y > 0:
+ if isOddInt(y) {
+ return x
+ }
+ return 0
+ }
+ case IsInf(y, 0):
+ switch {
+ case x == -1:
+ return 1
+ case (Abs(x) < 1) == IsInf(y, 1):
+ return 0
+ default:
+ return Inf(1)
+ }
+ case IsInf(x, 0):
+ if IsInf(x, -1) {
+ return Pow(1/x, -y) // Pow(-0, -y)
+ }
+ switch {
+ case y < 0:
+ return 0
+ case y > 0:
+ return Inf(1)
+ }
+ case y == 0.5:
+ return Sqrt(x)
+ case y == -0.5:
+ return 1 / Sqrt(x)
+ }
+
+ yi, yf := Modf(Abs(y))
+ if yf != 0 && x < 0 {
+ return NaN()
+ }
+ if yi >= 1<<63 {
+ // yi is a large even int that will lead to overflow (or underflow to 0)
+ // for all x except -1 (x == 1 was handled earlier)
+ switch {
+ case x == -1:
+ return 1
+ case (Abs(x) < 1) == (y > 0):
+ return 0
+ default:
+ return Inf(1)
+ }
+ }
+
+ // ans = a1 * 2**ae (= 1 for now).
+ a1 := 1.0
+ ae := 0
+
+ // ans *= x**yf
+ if yf != 0 {
+ if yf > 0.5 {
+ yf--
+ yi++
+ }
+ a1 = Exp(yf * Log(x))
+ }
+
+ // ans *= x**yi
+ // by multiplying in successive squarings
+ // of x according to bits of yi.
+ // accumulate powers of two into exp.
+ x1, xe := Frexp(x)
+ for i := int64(yi); i != 0; i >>= 1 {
+ if xe < -1<<12 || 1<<12 < xe {
+ // catch xe before it overflows the left shift below
+ // Since i !=0 it has at least one bit still set, so ae will accumulate xe
+ // on at least one more iteration, ae += xe is a lower bound on ae
+ // the lower bound on ae exceeds the size of a float64 exp
+ // so the final call to Ldexp will produce under/overflow (0/Inf)
+ ae += xe
+ break
+ }
+ if i&1 == 1 {
+ a1 *= x1
+ ae += xe
+ }
+ x1 *= x1
+ xe <<= 1
+ if x1 < .5 {
+ x1 += x1
+ xe--
+ }
+ }
+
+ // ans = a1*2**ae
+ // if y < 0 { ans = 1 / ans }
+ // but in the opposite order
+ if y < 0 {
+ a1 = 1 / a1
+ ae = -ae
+ }
+ return Ldexp(a1, ae)
+}
diff --git a/src/math/pow10.go b/src/math/pow10.go
new file mode 100644
index 0000000..1234e20
--- /dev/null
+++ b/src/math/pow10.go
@@ -0,0 +1,46 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// pow10tab stores the pre-computed values 10**i for i < 32.
+var pow10tab = [...]float64{
+ 1e00, 1e01, 1e02, 1e03, 1e04, 1e05, 1e06, 1e07, 1e08, 1e09,
+ 1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
+ 1e20, 1e21, 1e22, 1e23, 1e24, 1e25, 1e26, 1e27, 1e28, 1e29,
+ 1e30, 1e31,
+}
+
+// pow10postab32 stores the pre-computed value for 10**(i*32) at index i.
+var pow10postab32 = [...]float64{
+ 1e00, 1e32, 1e64, 1e96, 1e128, 1e160, 1e192, 1e224, 1e256, 1e288,
+}
+
+// pow10negtab32 stores the pre-computed value for 10**(-i*32) at index i.
+var pow10negtab32 = [...]float64{
+ 1e-00, 1e-32, 1e-64, 1e-96, 1e-128, 1e-160, 1e-192, 1e-224, 1e-256, 1e-288, 1e-320,
+}
+
+// Pow10 returns 10**n, the base-10 exponential of n.
+//
+// Special cases are:
+// Pow10(n) = 0 for n < -323
+// Pow10(n) = +Inf for n > 308
+func Pow10(n int) float64 {
+ if 0 <= n && n <= 308 {
+ return pow10postab32[uint(n)/32] * pow10tab[uint(n)%32]
+ }
+
+ if -323 <= n && n <= 0 {
+ return pow10negtab32[uint(-n)/32] / pow10tab[uint(-n)%32]
+ }
+
+ // n < -323 || 308 < n
+ if n > 0 {
+ return Inf(1)
+ }
+
+ // n < -323
+ return 0
+}
diff --git a/src/math/pow_s390x.s b/src/math/pow_s390x.s
new file mode 100644
index 0000000..9a0fff3
--- /dev/null
+++ b/src/math/pow_s390x.s
@@ -0,0 +1,634 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define PosInf 0x7FF0000000000000
+#define NaN 0x7FF8000000000001
+#define NegInf 0xFFF0000000000000
+#define PosOne 0x3FF0000000000000
+#define NegOne 0xBFF0000000000000
+#define NegZero 0x8000000000000000
+
+// Minimax polynomial approximation
+DATA ·powrodataL51<> + 0(SB)/8, $-1.0
+DATA ·powrodataL51<> + 8(SB)/8, $1.0
+DATA ·powrodataL51<> + 16(SB)/8, $0.24022650695910110361E+00
+DATA ·powrodataL51<> + 24(SB)/8, $0.69314718055994686185E+00
+DATA ·powrodataL51<> + 32(SB)/8, $0.96181291057109484809E-02
+DATA ·powrodataL51<> + 40(SB)/8, $0.15403814778342868389E-03
+DATA ·powrodataL51<> + 48(SB)/8, $0.55504108652095235601E-01
+DATA ·powrodataL51<> + 56(SB)/8, $0.13333818813168698658E-02
+DATA ·powrodataL51<> + 64(SB)/8, $0.68205322933914439200E-12
+DATA ·powrodataL51<> + 72(SB)/8, $-.18466496523378731640E-01
+DATA ·powrodataL51<> + 80(SB)/8, $0.19697596291603973706E-02
+DATA ·powrodataL51<> + 88(SB)/8, $0.23083120654155209200E+00
+DATA ·powrodataL51<> + 96(SB)/8, $0.55324356012093416771E-06
+DATA ·powrodataL51<> + 104(SB)/8, $-.40340677224649339048E-05
+DATA ·powrodataL51<> + 112(SB)/8, $0.30255507904062541562E-04
+DATA ·powrodataL51<> + 120(SB)/8, $-.77453979912413008787E-07
+DATA ·powrodataL51<> + 128(SB)/8, $-.23637115549923464737E-03
+DATA ·powrodataL51<> + 136(SB)/8, $0.11016119077267717198E-07
+DATA ·powrodataL51<> + 144(SB)/8, $0.22608272174486123035E-09
+DATA ·powrodataL51<> + 152(SB)/8, $-.15895808101370190382E-08
+DATA ·powrodataL51<> + 160(SB)/8, $0x4540190000000000
+GLOBL ·powrodataL51<> + 0(SB), RODATA, $168
+
+// Constants
+DATA ·pow_x001a<> + 0(SB)/8, $0x1a000000000000
+GLOBL ·pow_x001a<> + 0(SB), RODATA, $8
+DATA ·pow_xinf<> + 0(SB)/8, $0x7ff0000000000000 //+Inf
+GLOBL ·pow_xinf<> + 0(SB), RODATA, $8
+DATA ·pow_xnan<> + 0(SB)/8, $0x7ff8000000000000 //NaN
+GLOBL ·pow_xnan<> + 0(SB), RODATA, $8
+DATA ·pow_x434<> + 0(SB)/8, $0x4340000000000000
+GLOBL ·pow_x434<> + 0(SB), RODATA, $8
+DATA ·pow_x433<> + 0(SB)/8, $0x4330000000000000
+GLOBL ·pow_x433<> + 0(SB), RODATA, $8
+DATA ·pow_x43f<> + 0(SB)/8, $0x43f0000000000000
+GLOBL ·pow_x43f<> + 0(SB), RODATA, $8
+DATA ·pow_xadd<> + 0(SB)/8, $0xc2f0000100003fef
+GLOBL ·pow_xadd<> + 0(SB), RODATA, $8
+DATA ·pow_xa<> + 0(SB)/8, $0x4019000000000000
+GLOBL ·pow_xa<> + 0(SB), RODATA, $8
+
+// Scale correction tables
+DATA powiadd<> + 0(SB)/8, $0xf000000000000000
+DATA powiadd<> + 8(SB)/8, $0x1000000000000000
+GLOBL powiadd<> + 0(SB), RODATA, $16
+DATA powxscale<> + 0(SB)/8, $0x4ff0000000000000
+DATA powxscale<> + 8(SB)/8, $0x2ff0000000000000
+GLOBL powxscale<> + 0(SB), RODATA, $16
+
+// Fractional powers of 2 table
+DATA ·powtexp<> + 0(SB)/8, $0.442737824274138381E-01
+DATA ·powtexp<> + 8(SB)/8, $0.263602189790660309E-01
+DATA ·powtexp<> + 16(SB)/8, $0.122565642281703586E-01
+DATA ·powtexp<> + 24(SB)/8, $0.143757052860721398E-02
+DATA ·powtexp<> + 32(SB)/8, $-.651375034121276075E-02
+DATA ·powtexp<> + 40(SB)/8, $-.119317678849450159E-01
+DATA ·powtexp<> + 48(SB)/8, $-.150868749549871069E-01
+DATA ·powtexp<> + 56(SB)/8, $-.161992609578469234E-01
+DATA ·powtexp<> + 64(SB)/8, $-.154492360403337917E-01
+DATA ·powtexp<> + 72(SB)/8, $-.129850717389178721E-01
+DATA ·powtexp<> + 80(SB)/8, $-.892902649276657891E-02
+DATA ·powtexp<> + 88(SB)/8, $-.338202636596794887E-02
+DATA ·powtexp<> + 96(SB)/8, $0.357266307045684762E-02
+DATA ·powtexp<> + 104(SB)/8, $0.118665304327406698E-01
+DATA ·powtexp<> + 112(SB)/8, $0.214434994118118914E-01
+DATA ·powtexp<> + 120(SB)/8, $0.322580645161290314E-01
+GLOBL ·powtexp<> + 0(SB), RODATA, $128
+
+// Log multiplier tables
+DATA ·powtl<> + 0(SB)/8, $0xbdf9723a80db6a05
+DATA ·powtl<> + 8(SB)/8, $0x3e0cfe4a0babe862
+DATA ·powtl<> + 16(SB)/8, $0xbe163b42dd33dada
+DATA ·powtl<> + 24(SB)/8, $0xbe0cdf9de2a8429c
+DATA ·powtl<> + 32(SB)/8, $0xbde9723a80db6a05
+DATA ·powtl<> + 40(SB)/8, $0xbdb37fcae081745e
+DATA ·powtl<> + 48(SB)/8, $0xbdd8b2f901ac662c
+DATA ·powtl<> + 56(SB)/8, $0xbde867dc68c36cc9
+DATA ·powtl<> + 64(SB)/8, $0xbdd23e36b47256b7
+DATA ·powtl<> + 72(SB)/8, $0xbde4c9b89fcc7933
+DATA ·powtl<> + 80(SB)/8, $0xbdd16905cad7cf66
+DATA ·powtl<> + 88(SB)/8, $0x3ddb417414aa5529
+DATA ·powtl<> + 96(SB)/8, $0xbdce046f2889983c
+DATA ·powtl<> + 104(SB)/8, $0x3dc2c3865d072897
+DATA ·powtl<> + 112(SB)/8, $0x8000000000000000
+DATA ·powtl<> + 120(SB)/8, $0x3dc1ca48817f8afe
+DATA ·powtl<> + 128(SB)/8, $0xbdd703518a88bfb7
+DATA ·powtl<> + 136(SB)/8, $0x3dc64afcc46942ce
+DATA ·powtl<> + 144(SB)/8, $0xbd9d79191389891a
+DATA ·powtl<> + 152(SB)/8, $0x3ddd563044da4fa0
+DATA ·powtl<> + 160(SB)/8, $0x3e0f42b5e5f8f4b6
+DATA ·powtl<> + 168(SB)/8, $0x3e0dfa2c2cbf6ead
+DATA ·powtl<> + 176(SB)/8, $0x3e14e25e91661293
+DATA ·powtl<> + 184(SB)/8, $0x3e0aac461509e20c
+GLOBL ·powtl<> + 0(SB), RODATA, $192
+
+DATA ·powtm<> + 0(SB)/8, $0x3da69e13
+DATA ·powtm<> + 8(SB)/8, $0x100003d66fcb6
+DATA ·powtm<> + 16(SB)/8, $0x200003d1538df
+DATA ·powtm<> + 24(SB)/8, $0x300003cab729e
+DATA ·powtm<> + 32(SB)/8, $0x400003c1a784c
+DATA ·powtm<> + 40(SB)/8, $0x500003ac9b074
+DATA ·powtm<> + 48(SB)/8, $0x60000bb498d22
+DATA ·powtm<> + 56(SB)/8, $0x68000bb8b29a2
+DATA ·powtm<> + 64(SB)/8, $0x70000bb9a32d4
+DATA ·powtm<> + 72(SB)/8, $0x74000bb9946bb
+DATA ·powtm<> + 80(SB)/8, $0x78000bb92e34b
+DATA ·powtm<> + 88(SB)/8, $0x80000bb6c57dc
+DATA ·powtm<> + 96(SB)/8, $0x84000bb4020f7
+DATA ·powtm<> + 104(SB)/8, $0x8c000ba93832d
+DATA ·powtm<> + 112(SB)/8, $0x9000080000000
+DATA ·powtm<> + 120(SB)/8, $0x940003aa66c4c
+DATA ·powtm<> + 128(SB)/8, $0x980003b2fb12a
+DATA ·powtm<> + 136(SB)/8, $0xa00003bc1def6
+DATA ·powtm<> + 144(SB)/8, $0xa80003c1eb0eb
+DATA ·powtm<> + 152(SB)/8, $0xb00003c64dcec
+DATA ·powtm<> + 160(SB)/8, $0xc00003cc49e4e
+DATA ·powtm<> + 168(SB)/8, $0xd00003d12f1de
+DATA ·powtm<> + 176(SB)/8, $0xe00003d4a9c6f
+DATA ·powtm<> + 184(SB)/8, $0xf00003d846c66
+GLOBL ·powtm<> + 0(SB), RODATA, $192
+
+// Table of indeces into multiplier tables
+// Adjusted from asm to remove offset and convert
+DATA ·powtabi<> + 0(SB)/8, $0x1010101
+DATA ·powtabi<> + 8(SB)/8, $0x101020202020203
+DATA ·powtabi<> + 16(SB)/8, $0x303030404040405
+DATA ·powtabi<> + 24(SB)/8, $0x505050606060708
+DATA ·powtabi<> + 32(SB)/8, $0x90a0b0c0d0e0f10
+DATA ·powtabi<> + 40(SB)/8, $0x1011111212121313
+DATA ·powtabi<> + 48(SB)/8, $0x1314141414151515
+DATA ·powtabi<> + 56(SB)/8, $0x1516161617171717
+GLOBL ·powtabi<> + 0(SB), RODATA, $64
+
+// Pow returns x**y, the base-x exponential of y.
+//
+// Special cases are (in order):
+// Pow(x, ±0) = 1 for any x
+// Pow(1, y) = 1 for any y
+// Pow(x, 1) = x for any x
+// Pow(NaN, y) = NaN
+// Pow(x, NaN) = NaN
+// Pow(±0, y) = ±Inf for y an odd integer < 0
+// Pow(±0, -Inf) = +Inf
+// Pow(±0, +Inf) = +0
+// Pow(±0, y) = +Inf for finite y < 0 and not an odd integer
+// Pow(±0, y) = ±0 for y an odd integer > 0
+// Pow(±0, y) = +0 for finite y > 0 and not an odd integer
+// Pow(-1, ±Inf) = 1
+// Pow(x, +Inf) = +Inf for |x| > 1
+// Pow(x, -Inf) = +0 for |x| > 1
+// Pow(x, +Inf) = +0 for |x| < 1
+// Pow(x, -Inf) = +Inf for |x| < 1
+// Pow(+Inf, y) = +Inf for y > 0
+// Pow(+Inf, y) = +0 for y < 0
+// Pow(-Inf, y) = Pow(-0, -y)
+// Pow(x, y) = NaN for finite x < 0 and finite non-integer y
+
+TEXT ·powAsm(SB), NOSPLIT, $0-24
+ // special case
+ MOVD x+0(FP), R1
+ MOVD y+8(FP), R2
+
+ // special case Pow(1, y) = 1 for any y
+ MOVD $PosOne, R3
+ CMPUBEQ R1, R3, xIsOne
+
+ // special case Pow(x, 1) = x for any x
+ MOVD $PosOne, R4
+ CMPUBEQ R2, R4, yIsOne
+
+ // special case Pow(x, NaN) = NaN for any x
+ MOVD $~(1<<63), R5
+ AND R2, R5 // y = |y|
+ MOVD $PosInf, R4
+ CMPUBLT R4, R5, yIsNan
+
+ MOVD $NegInf, R3
+ CMPUBEQ R1, R3, xIsNegInf
+
+ MOVD $NegOne, R3
+ CMPUBEQ R1, R3, xIsNegOne
+
+ MOVD $PosInf, R3
+ CMPUBEQ R1, R3, xIsPosInf
+
+ MOVD $NegZero, R3
+ CMPUBEQ R1, R3, xIsNegZero
+
+ MOVD $PosInf, R4
+ CMPUBEQ R2, R4, yIsPosInf
+
+ MOVD $0x0, R3
+ CMPUBEQ R1, R3, xIsPosZero
+ CMPBLT R1, R3, xLtZero
+ BR Normal
+xIsPosInf:
+ // special case Pow(+Inf, y) = +Inf for y > 0
+ MOVD $0x0, R4
+ CMPBGT R2, R4, posInfGeZero
+ BR Normal
+xIsNegInf:
+ //Pow(-Inf, y) = Pow(-0, -y)
+ FMOVD y+8(FP), F2
+ FNEG F2, F2 // y = -y
+ BR negZeroNegY // call Pow(-0, -y)
+xIsNegOne:
+ // special case Pow(-1, ±Inf) = 1
+ MOVD $PosInf, R4
+ CMPUBEQ R2, R4, negOnePosInf
+ MOVD $NegInf, R4
+ CMPUBEQ R2, R4, negOneNegInf
+ BR Normal
+xIsPosZero:
+ // special case Pow(+0, -Inf) = +Inf
+ MOVD $NegInf, R4
+ CMPUBEQ R2, R4, zeroNegInf
+
+ // special case Pow(+0, y < 0) = +Inf
+ FMOVD y+8(FP), F2
+ FMOVD $(0.0), F4
+ FCMPU F2, F4
+ BLT posZeroLtZero //y < 0.0
+ BR Normal
+xIsNegZero:
+ // special case Pow(-0, -Inf) = +Inf
+ MOVD $NegInf, R4
+ CMPUBEQ R2, R4, zeroNegInf
+ FMOVD y+8(FP), F2
+negZeroNegY:
+ // special case Pow(x, ±0) = 1 for any x
+ FMOVD $(0.0), F4
+ FCMPU F4, F2
+ BLT negZeroGtZero // y > 0.0
+ BEQ yIsZero // y = 0.0
+
+ FMOVD $(-0.0), F4
+ FCMPU F4, F2
+ BLT negZeroGtZero // y > -0.0
+ BEQ yIsZero // y = -0.0
+
+ // special case Pow(-0, y) = -Inf for y an odd integer < 0
+ // special case Pow(-0, y) = +Inf for finite y < 0 and not an odd integer
+ FIDBR $5, F2, F4 //F2 translate to integer F4
+ FCMPU F2, F4
+ BNE zeroNotOdd // y is not an (odd) integer and y < 0
+ FMOVD $(2.0), F4
+ FDIV F4, F2 // F2 = F2 / 2.0
+ FIDBR $5, F2, F4 //F2 translate to integer F4
+ FCMPU F2, F4
+ BNE negZeroOddInt // y is an odd integer and y < 0
+ BR zeroNotOdd // y is not an (odd) integer and y < 0
+
+negZeroGtZero:
+ // special case Pow(-0, y) = -0 for y an odd integer > 0
+ // special case Pow(±0, y) = +0 for finite y > 0 and not an odd integer
+ FIDBR $5, F2, F4 //F2 translate to integer F4
+ FCMPU F2, F4
+ BNE zeroNotOddGtZero // y is not an (odd) integer and y > 0
+ FMOVD $(2.0), F4
+ FDIV F4, F2 // F2 = F2 / 2.0
+ FIDBR $5, F2, F4 //F2 translate to integer F4
+ FCMPU F2, F4
+ BNE negZeroOddIntGtZero // y is an odd integer and y > 0
+ BR zeroNotOddGtZero // y is not an (odd) integer
+
+xLtZero:
+ // special case Pow(x, y) = NaN for finite x < 0 and finite non-integer y
+ FMOVD y+8(FP), F2
+ FIDBR $5, F2, F4
+ FCMPU F2, F4
+ BNE ltZeroInt
+ BR Normal
+yIsPosInf:
+ // special case Pow(x, +Inf) = +Inf for |x| > 1
+ FMOVD x+0(FP), F1
+ FMOVD $(1.0), F3
+ FCMPU F1, F3
+ BGT gtOnePosInf
+ FMOVD $(-1.0), F3
+ FCMPU F1, F3
+ BLT ltNegOnePosInf
+Normal:
+ FMOVD x+0(FP), F0
+ FMOVD y+8(FP), F2
+ MOVD $·powrodataL51<>+0(SB), R9
+ LGDR F0, R3
+ WORD $0xC0298009 //iilf %r2,2148095317
+ BYTE $0x55
+ BYTE $0x55
+ RISBGNZ $32, $63, $32, R3, R1
+ SUBW R1, R2
+ RISBGNZ $58, $63, $50, R2, R3
+ BYTE $0x18 //lr %r5,%r1
+ BYTE $0x51
+ MOVD $·powtabi<>+0(SB), R12
+ WORD $0xE303C000 //llgc %r0,0(%r3,%r12)
+ BYTE $0x00
+ BYTE $0x90
+ SUBW $0x1A0000, R5
+ SLD $3, R0, R3
+ MOVD $·powtm<>+0(SB), R4
+ MOVH $0x0, R8
+ ANDW $0x7FF00000, R2
+ ORW R5, R1
+ WORD $0x5A234000 //a %r2,0(%r3,%r4)
+ MOVD $0x3FF0000000000000, R5
+ RISBGZ $40, $63, $56, R2, R3
+ RISBGN $0, $31, $32, R2, R8
+ ORW $0x45000000, R3
+ MOVW R1, R6
+ CMPBLT R6, $0, L42
+ FMOVD F0, F4
+L2:
+ VLVGF $0, R3, V1
+ MOVD $·pow_xa<>+0(SB), R2
+ WORD $0xED3090A0 //lde %f3,.L52-.L51(%r9)
+ BYTE $0x00
+ BYTE $0x24
+ FMOVD 0(R2), F6
+ FSUBS F1, F3
+ LDGR R8, F1
+ WFMSDB V4, V1, V6, V4
+ FMOVD 152(R9), F6
+ WFMDB V4, V4, V7
+ FMOVD 144(R9), F1
+ FMOVD 136(R9), F5
+ WFMADB V4, V1, V6, V1
+ VLEG $0, 128(R9), V16
+ FMOVD 120(R9), F6
+ WFMADB V4, V5, V6, V5
+ FMOVD 112(R9), F6
+ WFMADB V1, V7, V5, V1
+ WFMADB V4, V6, V16, V16
+ SLD $3, R0, R2
+ FMOVD 104(R9), F5
+ WORD $0xED824004 //ldeb %f8,4(%r2,%r4)
+ BYTE $0x00
+ BYTE $0x04
+ LDEBR F3, F3
+ FMOVD 96(R9), F6
+ WFMADB V4, V6, V5, V6
+ FADD F8, F3
+ WFMADB V7, V6, V16, V6
+ FMUL F7, F7
+ FMOVD 88(R9), F5
+ FMADD F7, F1, F6
+ WFMADB V4, V5, V3, V16
+ FMOVD 80(R9), F1
+ WFSDB V16, V3, V3
+ MOVD $·powtl<>+0(SB), R3
+ WFMADB V4, V6, V1, V6
+ FMADD F5, F4, F3
+ FMOVD 72(R9), F1
+ WFMADB V4, V6, V1, V6
+ WORD $0xED323000 //adb %f3,0(%r2,%r3)
+ BYTE $0x00
+ BYTE $0x1A
+ FMOVD 64(R9), F1
+ WFMADB V4, V6, V1, V6
+ MOVD $·pow_xadd<>+0(SB), R2
+ WFMADB V4, V6, V3, V4
+ FMOVD 0(R2), F5
+ WFADB V4, V16, V3
+ VLEG $0, 56(R9), V20
+ WFMSDB V2, V3, V5, V3
+ VLEG $0, 48(R9), V18
+ WFADB V3, V5, V6
+ LGDR F3, R2
+ WFMSDB V2, V16, V6, V16
+ FMOVD 40(R9), F1
+ WFMADB V2, V4, V16, V4
+ FMOVD 32(R9), F7
+ WFMDB V4, V4, V3
+ WFMADB V4, V1, V20, V1
+ WFMADB V4, V7, V18, V7
+ VLEG $0, 24(R9), V16
+ WFMADB V1, V3, V7, V1
+ FMOVD 16(R9), F5
+ WFMADB V4, V5, V16, V5
+ RISBGZ $57, $60, $3, R2, R4
+ WFMADB V3, V1, V5, V1
+ MOVD $·powtexp<>+0(SB), R3
+ WORD $0x68343000 //ld %f3,0(%r4,%r3)
+ FMADD F3, F4, F4
+ RISBGN $0, $15, $48, R2, R5
+ WFMADB V4, V1, V3, V4
+ LGDR F6, R2
+ LDGR R5, F1
+ SRAD $48, R2, R2
+ FMADD F1, F4, F1
+ RLL $16, R2, R2
+ ANDW $0x7FFF0000, R2
+ WORD $0xC22B3F71 //alfi %r2,1064370176
+ BYTE $0x00
+ BYTE $0x00
+ ORW R2, R1, R3
+ MOVW R3, R6
+ CMPBLT R6, $0, L43
+L1:
+ FMOVD F1, ret+16(FP)
+ RET
+L43:
+ LTDBR F0, F0
+ BLTU L44
+ FMOVD F0, F3
+L7:
+ MOVD $·pow_xinf<>+0(SB), R3
+ FMOVD 0(R3), F5
+ WFCEDBS V3, V5, V7
+ BVS L8
+ WFMDB V3, V2, V6
+L8:
+ WFCEDBS V2, V2, V3
+ BVS L9
+ LTDBR F2, F2
+ BEQ L26
+ MOVW R1, R6
+ CMPBLT R6, $0, L45
+L11:
+ WORD $0xC0190003 //iilf %r1,262143
+ BYTE $0xFF
+ BYTE $0xFF
+ MOVW R2, R7
+ MOVW R1, R6
+ CMPBLE R7, R6, L34
+ RISBGNZ $32, $63, $32, R5, R1
+ LGDR F6, R2
+ MOVD $powiadd<>+0(SB), R3
+ RISBGZ $60, $60, $4, R2, R2
+ WORD $0x5A123000 //a %r1,0(%r2,%r3)
+ RISBGN $0, $31, $32, R1, R5
+ LDGR R5, F1
+ FMADD F1, F4, F1
+ MOVD $powxscale<>+0(SB), R1
+ WORD $0xED121000 //mdb %f1,0(%r2,%r1)
+ BYTE $0x00
+ BYTE $0x1C
+ BR L1
+L42:
+ LTDBR F0, F0
+ BLTU L46
+ FMOVD F0, F4
+L3:
+ MOVD $·pow_x001a<>+0(SB), R2
+ WORD $0xED402000 //cdb %f4,0(%r2)
+ BYTE $0x00
+ BYTE $0x19
+ BGE L2
+ BVS L2
+ MOVD $·pow_x43f<>+0(SB), R2
+ WORD $0xED402000 //mdb %f4,0(%r2)
+ BYTE $0x00
+ BYTE $0x1C
+ WORD $0xC0298009 //iilf %r2,2148095317
+ BYTE $0x55
+ BYTE $0x55
+ LGDR F4, R3
+ RISBGNZ $32, $63, $32, R3, R3
+ SUBW R3, R2, R3
+ RISBGZ $33, $43, $0, R3, R2
+ RISBGNZ $58, $63, $50, R3, R3
+ WORD $0xE303C000 //llgc %r0,0(%r3,%r12)
+ BYTE $0x00
+ BYTE $0x90
+ SLD $3, R0, R3
+ WORD $0x5A234000 //a %r2,0(%r3,%r4)
+ BYTE $0x18 //lr %r3,%r2
+ BYTE $0x32
+ RISBGN $0, $31, $32, R3, R8
+ ADDW $0x4000000, R3
+ BLEU L5
+ RISBGZ $40, $63, $56, R3, R3
+ ORW $0x45000000, R3
+ BR L2
+L9:
+ WFCEDBS V0, V0, V4
+ BVS L35
+ FMOVD F2, F1
+ BR L1
+L46:
+ WORD $0xB3130040 //lcdbr %f4,%f0
+ BR L3
+L44:
+ WORD $0xB3130030 //lcdbr %f3,%f0
+ BR L7
+L35:
+ FMOVD F0, F1
+ BR L1
+L26:
+ FMOVD 8(R9), F1
+ BR L1
+L34:
+ FMOVD 8(R9), F4
+L19:
+ LTDBR F6, F6
+ BLEU L47
+L18:
+ WFMDB V4, V5, V1
+ BR L1
+L5:
+ RISBGZ $33, $50, $63, R3, R3
+ WORD $0xC23B4000 //alfi %r3,1073741824
+ BYTE $0x00
+ BYTE $0x00
+ RLL $24, R3, R3
+ ORW $0x45000000, R3
+ BR L2
+L45:
+ WFCEDBS V0, V0, V4
+ BVS L35
+ LTDBR F0, F0
+ BLEU L48
+ FMOVD 8(R9), F4
+L12:
+ MOVW R2, R6
+ CMPBLT R6, $0, L19
+ FMUL F4, F1
+ BR L1
+L47:
+ BLT L40
+ WFCEDBS V0, V0, V2
+ BVS L49
+L16:
+ MOVD ·pow_xnan<>+0(SB), R1
+ LDGR R1, F0
+ WFMDB V4, V0, V1
+ BR L1
+L48:
+ LGDR F0, R3
+ RISBGNZ $32, $63, $32, R3, R1
+ MOVW R1, R6
+ CMPBEQ R6, $0, L29
+ LTDBR F2, F2
+ BLTU L50
+ FMOVD F2, F4
+L14:
+ MOVD $·pow_x433<>+0(SB), R1
+ FMOVD 0(R1), F7
+ WFCHDBS V4, V7, V3
+ BEQ L15
+ WFADB V7, V4, V3
+ FSUB F7, F3
+ WFCEDBS V4, V3, V3
+ BEQ L15
+ LTDBR F0, F0
+ FMOVD 8(R9), F4
+ BNE L16
+L13:
+ LTDBR F2, F2
+ BLT L18
+L40:
+ FMOVD $0, F0
+ WFMDB V4, V0, V1
+ BR L1
+L49:
+ WFMDB V0, V4, V1
+ BR L1
+L29:
+ FMOVD 8(R9), F4
+ BR L13
+L15:
+ MOVD $·pow_x434<>+0(SB), R1
+ FMOVD 0(R1), F7
+ WFCHDBS V4, V7, V3
+ BEQ L32
+ WFADB V7, V4, V3
+ FSUB F7, F3
+ WFCEDBS V4, V3, V4
+ BEQ L32
+ FMOVD 0(R9), F4
+L17:
+ LTDBR F0, F0
+ BNE L12
+ BR L13
+L32:
+ FMOVD 8(R9), F4
+ BR L17
+L50:
+ WORD $0xB3130042 //lcdbr %f4,%f2
+ BR L14
+xIsOne: // Pow(1, y) = 1 for any y
+yIsOne: // Pow(x, 1) = x for any x
+posInfGeZero: // Pow(+Inf, y) = +Inf for y > 0
+ MOVD R1, ret+16(FP)
+ RET
+yIsNan: // Pow(NaN, y) = NaN
+ltZeroInt: // Pow(x, y) = NaN for finite x < 0 and finite non-integer y
+ MOVD $NaN, R2
+ MOVD R2, ret+16(FP)
+ RET
+negOnePosInf: // Pow(-1, ±Inf) = 1
+negOneNegInf:
+ MOVD $PosOne, R3
+ MOVD R3, ret+16(FP)
+ RET
+negZeroOddInt:
+ MOVD $NegInf, R3
+ MOVD R3, ret+16(FP)
+ RET
+zeroNotOdd: // Pow(±0, y) = +Inf for finite y < 0 and not an odd integer
+posZeroLtZero: // special case Pow(+0, y < 0) = +Inf
+zeroNegInf: // Pow(±0, -Inf) = +Inf
+ MOVD $PosInf, R3
+ MOVD R3, ret+16(FP)
+ RET
+gtOnePosInf: //Pow(x, +Inf) = +Inf for |x| > 1
+ltNegOnePosInf:
+ MOVD R2, ret+16(FP)
+ RET
+yIsZero: //Pow(x, ±0) = 1 for any x
+ MOVD $PosOne, R4
+ MOVD R4, ret+16(FP)
+ RET
+negZeroOddIntGtZero: // Pow(-0, y) = -0 for y an odd integer > 0
+ MOVD $NegZero, R3
+ MOVD R3, ret+16(FP)
+ RET
+zeroNotOddGtZero: // Pow(±0, y) = +0 for finite y > 0 and not an odd integer
+ MOVD $0, ret+16(FP)
+ RET
diff --git a/src/math/rand/example_test.go b/src/math/rand/example_test.go
new file mode 100644
index 0000000..4107613
--- /dev/null
+++ b/src/math/rand/example_test.go
@@ -0,0 +1,157 @@
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand_test
+
+import (
+ "fmt"
+ "math/rand"
+ "os"
+ "strings"
+ "text/tabwriter"
+)
+
+// These tests serve as an example but also make sure we don't change
+// the output of the random number generator when given a fixed seed.
+
+func Example() {
+ // Seeding with the same value results in the same random sequence each run.
+ // For different numbers, seed with a different value, such as
+ // time.Now().UnixNano(), which yields a constantly-changing number.
+ rand.Seed(42)
+ answers := []string{
+ "It is certain",
+ "It is decidedly so",
+ "Without a doubt",
+ "Yes definitely",
+ "You may rely on it",
+ "As I see it yes",
+ "Most likely",
+ "Outlook good",
+ "Yes",
+ "Signs point to yes",
+ "Reply hazy try again",
+ "Ask again later",
+ "Better not tell you now",
+ "Cannot predict now",
+ "Concentrate and ask again",
+ "Don't count on it",
+ "My reply is no",
+ "My sources say no",
+ "Outlook not so good",
+ "Very doubtful",
+ }
+ fmt.Println("Magic 8-Ball says:", answers[rand.Intn(len(answers))])
+ // Output: Magic 8-Ball says: As I see it yes
+}
+
+// This example shows the use of each of the methods on a *Rand.
+// The use of the global functions is the same, without the receiver.
+func Example_rand() {
+ // Create and seed the generator.
+ // Typically a non-fixed seed should be used, such as time.Now().UnixNano().
+ // Using a fixed seed will produce the same output on every run.
+ r := rand.New(rand.NewSource(99))
+
+ // The tabwriter here helps us generate aligned output.
+ w := tabwriter.NewWriter(os.Stdout, 1, 1, 1, ' ', 0)
+ defer w.Flush()
+ show := func(name string, v1, v2, v3 interface{}) {
+ fmt.Fprintf(w, "%s\t%v\t%v\t%v\n", name, v1, v2, v3)
+ }
+
+ // Float32 and Float64 values are in [0, 1).
+ show("Float32", r.Float32(), r.Float32(), r.Float32())
+ show("Float64", r.Float64(), r.Float64(), r.Float64())
+
+ // ExpFloat64 values have an average of 1 but decay exponentially.
+ show("ExpFloat64", r.ExpFloat64(), r.ExpFloat64(), r.ExpFloat64())
+
+ // NormFloat64 values have an average of 0 and a standard deviation of 1.
+ show("NormFloat64", r.NormFloat64(), r.NormFloat64(), r.NormFloat64())
+
+ // Int31, Int63, and Uint32 generate values of the given width.
+ // The Int method (not shown) is like either Int31 or Int63
+ // depending on the size of 'int'.
+ show("Int31", r.Int31(), r.Int31(), r.Int31())
+ show("Int63", r.Int63(), r.Int63(), r.Int63())
+ show("Uint32", r.Uint32(), r.Uint32(), r.Uint32())
+
+ // Intn, Int31n, and Int63n limit their output to be < n.
+ // They do so more carefully than using r.Int()%n.
+ show("Intn(10)", r.Intn(10), r.Intn(10), r.Intn(10))
+ show("Int31n(10)", r.Int31n(10), r.Int31n(10), r.Int31n(10))
+ show("Int63n(10)", r.Int63n(10), r.Int63n(10), r.Int63n(10))
+
+ // Perm generates a random permutation of the numbers [0, n).
+ show("Perm", r.Perm(5), r.Perm(5), r.Perm(5))
+ // Output:
+ // Float32 0.2635776 0.6358173 0.6718283
+ // Float64 0.628605430454327 0.4504798828572669 0.9562755949377957
+ // ExpFloat64 0.3362240648200941 1.4256072328483647 0.24354758816173044
+ // NormFloat64 0.17233959114940064 1.577014951434847 0.04259129641113857
+ // Int31 1501292890 1486668269 182840835
+ // Int63 3546343826724305832 5724354148158589552 5239846799706671610
+ // Uint32 2760229429 296659907 1922395059
+ // Intn(10) 1 2 5
+ // Int31n(10) 4 7 8
+ // Int63n(10) 7 6 3
+ // Perm [1 4 2 3 0] [4 2 1 3 0] [1 2 4 0 3]
+}
+
+func ExamplePerm() {
+ for _, value := range rand.Perm(3) {
+ fmt.Println(value)
+ }
+
+ // Unordered output: 1
+ // 2
+ // 0
+}
+
+func ExampleShuffle() {
+ words := strings.Fields("ink runs from the corners of my mouth")
+ rand.Shuffle(len(words), func(i, j int) {
+ words[i], words[j] = words[j], words[i]
+ })
+ fmt.Println(words)
+
+ // Output:
+ // [mouth my the of runs corners from ink]
+}
+
+func ExampleShuffle_slicesInUnison() {
+ numbers := []byte("12345")
+ letters := []byte("ABCDE")
+ // Shuffle numbers, swapping corresponding entries in letters at the same time.
+ rand.Shuffle(len(numbers), func(i, j int) {
+ numbers[i], numbers[j] = numbers[j], numbers[i]
+ letters[i], letters[j] = letters[j], letters[i]
+ })
+ for i := range numbers {
+ fmt.Printf("%c: %c\n", letters[i], numbers[i])
+ }
+
+ // Output:
+ // C: 3
+ // D: 4
+ // A: 1
+ // E: 5
+ // B: 2
+}
+
+func ExampleIntn() {
+ // Seeding with the same value results in the same random sequence each run.
+ // For different numbers, seed with a different value, such as
+ // time.Now().UnixNano(), which yields a constantly-changing number.
+ rand.Seed(86)
+ fmt.Println(rand.Intn(100))
+ fmt.Println(rand.Intn(100))
+ fmt.Println(rand.Intn(100))
+
+ // Output:
+ // 42
+ // 76
+ // 30
+}
diff --git a/src/math/rand/exp.go b/src/math/rand/exp.go
new file mode 100644
index 0000000..5a8d946
--- /dev/null
+++ b/src/math/rand/exp.go
@@ -0,0 +1,222 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand
+
+import (
+ "math"
+)
+
+/*
+ * Exponential distribution
+ *
+ * See "The Ziggurat Method for Generating Random Variables"
+ * (Marsaglia & Tsang, 2000)
+ * https://www.jstatsoft.org/v05/i08/paper [pdf]
+ */
+
+const (
+ re = 7.69711747013104972
+)
+
+// ExpFloat64 returns an exponentially distributed float64 in the range
+// (0, +math.MaxFloat64] with an exponential distribution whose rate parameter
+// (lambda) is 1 and whose mean is 1/lambda (1).
+// To produce a distribution with a different rate parameter,
+// callers can adjust the output using:
+//
+// sample = ExpFloat64() / desiredRateParameter
+//
+func (r *Rand) ExpFloat64() float64 {
+ for {
+ j := r.Uint32()
+ i := j & 0xFF
+ x := float64(j) * float64(we[i])
+ if j < ke[i] {
+ return x
+ }
+ if i == 0 {
+ return re - math.Log(r.Float64())
+ }
+ if fe[i]+float32(r.Float64())*(fe[i-1]-fe[i]) < float32(math.Exp(-x)) {
+ return x
+ }
+ }
+}
+
+var ke = [256]uint32{
+ 0xe290a139, 0x0, 0x9beadebc, 0xc377ac71, 0xd4ddb990,
+ 0xde893fb8, 0xe4a8e87c, 0xe8dff16a, 0xebf2deab, 0xee49a6e8,
+ 0xf0204efd, 0xf19bdb8e, 0xf2d458bb, 0xf3da104b, 0xf4b86d78,
+ 0xf577ad8a, 0xf61de83d, 0xf6afb784, 0xf730a573, 0xf7a37651,
+ 0xf80a5bb6, 0xf867189d, 0xf8bb1b4f, 0xf9079062, 0xf94d70ca,
+ 0xf98d8c7d, 0xf9c8928a, 0xf9ff175b, 0xfa319996, 0xfa6085f8,
+ 0xfa8c3a62, 0xfab5084e, 0xfadb36c8, 0xfaff0410, 0xfb20a6ea,
+ 0xfb404fb4, 0xfb5e2951, 0xfb7a59e9, 0xfb95038c, 0xfbae44ba,
+ 0xfbc638d8, 0xfbdcf892, 0xfbf29a30, 0xfc0731df, 0xfc1ad1ed,
+ 0xfc2d8b02, 0xfc3f6c4d, 0xfc5083ac, 0xfc60ddd1, 0xfc708662,
+ 0xfc7f8810, 0xfc8decb4, 0xfc9bbd62, 0xfca9027c, 0xfcb5c3c3,
+ 0xfcc20864, 0xfccdd70a, 0xfcd935e3, 0xfce42ab0, 0xfceebace,
+ 0xfcf8eb3b, 0xfd02c0a0, 0xfd0c3f59, 0xfd156b7b, 0xfd1e48d6,
+ 0xfd26daff, 0xfd2f2552, 0xfd372af7, 0xfd3eeee5, 0xfd4673e7,
+ 0xfd4dbc9e, 0xfd54cb85, 0xfd5ba2f2, 0xfd62451b, 0xfd68b415,
+ 0xfd6ef1da, 0xfd750047, 0xfd7ae120, 0xfd809612, 0xfd8620b4,
+ 0xfd8b8285, 0xfd90bcf5, 0xfd95d15e, 0xfd9ac10b, 0xfd9f8d36,
+ 0xfda43708, 0xfda8bf9e, 0xfdad2806, 0xfdb17141, 0xfdb59c46,
+ 0xfdb9a9fd, 0xfdbd9b46, 0xfdc170f6, 0xfdc52bd8, 0xfdc8ccac,
+ 0xfdcc542d, 0xfdcfc30b, 0xfdd319ef, 0xfdd6597a, 0xfdd98245,
+ 0xfddc94e5, 0xfddf91e6, 0xfde279ce, 0xfde54d1f, 0xfde80c52,
+ 0xfdeab7de, 0xfded5034, 0xfdefd5be, 0xfdf248e3, 0xfdf4aa06,
+ 0xfdf6f984, 0xfdf937b6, 0xfdfb64f4, 0xfdfd818d, 0xfdff8dd0,
+ 0xfe018a08, 0xfe03767a, 0xfe05536c, 0xfe07211c, 0xfe08dfc9,
+ 0xfe0a8fab, 0xfe0c30fb, 0xfe0dc3ec, 0xfe0f48b1, 0xfe10bf76,
+ 0xfe122869, 0xfe1383b4, 0xfe14d17c, 0xfe1611e7, 0xfe174516,
+ 0xfe186b2a, 0xfe19843e, 0xfe1a9070, 0xfe1b8fd6, 0xfe1c8289,
+ 0xfe1d689b, 0xfe1e4220, 0xfe1f0f26, 0xfe1fcfbc, 0xfe2083ed,
+ 0xfe212bc3, 0xfe21c745, 0xfe225678, 0xfe22d95f, 0xfe234ffb,
+ 0xfe23ba4a, 0xfe241849, 0xfe2469f2, 0xfe24af3c, 0xfe24e81e,
+ 0xfe25148b, 0xfe253474, 0xfe2547c7, 0xfe254e70, 0xfe25485a,
+ 0xfe25356a, 0xfe251586, 0xfe24e88f, 0xfe24ae64, 0xfe2466e1,
+ 0xfe2411df, 0xfe23af34, 0xfe233eb4, 0xfe22c02c, 0xfe22336b,
+ 0xfe219838, 0xfe20ee58, 0xfe20358c, 0xfe1f6d92, 0xfe1e9621,
+ 0xfe1daef0, 0xfe1cb7ac, 0xfe1bb002, 0xfe1a9798, 0xfe196e0d,
+ 0xfe1832fd, 0xfe16e5fe, 0xfe15869d, 0xfe141464, 0xfe128ed3,
+ 0xfe10f565, 0xfe0f478c, 0xfe0d84b1, 0xfe0bac36, 0xfe09bd73,
+ 0xfe07b7b5, 0xfe059a40, 0xfe03644c, 0xfe011504, 0xfdfeab88,
+ 0xfdfc26e9, 0xfdf98629, 0xfdf6c83b, 0xfdf3ec01, 0xfdf0f04a,
+ 0xfdedd3d1, 0xfdea953d, 0xfde7331e, 0xfde3abe9, 0xfddffdfb,
+ 0xfddc2791, 0xfdd826cd, 0xfdd3f9a8, 0xfdcf9dfc, 0xfdcb1176,
+ 0xfdc65198, 0xfdc15bb3, 0xfdbc2ce2, 0xfdb6c206, 0xfdb117be,
+ 0xfdab2a63, 0xfda4f5fd, 0xfd9e7640, 0xfd97a67a, 0xfd908192,
+ 0xfd8901f2, 0xfd812182, 0xfd78d98e, 0xfd7022bb, 0xfd66f4ed,
+ 0xfd5d4732, 0xfd530f9c, 0xfd48432b, 0xfd3cd59a, 0xfd30b936,
+ 0xfd23dea4, 0xfd16349e, 0xfd07a7a3, 0xfcf8219b, 0xfce7895b,
+ 0xfcd5c220, 0xfcc2aadb, 0xfcae1d5e, 0xfc97ed4e, 0xfc7fe6d4,
+ 0xfc65ccf3, 0xfc495762, 0xfc2a2fc8, 0xfc07ee19, 0xfbe213c1,
+ 0xfbb8051a, 0xfb890078, 0xfb5411a5, 0xfb180005, 0xfad33482,
+ 0xfa839276, 0xfa263b32, 0xf9b72d1c, 0xf930a1a2, 0xf889f023,
+ 0xf7b577d2, 0xf69c650c, 0xf51530f0, 0xf2cb0e3c, 0xeeefb15d,
+ 0xe6da6ecf,
+}
+var we = [256]float32{
+ 2.0249555e-09, 1.486674e-11, 2.4409617e-11, 3.1968806e-11,
+ 3.844677e-11, 4.4228204e-11, 4.9516443e-11, 5.443359e-11,
+ 5.905944e-11, 6.344942e-11, 6.7643814e-11, 7.1672945e-11,
+ 7.556032e-11, 7.932458e-11, 8.298079e-11, 8.654132e-11,
+ 9.0016515e-11, 9.3415074e-11, 9.674443e-11, 1.0001099e-10,
+ 1.03220314e-10, 1.06377254e-10, 1.09486115e-10, 1.1255068e-10,
+ 1.1557435e-10, 1.1856015e-10, 1.2151083e-10, 1.2442886e-10,
+ 1.2731648e-10, 1.3017575e-10, 1.3300853e-10, 1.3581657e-10,
+ 1.3860142e-10, 1.4136457e-10, 1.4410738e-10, 1.4683108e-10,
+ 1.4953687e-10, 1.5222583e-10, 1.54899e-10, 1.5755733e-10,
+ 1.6020171e-10, 1.6283301e-10, 1.6545203e-10, 1.6805951e-10,
+ 1.7065617e-10, 1.732427e-10, 1.7581973e-10, 1.7838787e-10,
+ 1.8094774e-10, 1.8349985e-10, 1.8604476e-10, 1.8858298e-10,
+ 1.9111498e-10, 1.9364126e-10, 1.9616223e-10, 1.9867835e-10,
+ 2.0119004e-10, 2.0369768e-10, 2.0620168e-10, 2.087024e-10,
+ 2.1120022e-10, 2.136955e-10, 2.1618855e-10, 2.1867974e-10,
+ 2.2116936e-10, 2.2365775e-10, 2.261452e-10, 2.2863202e-10,
+ 2.311185e-10, 2.3360494e-10, 2.360916e-10, 2.3857874e-10,
+ 2.4106667e-10, 2.4355562e-10, 2.4604588e-10, 2.485377e-10,
+ 2.5103128e-10, 2.5352695e-10, 2.560249e-10, 2.585254e-10,
+ 2.6102867e-10, 2.6353494e-10, 2.6604446e-10, 2.6855745e-10,
+ 2.7107416e-10, 2.7359479e-10, 2.761196e-10, 2.7864877e-10,
+ 2.8118255e-10, 2.8372119e-10, 2.8626485e-10, 2.888138e-10,
+ 2.9136826e-10, 2.939284e-10, 2.9649452e-10, 2.9906677e-10,
+ 3.016454e-10, 3.0423064e-10, 3.0682268e-10, 3.0942177e-10,
+ 3.1202813e-10, 3.1464195e-10, 3.1726352e-10, 3.19893e-10,
+ 3.2253064e-10, 3.251767e-10, 3.2783135e-10, 3.3049485e-10,
+ 3.3316744e-10, 3.3584938e-10, 3.3854083e-10, 3.4124212e-10,
+ 3.4395342e-10, 3.46675e-10, 3.4940711e-10, 3.5215003e-10,
+ 3.5490397e-10, 3.5766917e-10, 3.6044595e-10, 3.6323455e-10,
+ 3.660352e-10, 3.6884823e-10, 3.7167386e-10, 3.745124e-10,
+ 3.773641e-10, 3.802293e-10, 3.8310827e-10, 3.860013e-10,
+ 3.8890866e-10, 3.918307e-10, 3.9476775e-10, 3.9772008e-10,
+ 4.0068804e-10, 4.0367196e-10, 4.0667217e-10, 4.09689e-10,
+ 4.1272286e-10, 4.1577405e-10, 4.1884296e-10, 4.2192994e-10,
+ 4.250354e-10, 4.281597e-10, 4.313033e-10, 4.3446652e-10,
+ 4.3764986e-10, 4.408537e-10, 4.4407847e-10, 4.4732465e-10,
+ 4.5059267e-10, 4.5388301e-10, 4.571962e-10, 4.6053267e-10,
+ 4.6389292e-10, 4.6727755e-10, 4.70687e-10, 4.741219e-10,
+ 4.7758275e-10, 4.810702e-10, 4.845848e-10, 4.8812715e-10,
+ 4.9169796e-10, 4.9529775e-10, 4.989273e-10, 5.0258725e-10,
+ 5.0627835e-10, 5.100013e-10, 5.1375687e-10, 5.1754584e-10,
+ 5.21369e-10, 5.2522725e-10, 5.2912136e-10, 5.330522e-10,
+ 5.370208e-10, 5.4102806e-10, 5.45075e-10, 5.491625e-10,
+ 5.532918e-10, 5.5746385e-10, 5.616799e-10, 5.6594107e-10,
+ 5.7024857e-10, 5.746037e-10, 5.7900773e-10, 5.834621e-10,
+ 5.8796823e-10, 5.925276e-10, 5.971417e-10, 6.018122e-10,
+ 6.065408e-10, 6.113292e-10, 6.1617933e-10, 6.2109295e-10,
+ 6.260722e-10, 6.3111916e-10, 6.3623595e-10, 6.4142497e-10,
+ 6.4668854e-10, 6.5202926e-10, 6.5744976e-10, 6.6295286e-10,
+ 6.6854156e-10, 6.742188e-10, 6.79988e-10, 6.858526e-10,
+ 6.9181616e-10, 6.978826e-10, 7.04056e-10, 7.103407e-10,
+ 7.167412e-10, 7.2326256e-10, 7.2990985e-10, 7.366886e-10,
+ 7.4360473e-10, 7.5066453e-10, 7.5787476e-10, 7.6524265e-10,
+ 7.7277595e-10, 7.80483e-10, 7.883728e-10, 7.9645507e-10,
+ 8.047402e-10, 8.1323964e-10, 8.219657e-10, 8.309319e-10,
+ 8.401528e-10, 8.496445e-10, 8.594247e-10, 8.6951274e-10,
+ 8.799301e-10, 8.9070046e-10, 9.018503e-10, 9.134092e-10,
+ 9.254101e-10, 9.378904e-10, 9.508923e-10, 9.644638e-10,
+ 9.786603e-10, 9.935448e-10, 1.0091913e-09, 1.025686e-09,
+ 1.0431306e-09, 1.0616465e-09, 1.08138e-09, 1.1025096e-09,
+ 1.1252564e-09, 1.1498986e-09, 1.1767932e-09, 1.206409e-09,
+ 1.2393786e-09, 1.276585e-09, 1.3193139e-09, 1.3695435e-09,
+ 1.4305498e-09, 1.508365e-09, 1.6160854e-09, 1.7921248e-09,
+}
+var fe = [256]float32{
+ 1, 0.9381437, 0.90046996, 0.87170434, 0.8477855, 0.8269933,
+ 0.8084217, 0.7915276, 0.77595687, 0.7614634, 0.7478686,
+ 0.7350381, 0.72286767, 0.71127474, 0.70019263, 0.6895665,
+ 0.67935055, 0.6695063, 0.66000086, 0.65080583, 0.6418967,
+ 0.63325197, 0.6248527, 0.6166822, 0.60872537, 0.60096896,
+ 0.5934009, 0.58601034, 0.5787874, 0.57172304, 0.5648092,
+ 0.5580383, 0.5514034, 0.5448982, 0.5385169, 0.53225386,
+ 0.5261042, 0.52006316, 0.5141264, 0.50828975, 0.5025495,
+ 0.496902, 0.49134386, 0.485872, 0.48048335, 0.4751752,
+ 0.46994483, 0.46478975, 0.45970762, 0.45469615, 0.44975325,
+ 0.44487688, 0.44006512, 0.43531612, 0.43062815, 0.42599955,
+ 0.42142874, 0.4169142, 0.41245446, 0.40804818, 0.403694,
+ 0.3993907, 0.39513698, 0.39093173, 0.38677382, 0.38266218,
+ 0.37859577, 0.37457356, 0.37059465, 0.3666581, 0.362763,
+ 0.35890847, 0.35509375, 0.351318, 0.3475805, 0.34388044,
+ 0.34021714, 0.3365899, 0.33299807, 0.32944095, 0.32591796,
+ 0.3224285, 0.3189719, 0.31554767, 0.31215525, 0.30879408,
+ 0.3054636, 0.3021634, 0.29889292, 0.2956517, 0.29243928,
+ 0.28925523, 0.28609908, 0.28297043, 0.27986884, 0.27679393,
+ 0.2737453, 0.2707226, 0.2677254, 0.26475343, 0.26180625,
+ 0.25888354, 0.25598502, 0.2531103, 0.25025907, 0.24743107,
+ 0.24462597, 0.24184346, 0.23908329, 0.23634516, 0.23362878,
+ 0.23093392, 0.2282603, 0.22560766, 0.22297576, 0.22036438,
+ 0.21777324, 0.21520215, 0.21265087, 0.21011916, 0.20760682,
+ 0.20511365, 0.20263945, 0.20018397, 0.19774707, 0.19532852,
+ 0.19292815, 0.19054577, 0.1881812, 0.18583426, 0.18350479,
+ 0.1811926, 0.17889754, 0.17661946, 0.17435817, 0.17211354,
+ 0.1698854, 0.16767362, 0.16547804, 0.16329853, 0.16113494,
+ 0.15898713, 0.15685499, 0.15473837, 0.15263714, 0.15055119,
+ 0.14848037, 0.14642459, 0.14438373, 0.14235765, 0.14034624,
+ 0.13834943, 0.13636707, 0.13439907, 0.13244532, 0.13050574,
+ 0.1285802, 0.12666863, 0.12477092, 0.12288698, 0.12101672,
+ 0.119160056, 0.1173169, 0.115487166, 0.11367077, 0.11186763,
+ 0.11007768, 0.10830083, 0.10653701, 0.10478614, 0.10304816,
+ 0.101323, 0.09961058, 0.09791085, 0.09622374, 0.09454919,
+ 0.09288713, 0.091237515, 0.08960028, 0.087975375, 0.08636274,
+ 0.08476233, 0.083174095, 0.081597984, 0.08003395, 0.07848195,
+ 0.076941945, 0.07541389, 0.07389775, 0.072393484, 0.07090106,
+ 0.069420435, 0.06795159, 0.066494495, 0.06504912, 0.063615434,
+ 0.062193416, 0.060783047, 0.059384305, 0.057997175,
+ 0.05662164, 0.05525769, 0.053905312, 0.052564494, 0.051235236,
+ 0.049917534, 0.048611384, 0.047316793, 0.046033762, 0.0447623,
+ 0.043502413, 0.042254124, 0.041017443, 0.039792392,
+ 0.038578995, 0.037377283, 0.036187284, 0.035009038,
+ 0.033842582, 0.032687962, 0.031545233, 0.030414443, 0.02929566,
+ 0.02818895, 0.027094385, 0.026012046, 0.024942026, 0.023884421,
+ 0.022839336, 0.021806888, 0.020787204, 0.019780423, 0.0187867,
+ 0.0178062, 0.016839107, 0.015885621, 0.014945968, 0.014020392,
+ 0.013109165, 0.012212592, 0.011331013, 0.01046481, 0.009614414,
+ 0.008780315, 0.007963077, 0.0071633533, 0.006381906,
+ 0.0056196423, 0.0048776558, 0.004157295, 0.0034602648,
+ 0.0027887989, 0.0021459677, 0.0015362998, 0.0009672693,
+ 0.00045413437,
+}
diff --git a/src/math/rand/gen_cooked.go b/src/math/rand/gen_cooked.go
new file mode 100644
index 0000000..0afc10d
--- /dev/null
+++ b/src/math/rand/gen_cooked.go
@@ -0,0 +1,89 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build ignore
+
+// This program computes the value of rngCooked in rng.go,
+// which is used for seeding all instances of rand.Source.
+// a 64bit and a 63bit version of the array is printed to
+// the standard output.
+
+package main
+
+import "fmt"
+
+const (
+ length = 607
+ tap = 273
+ mask = (1 << 63) - 1
+ a = 48271
+ m = (1 << 31) - 1
+ q = 44488
+ r = 3399
+)
+
+var (
+ rngVec [length]int64
+ rngTap, rngFeed int
+)
+
+func seedrand(x int32) int32 {
+ hi := x / q
+ lo := x % q
+ x = a*lo - r*hi
+ if x < 0 {
+ x += m
+ }
+ return x
+}
+
+func srand(seed int32) {
+ rngTap = 0
+ rngFeed = length - tap
+ seed %= m
+ if seed < 0 {
+ seed += m
+ } else if seed == 0 {
+ seed = 89482311
+ }
+ x := seed
+ for i := -20; i < length; i++ {
+ x = seedrand(x)
+ if i >= 0 {
+ var u int64
+ u = int64(x) << 20
+ x = seedrand(x)
+ u ^= int64(x) << 10
+ x = seedrand(x)
+ u ^= int64(x)
+ rngVec[i] = u
+ }
+ }
+}
+
+func vrand() int64 {
+ rngTap--
+ if rngTap < 0 {
+ rngTap += length
+ }
+ rngFeed--
+ if rngFeed < 0 {
+ rngFeed += length
+ }
+ x := (rngVec[rngFeed] + rngVec[rngTap])
+ rngVec[rngFeed] = x
+ return x
+}
+
+func main() {
+ srand(1)
+ for i := uint64(0); i < 7.8e12; i++ {
+ vrand()
+ }
+ fmt.Printf("rngVec after 7.8e12 calls to vrand:\n%#v\n", rngVec)
+ for i := range rngVec {
+ rngVec[i] &= mask
+ }
+ fmt.Printf("lower 63bit of rngVec after 7.8e12 calls to vrand:\n%#v\n", rngVec)
+}
diff --git a/src/math/rand/normal.go b/src/math/rand/normal.go
new file mode 100644
index 0000000..2c5a7aa
--- /dev/null
+++ b/src/math/rand/normal.go
@@ -0,0 +1,157 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand
+
+import (
+ "math"
+)
+
+/*
+ * Normal distribution
+ *
+ * See "The Ziggurat Method for Generating Random Variables"
+ * (Marsaglia & Tsang, 2000)
+ * http://www.jstatsoft.org/v05/i08/paper [pdf]
+ */
+
+const (
+ rn = 3.442619855899
+)
+
+func absInt32(i int32) uint32 {
+ if i < 0 {
+ return uint32(-i)
+ }
+ return uint32(i)
+}
+
+// NormFloat64 returns a normally distributed float64 in
+// the range -math.MaxFloat64 through +math.MaxFloat64 inclusive,
+// with standard normal distribution (mean = 0, stddev = 1).
+// To produce a different normal distribution, callers can
+// adjust the output using:
+//
+// sample = NormFloat64() * desiredStdDev + desiredMean
+//
+func (r *Rand) NormFloat64() float64 {
+ for {
+ j := int32(r.Uint32()) // Possibly negative
+ i := j & 0x7F
+ x := float64(j) * float64(wn[i])
+ if absInt32(j) < kn[i] {
+ // This case should be hit better than 99% of the time.
+ return x
+ }
+
+ if i == 0 {
+ // This extra work is only required for the base strip.
+ for {
+ x = -math.Log(r.Float64()) * (1.0 / rn)
+ y := -math.Log(r.Float64())
+ if y+y >= x*x {
+ break
+ }
+ }
+ if j > 0 {
+ return rn + x
+ }
+ return -rn - x
+ }
+ if fn[i]+float32(r.Float64())*(fn[i-1]-fn[i]) < float32(math.Exp(-.5*x*x)) {
+ return x
+ }
+ }
+}
+
+var kn = [128]uint32{
+ 0x76ad2212, 0x0, 0x600f1b53, 0x6ce447a6, 0x725b46a2,
+ 0x7560051d, 0x774921eb, 0x789a25bd, 0x799045c3, 0x7a4bce5d,
+ 0x7adf629f, 0x7b5682a6, 0x7bb8a8c6, 0x7c0ae722, 0x7c50cce7,
+ 0x7c8cec5b, 0x7cc12cd6, 0x7ceefed2, 0x7d177e0b, 0x7d3b8883,
+ 0x7d5bce6c, 0x7d78dd64, 0x7d932886, 0x7dab0e57, 0x7dc0dd30,
+ 0x7dd4d688, 0x7de73185, 0x7df81cea, 0x7e07c0a3, 0x7e163efa,
+ 0x7e23b587, 0x7e303dfd, 0x7e3beec2, 0x7e46db77, 0x7e51155d,
+ 0x7e5aabb3, 0x7e63abf7, 0x7e6c222c, 0x7e741906, 0x7e7b9a18,
+ 0x7e82adfa, 0x7e895c63, 0x7e8fac4b, 0x7e95a3fb, 0x7e9b4924,
+ 0x7ea0a0ef, 0x7ea5b00d, 0x7eaa7ac3, 0x7eaf04f3, 0x7eb3522a,
+ 0x7eb765a5, 0x7ebb4259, 0x7ebeeafd, 0x7ec2620a, 0x7ec5a9c4,
+ 0x7ec8c441, 0x7ecbb365, 0x7ece78ed, 0x7ed11671, 0x7ed38d62,
+ 0x7ed5df12, 0x7ed80cb4, 0x7eda175c, 0x7edc0005, 0x7eddc78e,
+ 0x7edf6ebf, 0x7ee0f647, 0x7ee25ebe, 0x7ee3a8a9, 0x7ee4d473,
+ 0x7ee5e276, 0x7ee6d2f5, 0x7ee7a620, 0x7ee85c10, 0x7ee8f4cd,
+ 0x7ee97047, 0x7ee9ce59, 0x7eea0eca, 0x7eea3147, 0x7eea3568,
+ 0x7eea1aab, 0x7ee9e071, 0x7ee98602, 0x7ee90a88, 0x7ee86d08,
+ 0x7ee7ac6a, 0x7ee6c769, 0x7ee5bc9c, 0x7ee48a67, 0x7ee32efc,
+ 0x7ee1a857, 0x7edff42f, 0x7ede0ffa, 0x7edbf8d9, 0x7ed9ab94,
+ 0x7ed7248d, 0x7ed45fae, 0x7ed1585c, 0x7ece095f, 0x7eca6ccb,
+ 0x7ec67be2, 0x7ec22eee, 0x7ebd7d1a, 0x7eb85c35, 0x7eb2c075,
+ 0x7eac9c20, 0x7ea5df27, 0x7e9e769f, 0x7e964c16, 0x7e8d44ba,
+ 0x7e834033, 0x7e781728, 0x7e6b9933, 0x7e5d8a1a, 0x7e4d9ded,
+ 0x7e3b737a, 0x7e268c2f, 0x7e0e3ff5, 0x7df1aa5d, 0x7dcf8c72,
+ 0x7da61a1e, 0x7d72a0fb, 0x7d30e097, 0x7cd9b4ab, 0x7c600f1a,
+ 0x7ba90bdc, 0x7a722176, 0x77d664e5,
+}
+var wn = [128]float32{
+ 1.7290405e-09, 1.2680929e-10, 1.6897518e-10, 1.9862688e-10,
+ 2.2232431e-10, 2.4244937e-10, 2.601613e-10, 2.7611988e-10,
+ 2.9073963e-10, 3.042997e-10, 3.1699796e-10, 3.289802e-10,
+ 3.4035738e-10, 3.5121603e-10, 3.616251e-10, 3.7164058e-10,
+ 3.8130857e-10, 3.9066758e-10, 3.9975012e-10, 4.08584e-10,
+ 4.1719309e-10, 4.2559822e-10, 4.338176e-10, 4.418672e-10,
+ 4.497613e-10, 4.5751258e-10, 4.651324e-10, 4.7263105e-10,
+ 4.8001775e-10, 4.87301e-10, 4.944885e-10, 5.015873e-10,
+ 5.0860405e-10, 5.155446e-10, 5.2241467e-10, 5.2921934e-10,
+ 5.359635e-10, 5.426517e-10, 5.4928817e-10, 5.5587696e-10,
+ 5.624219e-10, 5.6892646e-10, 5.753941e-10, 5.818282e-10,
+ 5.882317e-10, 5.946077e-10, 6.00959e-10, 6.072884e-10,
+ 6.135985e-10, 6.19892e-10, 6.2617134e-10, 6.3243905e-10,
+ 6.386974e-10, 6.449488e-10, 6.511956e-10, 6.5744005e-10,
+ 6.6368433e-10, 6.699307e-10, 6.7618144e-10, 6.824387e-10,
+ 6.8870465e-10, 6.949815e-10, 7.012715e-10, 7.075768e-10,
+ 7.1389966e-10, 7.202424e-10, 7.266073e-10, 7.329966e-10,
+ 7.394128e-10, 7.4585826e-10, 7.5233547e-10, 7.58847e-10,
+ 7.653954e-10, 7.719835e-10, 7.7861395e-10, 7.852897e-10,
+ 7.920138e-10, 7.987892e-10, 8.0561924e-10, 8.125073e-10,
+ 8.194569e-10, 8.2647167e-10, 8.3355556e-10, 8.407127e-10,
+ 8.479473e-10, 8.55264e-10, 8.6266755e-10, 8.7016316e-10,
+ 8.777562e-10, 8.8545243e-10, 8.932582e-10, 9.0117996e-10,
+ 9.09225e-10, 9.174008e-10, 9.2571584e-10, 9.341788e-10,
+ 9.427997e-10, 9.515889e-10, 9.605579e-10, 9.697193e-10,
+ 9.790869e-10, 9.88676e-10, 9.985036e-10, 1.0085882e-09,
+ 1.0189509e-09, 1.0296151e-09, 1.0406069e-09, 1.0519566e-09,
+ 1.063698e-09, 1.0758702e-09, 1.0885183e-09, 1.1016947e-09,
+ 1.1154611e-09, 1.1298902e-09, 1.1450696e-09, 1.1611052e-09,
+ 1.1781276e-09, 1.1962995e-09, 1.2158287e-09, 1.2369856e-09,
+ 1.2601323e-09, 1.2857697e-09, 1.3146202e-09, 1.347784e-09,
+ 1.3870636e-09, 1.4357403e-09, 1.5008659e-09, 1.6030948e-09,
+}
+var fn = [128]float32{
+ 1, 0.9635997, 0.9362827, 0.9130436, 0.89228165, 0.87324303,
+ 0.8555006, 0.8387836, 0.8229072, 0.8077383, 0.793177,
+ 0.7791461, 0.7655842, 0.7524416, 0.73967725, 0.7272569,
+ 0.7151515, 0.7033361, 0.69178915, 0.68049186, 0.6694277,
+ 0.658582, 0.6479418, 0.63749546, 0.6272325, 0.6171434,
+ 0.6072195, 0.5974532, 0.58783704, 0.5783647, 0.56903,
+ 0.5598274, 0.5507518, 0.54179835, 0.5329627, 0.52424055,
+ 0.5156282, 0.50712204, 0.49871865, 0.49041483, 0.48220766,
+ 0.4740943, 0.46607214, 0.4581387, 0.45029163, 0.44252872,
+ 0.43484783, 0.427247, 0.41972435, 0.41227803, 0.40490642,
+ 0.39760786, 0.3903808, 0.3832238, 0.37613547, 0.36911446,
+ 0.3621595, 0.35526937, 0.34844297, 0.34167916, 0.33497685,
+ 0.3283351, 0.3217529, 0.3152294, 0.30876362, 0.30235484,
+ 0.29600215, 0.28970486, 0.2834622, 0.2772735, 0.27113807,
+ 0.2650553, 0.25902456, 0.2530453, 0.24711695, 0.241239,
+ 0.23541094, 0.22963232, 0.2239027, 0.21822165, 0.21258877,
+ 0.20700371, 0.20146611, 0.19597565, 0.19053204, 0.18513499,
+ 0.17978427, 0.17447963, 0.1692209, 0.16400786, 0.15884037,
+ 0.15371831, 0.14864157, 0.14361008, 0.13862377, 0.13368265,
+ 0.12878671, 0.12393598, 0.119130544, 0.11437051, 0.10965602,
+ 0.104987256, 0.10036444, 0.095787846, 0.0912578, 0.08677467,
+ 0.0823389, 0.077950984, 0.073611505, 0.06932112, 0.06508058,
+ 0.06089077, 0.056752663, 0.0526674, 0.048636295, 0.044660863,
+ 0.040742867, 0.03688439, 0.033087887, 0.029356318,
+ 0.025693292, 0.022103304, 0.018592102, 0.015167298,
+ 0.011839478, 0.008624485, 0.005548995, 0.0026696292,
+}
diff --git a/src/math/rand/race_test.go b/src/math/rand/race_test.go
new file mode 100644
index 0000000..186c716
--- /dev/null
+++ b/src/math/rand/race_test.go
@@ -0,0 +1,48 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand
+
+import (
+ "sync"
+ "testing"
+)
+
+// TestConcurrent exercises the rand API concurrently, triggering situations
+// where the race detector is likely to detect issues.
+func TestConcurrent(t *testing.T) {
+ const (
+ numRoutines = 10
+ numCycles = 10
+ )
+ var wg sync.WaitGroup
+ defer wg.Wait()
+ wg.Add(numRoutines)
+ for i := 0; i < numRoutines; i++ {
+ go func(i int) {
+ defer wg.Done()
+ buf := make([]byte, 997)
+ for j := 0; j < numCycles; j++ {
+ var seed int64
+ seed += int64(ExpFloat64())
+ seed += int64(Float32())
+ seed += int64(Float64())
+ seed += int64(Intn(Int()))
+ seed += int64(Int31n(Int31()))
+ seed += int64(Int63n(Int63()))
+ seed += int64(NormFloat64())
+ seed += int64(Uint32())
+ seed += int64(Uint64())
+ for _, p := range Perm(10) {
+ seed += int64(p)
+ }
+ Read(buf)
+ for _, b := range buf {
+ seed += int64(b)
+ }
+ Seed(int64(i*j) * seed)
+ }
+ }(i)
+ }
+}
diff --git a/src/math/rand/rand.go b/src/math/rand/rand.go
new file mode 100644
index 0000000..d6422c9
--- /dev/null
+++ b/src/math/rand/rand.go
@@ -0,0 +1,421 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package rand implements pseudo-random number generators.
+//
+// Random numbers are generated by a Source. Top-level functions, such as
+// Float64 and Int, use a default shared Source that produces a deterministic
+// sequence of values each time a program is run. Use the Seed function to
+// initialize the default Source if different behavior is required for each run.
+// The default Source is safe for concurrent use by multiple goroutines, but
+// Sources created by NewSource are not.
+//
+// Mathematical interval notation such as [0, n) is used throughout the
+// documentation for this package.
+//
+// For random numbers suitable for security-sensitive work, see the crypto/rand
+// package.
+package rand
+
+import "sync"
+
+// A Source represents a source of uniformly-distributed
+// pseudo-random int64 values in the range [0, 1<<63).
+type Source interface {
+ Int63() int64
+ Seed(seed int64)
+}
+
+// A Source64 is a Source that can also generate
+// uniformly-distributed pseudo-random uint64 values in
+// the range [0, 1<<64) directly.
+// If a Rand r's underlying Source s implements Source64,
+// then r.Uint64 returns the result of one call to s.Uint64
+// instead of making two calls to s.Int63.
+type Source64 interface {
+ Source
+ Uint64() uint64
+}
+
+// NewSource returns a new pseudo-random Source seeded with the given value.
+// Unlike the default Source used by top-level functions, this source is not
+// safe for concurrent use by multiple goroutines.
+func NewSource(seed int64) Source {
+ var rng rngSource
+ rng.Seed(seed)
+ return &rng
+}
+
+// A Rand is a source of random numbers.
+type Rand struct {
+ src Source
+ s64 Source64 // non-nil if src is source64
+
+ // readVal contains remainder of 63-bit integer used for bytes
+ // generation during most recent Read call.
+ // It is saved so next Read call can start where the previous
+ // one finished.
+ readVal int64
+ // readPos indicates the number of low-order bytes of readVal
+ // that are still valid.
+ readPos int8
+}
+
+// New returns a new Rand that uses random values from src
+// to generate other random values.
+func New(src Source) *Rand {
+ s64, _ := src.(Source64)
+ return &Rand{src: src, s64: s64}
+}
+
+// Seed uses the provided seed value to initialize the generator to a deterministic state.
+// Seed should not be called concurrently with any other Rand method.
+func (r *Rand) Seed(seed int64) {
+ if lk, ok := r.src.(*lockedSource); ok {
+ lk.seedPos(seed, &r.readPos)
+ return
+ }
+
+ r.src.Seed(seed)
+ r.readPos = 0
+}
+
+// Int63 returns a non-negative pseudo-random 63-bit integer as an int64.
+func (r *Rand) Int63() int64 { return r.src.Int63() }
+
+// Uint32 returns a pseudo-random 32-bit value as a uint32.
+func (r *Rand) Uint32() uint32 { return uint32(r.Int63() >> 31) }
+
+// Uint64 returns a pseudo-random 64-bit value as a uint64.
+func (r *Rand) Uint64() uint64 {
+ if r.s64 != nil {
+ return r.s64.Uint64()
+ }
+ return uint64(r.Int63())>>31 | uint64(r.Int63())<<32
+}
+
+// Int31 returns a non-negative pseudo-random 31-bit integer as an int32.
+func (r *Rand) Int31() int32 { return int32(r.Int63() >> 32) }
+
+// Int returns a non-negative pseudo-random int.
+func (r *Rand) Int() int {
+ u := uint(r.Int63())
+ return int(u << 1 >> 1) // clear sign bit if int == int32
+}
+
+// Int63n returns, as an int64, a non-negative pseudo-random number in [0,n).
+// It panics if n <= 0.
+func (r *Rand) Int63n(n int64) int64 {
+ if n <= 0 {
+ panic("invalid argument to Int63n")
+ }
+ if n&(n-1) == 0 { // n is power of two, can mask
+ return r.Int63() & (n - 1)
+ }
+ max := int64((1 << 63) - 1 - (1<<63)%uint64(n))
+ v := r.Int63()
+ for v > max {
+ v = r.Int63()
+ }
+ return v % n
+}
+
+// Int31n returns, as an int32, a non-negative pseudo-random number in [0,n).
+// It panics if n <= 0.
+func (r *Rand) Int31n(n int32) int32 {
+ if n <= 0 {
+ panic("invalid argument to Int31n")
+ }
+ if n&(n-1) == 0 { // n is power of two, can mask
+ return r.Int31() & (n - 1)
+ }
+ max := int32((1 << 31) - 1 - (1<<31)%uint32(n))
+ v := r.Int31()
+ for v > max {
+ v = r.Int31()
+ }
+ return v % n
+}
+
+// int31n returns, as an int32, a non-negative pseudo-random number in [0,n).
+// n must be > 0, but int31n does not check this; the caller must ensure it.
+// int31n exists because Int31n is inefficient, but Go 1 compatibility
+// requires that the stream of values produced by math/rand remain unchanged.
+// int31n can thus only be used internally, by newly introduced APIs.
+//
+// For implementation details, see:
+// https://lemire.me/blog/2016/06/27/a-fast-alternative-to-the-modulo-reduction
+// https://lemire.me/blog/2016/06/30/fast-random-shuffling
+func (r *Rand) int31n(n int32) int32 {
+ v := r.Uint32()
+ prod := uint64(v) * uint64(n)
+ low := uint32(prod)
+ if low < uint32(n) {
+ thresh := uint32(-n) % uint32(n)
+ for low < thresh {
+ v = r.Uint32()
+ prod = uint64(v) * uint64(n)
+ low = uint32(prod)
+ }
+ }
+ return int32(prod >> 32)
+}
+
+// Intn returns, as an int, a non-negative pseudo-random number in [0,n).
+// It panics if n <= 0.
+func (r *Rand) Intn(n int) int {
+ if n <= 0 {
+ panic("invalid argument to Intn")
+ }
+ if n <= 1<<31-1 {
+ return int(r.Int31n(int32(n)))
+ }
+ return int(r.Int63n(int64(n)))
+}
+
+// Float64 returns, as a float64, a pseudo-random number in [0.0,1.0).
+func (r *Rand) Float64() float64 {
+ // A clearer, simpler implementation would be:
+ // return float64(r.Int63n(1<<53)) / (1<<53)
+ // However, Go 1 shipped with
+ // return float64(r.Int63()) / (1 << 63)
+ // and we want to preserve that value stream.
+ //
+ // There is one bug in the value stream: r.Int63() may be so close
+ // to 1<<63 that the division rounds up to 1.0, and we've guaranteed
+ // that the result is always less than 1.0.
+ //
+ // We tried to fix this by mapping 1.0 back to 0.0, but since float64
+ // values near 0 are much denser than near 1, mapping 1 to 0 caused
+ // a theoretically significant overshoot in the probability of returning 0.
+ // Instead of that, if we round up to 1, just try again.
+ // Getting 1 only happens 1/2⁵³ of the time, so most clients
+ // will not observe it anyway.
+again:
+ f := float64(r.Int63()) / (1 << 63)
+ if f == 1 {
+ goto again // resample; this branch is taken O(never)
+ }
+ return f
+}
+
+// Float32 returns, as a float32, a pseudo-random number in [0.0,1.0).
+func (r *Rand) Float32() float32 {
+ // Same rationale as in Float64: we want to preserve the Go 1 value
+ // stream except we want to fix it not to return 1.0
+ // This only happens 1/2²⁴ of the time (plus the 1/2⁵³ of the time in Float64).
+again:
+ f := float32(r.Float64())
+ if f == 1 {
+ goto again // resample; this branch is taken O(very rarely)
+ }
+ return f
+}
+
+// Perm returns, as a slice of n ints, a pseudo-random permutation of the integers [0,n).
+func (r *Rand) Perm(n int) []int {
+ m := make([]int, n)
+ // In the following loop, the iteration when i=0 always swaps m[0] with m[0].
+ // A change to remove this useless iteration is to assign 1 to i in the init
+ // statement. But Perm also effects r. Making this change will affect
+ // the final state of r. So this change can't be made for compatibility
+ // reasons for Go 1.
+ for i := 0; i < n; i++ {
+ j := r.Intn(i + 1)
+ m[i] = m[j]
+ m[j] = i
+ }
+ return m
+}
+
+// Shuffle pseudo-randomizes the order of elements.
+// n is the number of elements. Shuffle panics if n < 0.
+// swap swaps the elements with indexes i and j.
+func (r *Rand) Shuffle(n int, swap func(i, j int)) {
+ if n < 0 {
+ panic("invalid argument to Shuffle")
+ }
+
+ // Fisher-Yates shuffle: https://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle
+ // Shuffle really ought not be called with n that doesn't fit in 32 bits.
+ // Not only will it take a very long time, but with 2³¹! possible permutations,
+ // there's no way that any PRNG can have a big enough internal state to
+ // generate even a minuscule percentage of the possible permutations.
+ // Nevertheless, the right API signature accepts an int n, so handle it as best we can.
+ i := n - 1
+ for ; i > 1<<31-1-1; i-- {
+ j := int(r.Int63n(int64(i + 1)))
+ swap(i, j)
+ }
+ for ; i > 0; i-- {
+ j := int(r.int31n(int32(i + 1)))
+ swap(i, j)
+ }
+}
+
+// Read generates len(p) random bytes and writes them into p. It
+// always returns len(p) and a nil error.
+// Read should not be called concurrently with any other Rand method.
+func (r *Rand) Read(p []byte) (n int, err error) {
+ if lk, ok := r.src.(*lockedSource); ok {
+ return lk.read(p, &r.readVal, &r.readPos)
+ }
+ return read(p, r.src, &r.readVal, &r.readPos)
+}
+
+func read(p []byte, src Source, readVal *int64, readPos *int8) (n int, err error) {
+ pos := *readPos
+ val := *readVal
+ rng, _ := src.(*rngSource)
+ for n = 0; n < len(p); n++ {
+ if pos == 0 {
+ if rng != nil {
+ val = rng.Int63()
+ } else {
+ val = src.Int63()
+ }
+ pos = 7
+ }
+ p[n] = byte(val)
+ val >>= 8
+ pos--
+ }
+ *readPos = pos
+ *readVal = val
+ return
+}
+
+/*
+ * Top-level convenience functions
+ */
+
+var globalRand = New(&lockedSource{src: NewSource(1).(*rngSource)})
+
+// Type assert that globalRand's source is a lockedSource whose src is a *rngSource.
+var _ *rngSource = globalRand.src.(*lockedSource).src
+
+// Seed uses the provided seed value to initialize the default Source to a
+// deterministic state. If Seed is not called, the generator behaves as
+// if seeded by Seed(1). Seed values that have the same remainder when
+// divided by 2³¹-1 generate the same pseudo-random sequence.
+// Seed, unlike the Rand.Seed method, is safe for concurrent use.
+func Seed(seed int64) { globalRand.Seed(seed) }
+
+// Int63 returns a non-negative pseudo-random 63-bit integer as an int64
+// from the default Source.
+func Int63() int64 { return globalRand.Int63() }
+
+// Uint32 returns a pseudo-random 32-bit value as a uint32
+// from the default Source.
+func Uint32() uint32 { return globalRand.Uint32() }
+
+// Uint64 returns a pseudo-random 64-bit value as a uint64
+// from the default Source.
+func Uint64() uint64 { return globalRand.Uint64() }
+
+// Int31 returns a non-negative pseudo-random 31-bit integer as an int32
+// from the default Source.
+func Int31() int32 { return globalRand.Int31() }
+
+// Int returns a non-negative pseudo-random int from the default Source.
+func Int() int { return globalRand.Int() }
+
+// Int63n returns, as an int64, a non-negative pseudo-random number in [0,n)
+// from the default Source.
+// It panics if n <= 0.
+func Int63n(n int64) int64 { return globalRand.Int63n(n) }
+
+// Int31n returns, as an int32, a non-negative pseudo-random number in [0,n)
+// from the default Source.
+// It panics if n <= 0.
+func Int31n(n int32) int32 { return globalRand.Int31n(n) }
+
+// Intn returns, as an int, a non-negative pseudo-random number in [0,n)
+// from the default Source.
+// It panics if n <= 0.
+func Intn(n int) int { return globalRand.Intn(n) }
+
+// Float64 returns, as a float64, a pseudo-random number in [0.0,1.0)
+// from the default Source.
+func Float64() float64 { return globalRand.Float64() }
+
+// Float32 returns, as a float32, a pseudo-random number in [0.0,1.0)
+// from the default Source.
+func Float32() float32 { return globalRand.Float32() }
+
+// Perm returns, as a slice of n ints, a pseudo-random permutation of the integers [0,n)
+// from the default Source.
+func Perm(n int) []int { return globalRand.Perm(n) }
+
+// Shuffle pseudo-randomizes the order of elements using the default Source.
+// n is the number of elements. Shuffle panics if n < 0.
+// swap swaps the elements with indexes i and j.
+func Shuffle(n int, swap func(i, j int)) { globalRand.Shuffle(n, swap) }
+
+// Read generates len(p) random bytes from the default Source and
+// writes them into p. It always returns len(p) and a nil error.
+// Read, unlike the Rand.Read method, is safe for concurrent use.
+func Read(p []byte) (n int, err error) { return globalRand.Read(p) }
+
+// NormFloat64 returns a normally distributed float64 in the range
+// [-math.MaxFloat64, +math.MaxFloat64] with
+// standard normal distribution (mean = 0, stddev = 1)
+// from the default Source.
+// To produce a different normal distribution, callers can
+// adjust the output using:
+//
+// sample = NormFloat64() * desiredStdDev + desiredMean
+//
+func NormFloat64() float64 { return globalRand.NormFloat64() }
+
+// ExpFloat64 returns an exponentially distributed float64 in the range
+// (0, +math.MaxFloat64] with an exponential distribution whose rate parameter
+// (lambda) is 1 and whose mean is 1/lambda (1) from the default Source.
+// To produce a distribution with a different rate parameter,
+// callers can adjust the output using:
+//
+// sample = ExpFloat64() / desiredRateParameter
+//
+func ExpFloat64() float64 { return globalRand.ExpFloat64() }
+
+type lockedSource struct {
+ lk sync.Mutex
+ src *rngSource
+}
+
+func (r *lockedSource) Int63() (n int64) {
+ r.lk.Lock()
+ n = r.src.Int63()
+ r.lk.Unlock()
+ return
+}
+
+func (r *lockedSource) Uint64() (n uint64) {
+ r.lk.Lock()
+ n = r.src.Uint64()
+ r.lk.Unlock()
+ return
+}
+
+func (r *lockedSource) Seed(seed int64) {
+ r.lk.Lock()
+ r.src.Seed(seed)
+ r.lk.Unlock()
+}
+
+// seedPos implements Seed for a lockedSource without a race condition.
+func (r *lockedSource) seedPos(seed int64, readPos *int8) {
+ r.lk.Lock()
+ r.src.Seed(seed)
+ *readPos = 0
+ r.lk.Unlock()
+}
+
+// read implements Read for a lockedSource without a race condition.
+func (r *lockedSource) read(p []byte, readVal *int64, readPos *int8) (n int, err error) {
+ r.lk.Lock()
+ n, err = read(p, r.src, readVal, readPos)
+ r.lk.Unlock()
+ return
+}
diff --git a/src/math/rand/rand_test.go b/src/math/rand/rand_test.go
new file mode 100644
index 0000000..e037aae
--- /dev/null
+++ b/src/math/rand/rand_test.go
@@ -0,0 +1,681 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand
+
+import (
+ "bytes"
+ "errors"
+ "fmt"
+ "internal/testenv"
+ "io"
+ "math"
+ "os"
+ "runtime"
+ "testing"
+ "testing/iotest"
+)
+
+const (
+ numTestSamples = 10000
+)
+
+type statsResults struct {
+ mean float64
+ stddev float64
+ closeEnough float64
+ maxError float64
+}
+
+func max(a, b float64) float64 {
+ if a > b {
+ return a
+ }
+ return b
+}
+
+func nearEqual(a, b, closeEnough, maxError float64) bool {
+ absDiff := math.Abs(a - b)
+ if absDiff < closeEnough { // Necessary when one value is zero and one value is close to zero.
+ return true
+ }
+ return absDiff/max(math.Abs(a), math.Abs(b)) < maxError
+}
+
+var testSeeds = []int64{1, 1754801282, 1698661970, 1550503961}
+
+// checkSimilarDistribution returns success if the mean and stddev of the
+// two statsResults are similar.
+func (this *statsResults) checkSimilarDistribution(expected *statsResults) error {
+ if !nearEqual(this.mean, expected.mean, expected.closeEnough, expected.maxError) {
+ s := fmt.Sprintf("mean %v != %v (allowed error %v, %v)", this.mean, expected.mean, expected.closeEnough, expected.maxError)
+ fmt.Println(s)
+ return errors.New(s)
+ }
+ if !nearEqual(this.stddev, expected.stddev, expected.closeEnough, expected.maxError) {
+ s := fmt.Sprintf("stddev %v != %v (allowed error %v, %v)", this.stddev, expected.stddev, expected.closeEnough, expected.maxError)
+ fmt.Println(s)
+ return errors.New(s)
+ }
+ return nil
+}
+
+func getStatsResults(samples []float64) *statsResults {
+ res := new(statsResults)
+ var sum, squaresum float64
+ for _, s := range samples {
+ sum += s
+ squaresum += s * s
+ }
+ res.mean = sum / float64(len(samples))
+ res.stddev = math.Sqrt(squaresum/float64(len(samples)) - res.mean*res.mean)
+ return res
+}
+
+func checkSampleDistribution(t *testing.T, samples []float64, expected *statsResults) {
+ t.Helper()
+ actual := getStatsResults(samples)
+ err := actual.checkSimilarDistribution(expected)
+ if err != nil {
+ t.Errorf(err.Error())
+ }
+}
+
+func checkSampleSliceDistributions(t *testing.T, samples []float64, nslices int, expected *statsResults) {
+ t.Helper()
+ chunk := len(samples) / nslices
+ for i := 0; i < nslices; i++ {
+ low := i * chunk
+ var high int
+ if i == nslices-1 {
+ high = len(samples) - 1
+ } else {
+ high = (i + 1) * chunk
+ }
+ checkSampleDistribution(t, samples[low:high], expected)
+ }
+}
+
+//
+// Normal distribution tests
+//
+
+func generateNormalSamples(nsamples int, mean, stddev float64, seed int64) []float64 {
+ r := New(NewSource(seed))
+ samples := make([]float64, nsamples)
+ for i := range samples {
+ samples[i] = r.NormFloat64()*stddev + mean
+ }
+ return samples
+}
+
+func testNormalDistribution(t *testing.T, nsamples int, mean, stddev float64, seed int64) {
+ //fmt.Printf("testing nsamples=%v mean=%v stddev=%v seed=%v\n", nsamples, mean, stddev, seed);
+
+ samples := generateNormalSamples(nsamples, mean, stddev, seed)
+ errorScale := max(1.0, stddev) // Error scales with stddev
+ expected := &statsResults{mean, stddev, 0.10 * errorScale, 0.08 * errorScale}
+
+ // Make sure that the entire set matches the expected distribution.
+ checkSampleDistribution(t, samples, expected)
+
+ // Make sure that each half of the set matches the expected distribution.
+ checkSampleSliceDistributions(t, samples, 2, expected)
+
+ // Make sure that each 7th of the set matches the expected distribution.
+ checkSampleSliceDistributions(t, samples, 7, expected)
+}
+
+// Actual tests
+
+func TestStandardNormalValues(t *testing.T) {
+ for _, seed := range testSeeds {
+ testNormalDistribution(t, numTestSamples, 0, 1, seed)
+ }
+}
+
+func TestNonStandardNormalValues(t *testing.T) {
+ sdmax := 1000.0
+ mmax := 1000.0
+ if testing.Short() {
+ sdmax = 5
+ mmax = 5
+ }
+ for sd := 0.5; sd < sdmax; sd *= 2 {
+ for m := 0.5; m < mmax; m *= 2 {
+ for _, seed := range testSeeds {
+ testNormalDistribution(t, numTestSamples, m, sd, seed)
+ if testing.Short() {
+ break
+ }
+ }
+ }
+ }
+}
+
+//
+// Exponential distribution tests
+//
+
+func generateExponentialSamples(nsamples int, rate float64, seed int64) []float64 {
+ r := New(NewSource(seed))
+ samples := make([]float64, nsamples)
+ for i := range samples {
+ samples[i] = r.ExpFloat64() / rate
+ }
+ return samples
+}
+
+func testExponentialDistribution(t *testing.T, nsamples int, rate float64, seed int64) {
+ //fmt.Printf("testing nsamples=%v rate=%v seed=%v\n", nsamples, rate, seed);
+
+ mean := 1 / rate
+ stddev := mean
+
+ samples := generateExponentialSamples(nsamples, rate, seed)
+ errorScale := max(1.0, 1/rate) // Error scales with the inverse of the rate
+ expected := &statsResults{mean, stddev, 0.10 * errorScale, 0.20 * errorScale}
+
+ // Make sure that the entire set matches the expected distribution.
+ checkSampleDistribution(t, samples, expected)
+
+ // Make sure that each half of the set matches the expected distribution.
+ checkSampleSliceDistributions(t, samples, 2, expected)
+
+ // Make sure that each 7th of the set matches the expected distribution.
+ checkSampleSliceDistributions(t, samples, 7, expected)
+}
+
+// Actual tests
+
+func TestStandardExponentialValues(t *testing.T) {
+ for _, seed := range testSeeds {
+ testExponentialDistribution(t, numTestSamples, 1, seed)
+ }
+}
+
+func TestNonStandardExponentialValues(t *testing.T) {
+ for rate := 0.05; rate < 10; rate *= 2 {
+ for _, seed := range testSeeds {
+ testExponentialDistribution(t, numTestSamples, rate, seed)
+ if testing.Short() {
+ break
+ }
+ }
+ }
+}
+
+//
+// Table generation tests
+//
+
+func initNorm() (testKn []uint32, testWn, testFn []float32) {
+ const m1 = 1 << 31
+ var (
+ dn float64 = rn
+ tn = dn
+ vn float64 = 9.91256303526217e-3
+ )
+
+ testKn = make([]uint32, 128)
+ testWn = make([]float32, 128)
+ testFn = make([]float32, 128)
+
+ q := vn / math.Exp(-0.5*dn*dn)
+ testKn[0] = uint32((dn / q) * m1)
+ testKn[1] = 0
+ testWn[0] = float32(q / m1)
+ testWn[127] = float32(dn / m1)
+ testFn[0] = 1.0
+ testFn[127] = float32(math.Exp(-0.5 * dn * dn))
+ for i := 126; i >= 1; i-- {
+ dn = math.Sqrt(-2.0 * math.Log(vn/dn+math.Exp(-0.5*dn*dn)))
+ testKn[i+1] = uint32((dn / tn) * m1)
+ tn = dn
+ testFn[i] = float32(math.Exp(-0.5 * dn * dn))
+ testWn[i] = float32(dn / m1)
+ }
+ return
+}
+
+func initExp() (testKe []uint32, testWe, testFe []float32) {
+ const m2 = 1 << 32
+ var (
+ de float64 = re
+ te = de
+ ve float64 = 3.9496598225815571993e-3
+ )
+
+ testKe = make([]uint32, 256)
+ testWe = make([]float32, 256)
+ testFe = make([]float32, 256)
+
+ q := ve / math.Exp(-de)
+ testKe[0] = uint32((de / q) * m2)
+ testKe[1] = 0
+ testWe[0] = float32(q / m2)
+ testWe[255] = float32(de / m2)
+ testFe[0] = 1.0
+ testFe[255] = float32(math.Exp(-de))
+ for i := 254; i >= 1; i-- {
+ de = -math.Log(ve/de + math.Exp(-de))
+ testKe[i+1] = uint32((de / te) * m2)
+ te = de
+ testFe[i] = float32(math.Exp(-de))
+ testWe[i] = float32(de / m2)
+ }
+ return
+}
+
+// compareUint32Slices returns the first index where the two slices
+// disagree, or <0 if the lengths are the same and all elements
+// are identical.
+func compareUint32Slices(s1, s2 []uint32) int {
+ if len(s1) != len(s2) {
+ if len(s1) > len(s2) {
+ return len(s2) + 1
+ }
+ return len(s1) + 1
+ }
+ for i := range s1 {
+ if s1[i] != s2[i] {
+ return i
+ }
+ }
+ return -1
+}
+
+// compareFloat32Slices returns the first index where the two slices
+// disagree, or <0 if the lengths are the same and all elements
+// are identical.
+func compareFloat32Slices(s1, s2 []float32) int {
+ if len(s1) != len(s2) {
+ if len(s1) > len(s2) {
+ return len(s2) + 1
+ }
+ return len(s1) + 1
+ }
+ for i := range s1 {
+ if !nearEqual(float64(s1[i]), float64(s2[i]), 0, 1e-7) {
+ return i
+ }
+ }
+ return -1
+}
+
+func TestNormTables(t *testing.T) {
+ testKn, testWn, testFn := initNorm()
+ if i := compareUint32Slices(kn[0:], testKn); i >= 0 {
+ t.Errorf("kn disagrees at index %v; %v != %v", i, kn[i], testKn[i])
+ }
+ if i := compareFloat32Slices(wn[0:], testWn); i >= 0 {
+ t.Errorf("wn disagrees at index %v; %v != %v", i, wn[i], testWn[i])
+ }
+ if i := compareFloat32Slices(fn[0:], testFn); i >= 0 {
+ t.Errorf("fn disagrees at index %v; %v != %v", i, fn[i], testFn[i])
+ }
+}
+
+func TestExpTables(t *testing.T) {
+ testKe, testWe, testFe := initExp()
+ if i := compareUint32Slices(ke[0:], testKe); i >= 0 {
+ t.Errorf("ke disagrees at index %v; %v != %v", i, ke[i], testKe[i])
+ }
+ if i := compareFloat32Slices(we[0:], testWe); i >= 0 {
+ t.Errorf("we disagrees at index %v; %v != %v", i, we[i], testWe[i])
+ }
+ if i := compareFloat32Slices(fe[0:], testFe); i >= 0 {
+ t.Errorf("fe disagrees at index %v; %v != %v", i, fe[i], testFe[i])
+ }
+}
+
+func hasSlowFloatingPoint() bool {
+ switch runtime.GOARCH {
+ case "arm":
+ return os.Getenv("GOARM") == "5"
+ case "mips", "mipsle", "mips64", "mips64le":
+ // Be conservative and assume that all mips boards
+ // have emulated floating point.
+ // TODO: detect what it actually has.
+ return true
+ }
+ return false
+}
+
+func TestFloat32(t *testing.T) {
+ // For issue 6721, the problem came after 7533753 calls, so check 10e6.
+ num := int(10e6)
+ // But do the full amount only on builders (not locally).
+ // But ARM5 floating point emulation is slow (Issue 10749), so
+ // do less for that builder:
+ if testing.Short() && (testenv.Builder() == "" || hasSlowFloatingPoint()) {
+ num /= 100 // 1.72 seconds instead of 172 seconds
+ }
+
+ r := New(NewSource(1))
+ for ct := 0; ct < num; ct++ {
+ f := r.Float32()
+ if f >= 1 {
+ t.Fatal("Float32() should be in range [0,1). ct:", ct, "f:", f)
+ }
+ }
+}
+
+func testReadUniformity(t *testing.T, n int, seed int64) {
+ r := New(NewSource(seed))
+ buf := make([]byte, n)
+ nRead, err := r.Read(buf)
+ if err != nil {
+ t.Errorf("Read err %v", err)
+ }
+ if nRead != n {
+ t.Errorf("Read returned unexpected n; %d != %d", nRead, n)
+ }
+
+ // Expect a uniform distribution of byte values, which lie in [0, 255].
+ var (
+ mean = 255.0 / 2
+ stddev = 256.0 / math.Sqrt(12.0)
+ errorScale = stddev / math.Sqrt(float64(n))
+ )
+
+ expected := &statsResults{mean, stddev, 0.10 * errorScale, 0.08 * errorScale}
+
+ // Cast bytes as floats to use the common distribution-validity checks.
+ samples := make([]float64, n)
+ for i, val := range buf {
+ samples[i] = float64(val)
+ }
+ // Make sure that the entire set matches the expected distribution.
+ checkSampleDistribution(t, samples, expected)
+}
+
+func TestReadUniformity(t *testing.T) {
+ testBufferSizes := []int{
+ 2, 4, 7, 64, 1024, 1 << 16, 1 << 20,
+ }
+ for _, seed := range testSeeds {
+ for _, n := range testBufferSizes {
+ testReadUniformity(t, n, seed)
+ }
+ }
+}
+
+func TestReadEmpty(t *testing.T) {
+ r := New(NewSource(1))
+ buf := make([]byte, 0)
+ n, err := r.Read(buf)
+ if err != nil {
+ t.Errorf("Read err into empty buffer; %v", err)
+ }
+ if n != 0 {
+ t.Errorf("Read into empty buffer returned unexpected n of %d", n)
+ }
+}
+
+func TestReadByOneByte(t *testing.T) {
+ r := New(NewSource(1))
+ b1 := make([]byte, 100)
+ _, err := io.ReadFull(iotest.OneByteReader(r), b1)
+ if err != nil {
+ t.Errorf("read by one byte: %v", err)
+ }
+ r = New(NewSource(1))
+ b2 := make([]byte, 100)
+ _, err = r.Read(b2)
+ if err != nil {
+ t.Errorf("read: %v", err)
+ }
+ if !bytes.Equal(b1, b2) {
+ t.Errorf("read by one byte vs single read:\n%x\n%x", b1, b2)
+ }
+}
+
+func TestReadSeedReset(t *testing.T) {
+ r := New(NewSource(42))
+ b1 := make([]byte, 128)
+ _, err := r.Read(b1)
+ if err != nil {
+ t.Errorf("read: %v", err)
+ }
+ r.Seed(42)
+ b2 := make([]byte, 128)
+ _, err = r.Read(b2)
+ if err != nil {
+ t.Errorf("read: %v", err)
+ }
+ if !bytes.Equal(b1, b2) {
+ t.Errorf("mismatch after re-seed:\n%x\n%x", b1, b2)
+ }
+}
+
+func TestShuffleSmall(t *testing.T) {
+ // Check that Shuffle allows n=0 and n=1, but that swap is never called for them.
+ r := New(NewSource(1))
+ for n := 0; n <= 1; n++ {
+ r.Shuffle(n, func(i, j int) { t.Fatalf("swap called, n=%d i=%d j=%d", n, i, j) })
+ }
+}
+
+// encodePerm converts from a permuted slice of length n, such as Perm generates, to an int in [0, n!).
+// See https://en.wikipedia.org/wiki/Lehmer_code.
+// encodePerm modifies the input slice.
+func encodePerm(s []int) int {
+ // Convert to Lehmer code.
+ for i, x := range s {
+ r := s[i+1:]
+ for j, y := range r {
+ if y > x {
+ r[j]--
+ }
+ }
+ }
+ // Convert to int in [0, n!).
+ m := 0
+ fact := 1
+ for i := len(s) - 1; i >= 0; i-- {
+ m += s[i] * fact
+ fact *= len(s) - i
+ }
+ return m
+}
+
+// TestUniformFactorial tests several ways of generating a uniform value in [0, n!).
+func TestUniformFactorial(t *testing.T) {
+ r := New(NewSource(testSeeds[0]))
+ top := 6
+ if testing.Short() {
+ top = 3
+ }
+ for n := 3; n <= top; n++ {
+ t.Run(fmt.Sprintf("n=%d", n), func(t *testing.T) {
+ // Calculate n!.
+ nfact := 1
+ for i := 2; i <= n; i++ {
+ nfact *= i
+ }
+
+ // Test a few different ways to generate a uniform distribution.
+ p := make([]int, n) // re-usable slice for Shuffle generator
+ tests := [...]struct {
+ name string
+ fn func() int
+ }{
+ {name: "Int31n", fn: func() int { return int(r.Int31n(int32(nfact))) }},
+ {name: "int31n", fn: func() int { return int(r.int31n(int32(nfact))) }},
+ {name: "Perm", fn: func() int { return encodePerm(r.Perm(n)) }},
+ {name: "Shuffle", fn: func() int {
+ // Generate permutation using Shuffle.
+ for i := range p {
+ p[i] = i
+ }
+ r.Shuffle(n, func(i, j int) { p[i], p[j] = p[j], p[i] })
+ return encodePerm(p)
+ }},
+ }
+
+ for _, test := range tests {
+ t.Run(test.name, func(t *testing.T) {
+ // Gather chi-squared values and check that they follow
+ // the expected normal distribution given n!-1 degrees of freedom.
+ // See https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test and
+ // https://www.johndcook.com/Beautiful_Testing_ch10.pdf.
+ nsamples := 10 * nfact
+ if nsamples < 200 {
+ nsamples = 200
+ }
+ samples := make([]float64, nsamples)
+ for i := range samples {
+ // Generate some uniformly distributed values and count their occurrences.
+ const iters = 1000
+ counts := make([]int, nfact)
+ for i := 0; i < iters; i++ {
+ counts[test.fn()]++
+ }
+ // Calculate chi-squared and add to samples.
+ want := iters / float64(nfact)
+ var χ2 float64
+ for _, have := range counts {
+ err := float64(have) - want
+ χ2 += err * err
+ }
+ χ2 /= want
+ samples[i] = χ2
+ }
+
+ // Check that our samples approximate the appropriate normal distribution.
+ dof := float64(nfact - 1)
+ expected := &statsResults{mean: dof, stddev: math.Sqrt(2 * dof)}
+ errorScale := max(1.0, expected.stddev)
+ expected.closeEnough = 0.10 * errorScale
+ expected.maxError = 0.08 // TODO: What is the right value here? See issue 21211.
+ checkSampleDistribution(t, samples, expected)
+ })
+ }
+ })
+ }
+}
+
+// Benchmarks
+
+func BenchmarkInt63Threadsafe(b *testing.B) {
+ for n := b.N; n > 0; n-- {
+ Int63()
+ }
+}
+
+func BenchmarkInt63ThreadsafeParallel(b *testing.B) {
+ b.RunParallel(func(pb *testing.PB) {
+ for pb.Next() {
+ Int63()
+ }
+ })
+}
+
+func BenchmarkInt63Unthreadsafe(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Int63()
+ }
+}
+
+func BenchmarkIntn1000(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Intn(1000)
+ }
+}
+
+func BenchmarkInt63n1000(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Int63n(1000)
+ }
+}
+
+func BenchmarkInt31n1000(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Int31n(1000)
+ }
+}
+
+func BenchmarkFloat32(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Float32()
+ }
+}
+
+func BenchmarkFloat64(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Float64()
+ }
+}
+
+func BenchmarkPerm3(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Perm(3)
+ }
+}
+
+func BenchmarkPerm30(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Perm(30)
+ }
+}
+
+func BenchmarkPerm30ViaShuffle(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ p := make([]int, 30)
+ for i := range p {
+ p[i] = i
+ }
+ r.Shuffle(30, func(i, j int) { p[i], p[j] = p[j], p[i] })
+ }
+}
+
+// BenchmarkShuffleOverhead uses a minimal swap function
+// to measure just the shuffling overhead.
+func BenchmarkShuffleOverhead(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Shuffle(52, func(i, j int) {
+ if i < 0 || i >= 52 || j < 0 || j >= 52 {
+ b.Fatalf("bad swap(%d, %d)", i, j)
+ }
+ })
+ }
+}
+
+func BenchmarkRead3(b *testing.B) {
+ r := New(NewSource(1))
+ buf := make([]byte, 3)
+ b.ResetTimer()
+ for n := b.N; n > 0; n-- {
+ r.Read(buf)
+ }
+}
+
+func BenchmarkRead64(b *testing.B) {
+ r := New(NewSource(1))
+ buf := make([]byte, 64)
+ b.ResetTimer()
+ for n := b.N; n > 0; n-- {
+ r.Read(buf)
+ }
+}
+
+func BenchmarkRead1000(b *testing.B) {
+ r := New(NewSource(1))
+ buf := make([]byte, 1000)
+ b.ResetTimer()
+ for n := b.N; n > 0; n-- {
+ r.Read(buf)
+ }
+}
diff --git a/src/math/rand/regress_test.go b/src/math/rand/regress_test.go
new file mode 100644
index 0000000..1f30be8
--- /dev/null
+++ b/src/math/rand/regress_test.go
@@ -0,0 +1,404 @@
+// Copyright 2014 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Test that random number sequences generated by a specific seed
+// do not change from version to version.
+//
+// Do NOT make changes to the golden outputs. If bugs need to be fixed
+// in the underlying code, find ways to fix them that do not affect the
+// outputs.
+
+package rand_test
+
+import (
+ "flag"
+ "fmt"
+ . "math/rand"
+ "reflect"
+ "testing"
+)
+
+var printgolden = flag.Bool("printgolden", false, "print golden results for regression test")
+
+func TestRegress(t *testing.T) {
+ var int32s = []int32{1, 10, 32, 1 << 20, 1<<20 + 1, 1000000000, 1 << 30, 1<<31 - 2, 1<<31 - 1}
+ var int64s = []int64{1, 10, 32, 1 << 20, 1<<20 + 1, 1000000000, 1 << 30, 1<<31 - 2, 1<<31 - 1, 1000000000000000000, 1 << 60, 1<<63 - 2, 1<<63 - 1}
+ var permSizes = []int{0, 1, 5, 8, 9, 10, 16}
+ var readBufferSizes = []int{1, 7, 8, 9, 10}
+ r := New(NewSource(0))
+
+ rv := reflect.ValueOf(r)
+ n := rv.NumMethod()
+ p := 0
+ if *printgolden {
+ fmt.Printf("var regressGolden = []interface{}{\n")
+ }
+ for i := 0; i < n; i++ {
+ m := rv.Type().Method(i)
+ mv := rv.Method(i)
+ mt := mv.Type()
+ if mt.NumOut() == 0 {
+ continue
+ }
+ r.Seed(0)
+ for repeat := 0; repeat < 20; repeat++ {
+ var args []reflect.Value
+ var argstr string
+ if mt.NumIn() == 1 {
+ var x interface{}
+ switch mt.In(0).Kind() {
+ default:
+ t.Fatalf("unexpected argument type for r.%s", m.Name)
+
+ case reflect.Int:
+ if m.Name == "Perm" {
+ x = permSizes[repeat%len(permSizes)]
+ break
+ }
+ big := int64s[repeat%len(int64s)]
+ if int64(int(big)) != big {
+ r.Int63n(big) // what would happen on 64-bit machine, to keep stream in sync
+ if *printgolden {
+ fmt.Printf("\tskipped, // must run printgolden on 64-bit machine\n")
+ }
+ p++
+ continue
+ }
+ x = int(big)
+
+ case reflect.Int32:
+ x = int32s[repeat%len(int32s)]
+
+ case reflect.Int64:
+ x = int64s[repeat%len(int64s)]
+
+ case reflect.Slice:
+ if m.Name == "Read" {
+ n := readBufferSizes[repeat%len(readBufferSizes)]
+ x = make([]byte, n)
+ }
+ }
+ argstr = fmt.Sprint(x)
+ args = append(args, reflect.ValueOf(x))
+ }
+
+ var out interface{}
+ out = mv.Call(args)[0].Interface()
+ if m.Name == "Int" || m.Name == "Intn" {
+ out = int64(out.(int))
+ }
+ if m.Name == "Read" {
+ out = args[0].Interface().([]byte)
+ }
+ if *printgolden {
+ var val string
+ big := int64(1 << 60)
+ if int64(int(big)) != big && (m.Name == "Int" || m.Name == "Intn") {
+ // 32-bit machine cannot print 64-bit results
+ val = "truncated"
+ } else if reflect.TypeOf(out).Kind() == reflect.Slice {
+ val = fmt.Sprintf("%#v", out)
+ } else {
+ val = fmt.Sprintf("%T(%v)", out, out)
+ }
+ fmt.Printf("\t%s, // %s(%s)\n", val, m.Name, argstr)
+ } else {
+ want := regressGolden[p]
+ if m.Name == "Int" {
+ want = int64(int(uint(want.(int64)) << 1 >> 1))
+ }
+ if !reflect.DeepEqual(out, want) {
+ t.Errorf("r.%s(%s) = %v, want %v", m.Name, argstr, out, want)
+ }
+ }
+ p++
+ }
+ }
+ if *printgolden {
+ fmt.Printf("}\n")
+ }
+}
+
+var regressGolden = []interface{}{
+ float64(4.668112973579268), // ExpFloat64()
+ float64(0.1601593871172866), // ExpFloat64()
+ float64(3.0465834105636), // ExpFloat64()
+ float64(0.06385839451671879), // ExpFloat64()
+ float64(1.8578917487258961), // ExpFloat64()
+ float64(0.784676123472182), // ExpFloat64()
+ float64(0.11225477361256932), // ExpFloat64()
+ float64(0.20173283329802255), // ExpFloat64()
+ float64(0.3468619496201105), // ExpFloat64()
+ float64(0.35601103454384536), // ExpFloat64()
+ float64(0.888376329507869), // ExpFloat64()
+ float64(1.4081362450365698), // ExpFloat64()
+ float64(1.0077753823151994), // ExpFloat64()
+ float64(0.23594100766227588), // ExpFloat64()
+ float64(2.777245612300007), // ExpFloat64()
+ float64(0.5202997830662377), // ExpFloat64()
+ float64(1.2842705247770294), // ExpFloat64()
+ float64(0.030307408362776206), // ExpFloat64()
+ float64(2.204156824853721), // ExpFloat64()
+ float64(2.09891923895058), // ExpFloat64()
+ float32(0.94519615), // Float32()
+ float32(0.24496509), // Float32()
+ float32(0.65595627), // Float32()
+ float32(0.05434384), // Float32()
+ float32(0.3675872), // Float32()
+ float32(0.28948045), // Float32()
+ float32(0.1924386), // Float32()
+ float32(0.65533215), // Float32()
+ float32(0.8971697), // Float32()
+ float32(0.16735445), // Float32()
+ float32(0.28858566), // Float32()
+ float32(0.9026048), // Float32()
+ float32(0.84978026), // Float32()
+ float32(0.2730468), // Float32()
+ float32(0.6090802), // Float32()
+ float32(0.253656), // Float32()
+ float32(0.7746542), // Float32()
+ float32(0.017480763), // Float32()
+ float32(0.78707397), // Float32()
+ float32(0.7993937), // Float32()
+ float64(0.9451961492941164), // Float64()
+ float64(0.24496508529377975), // Float64()
+ float64(0.6559562651954052), // Float64()
+ float64(0.05434383959970039), // Float64()
+ float64(0.36758720663245853), // Float64()
+ float64(0.2894804331565928), // Float64()
+ float64(0.19243860967493215), // Float64()
+ float64(0.6553321508148324), // Float64()
+ float64(0.897169713149801), // Float64()
+ float64(0.16735444255905835), // Float64()
+ float64(0.2885856518054551), // Float64()
+ float64(0.9026048462705047), // Float64()
+ float64(0.8497802817628735), // Float64()
+ float64(0.2730468047134829), // Float64()
+ float64(0.6090801919903561), // Float64()
+ float64(0.25365600644283687), // Float64()
+ float64(0.7746542391859803), // Float64()
+ float64(0.017480762156647272), // Float64()
+ float64(0.7870739563039942), // Float64()
+ float64(0.7993936979594545), // Float64()
+ int64(8717895732742165505), // Int()
+ int64(2259404117704393152), // Int()
+ int64(6050128673802995827), // Int()
+ int64(501233450539197794), // Int()
+ int64(3390393562759376202), // Int()
+ int64(2669985732393126063), // Int()
+ int64(1774932891286980153), // Int()
+ int64(6044372234677422456), // Int()
+ int64(8274930044578894929), // Int()
+ int64(1543572285742637646), // Int()
+ int64(2661732831099943416), // Int()
+ int64(8325060299420976708), // Int()
+ int64(7837839688282259259), // Int()
+ int64(2518412263346885298), // Int()
+ int64(5617773211005988520), // Int()
+ int64(2339563716805116249), // Int()
+ int64(7144924247938981575), // Int()
+ int64(161231572858529631), // Int()
+ int64(7259475919510918339), // Int()
+ int64(7373105480197164748), // Int()
+ int32(2029793274), // Int31()
+ int32(526058514), // Int31()
+ int32(1408655353), // Int31()
+ int32(116702506), // Int31()
+ int32(789387515), // Int31()
+ int32(621654496), // Int31()
+ int32(413258767), // Int31()
+ int32(1407315077), // Int31()
+ int32(1926657288), // Int31()
+ int32(359390928), // Int31()
+ int32(619732968), // Int31()
+ int32(1938329147), // Int31()
+ int32(1824889259), // Int31()
+ int32(586363548), // Int31()
+ int32(1307989752), // Int31()
+ int32(544722126), // Int31()
+ int32(1663557311), // Int31()
+ int32(37539650), // Int31()
+ int32(1690228450), // Int31()
+ int32(1716684894), // Int31()
+ int32(0), // Int31n(1)
+ int32(4), // Int31n(10)
+ int32(25), // Int31n(32)
+ int32(310570), // Int31n(1048576)
+ int32(857611), // Int31n(1048577)
+ int32(621654496), // Int31n(1000000000)
+ int32(413258767), // Int31n(1073741824)
+ int32(1407315077), // Int31n(2147483646)
+ int32(1926657288), // Int31n(2147483647)
+ int32(0), // Int31n(1)
+ int32(8), // Int31n(10)
+ int32(27), // Int31n(32)
+ int32(367019), // Int31n(1048576)
+ int32(209005), // Int31n(1048577)
+ int32(307989752), // Int31n(1000000000)
+ int32(544722126), // Int31n(1073741824)
+ int32(1663557311), // Int31n(2147483646)
+ int32(37539650), // Int31n(2147483647)
+ int32(0), // Int31n(1)
+ int32(4), // Int31n(10)
+ int64(8717895732742165505), // Int63()
+ int64(2259404117704393152), // Int63()
+ int64(6050128673802995827), // Int63()
+ int64(501233450539197794), // Int63()
+ int64(3390393562759376202), // Int63()
+ int64(2669985732393126063), // Int63()
+ int64(1774932891286980153), // Int63()
+ int64(6044372234677422456), // Int63()
+ int64(8274930044578894929), // Int63()
+ int64(1543572285742637646), // Int63()
+ int64(2661732831099943416), // Int63()
+ int64(8325060299420976708), // Int63()
+ int64(7837839688282259259), // Int63()
+ int64(2518412263346885298), // Int63()
+ int64(5617773211005988520), // Int63()
+ int64(2339563716805116249), // Int63()
+ int64(7144924247938981575), // Int63()
+ int64(161231572858529631), // Int63()
+ int64(7259475919510918339), // Int63()
+ int64(7373105480197164748), // Int63()
+ int64(0), // Int63n(1)
+ int64(2), // Int63n(10)
+ int64(19), // Int63n(32)
+ int64(959842), // Int63n(1048576)
+ int64(688912), // Int63n(1048577)
+ int64(393126063), // Int63n(1000000000)
+ int64(89212473), // Int63n(1073741824)
+ int64(834026388), // Int63n(2147483646)
+ int64(1577188963), // Int63n(2147483647)
+ int64(543572285742637646), // Int63n(1000000000000000000)
+ int64(355889821886249464), // Int63n(1152921504606846976)
+ int64(8325060299420976708), // Int63n(9223372036854775806)
+ int64(7837839688282259259), // Int63n(9223372036854775807)
+ int64(0), // Int63n(1)
+ int64(0), // Int63n(10)
+ int64(25), // Int63n(32)
+ int64(679623), // Int63n(1048576)
+ int64(882178), // Int63n(1048577)
+ int64(510918339), // Int63n(1000000000)
+ int64(782454476), // Int63n(1073741824)
+ int64(0), // Intn(1)
+ int64(4), // Intn(10)
+ int64(25), // Intn(32)
+ int64(310570), // Intn(1048576)
+ int64(857611), // Intn(1048577)
+ int64(621654496), // Intn(1000000000)
+ int64(413258767), // Intn(1073741824)
+ int64(1407315077), // Intn(2147483646)
+ int64(1926657288), // Intn(2147483647)
+ int64(543572285742637646), // Intn(1000000000000000000)
+ int64(355889821886249464), // Intn(1152921504606846976)
+ int64(8325060299420976708), // Intn(9223372036854775806)
+ int64(7837839688282259259), // Intn(9223372036854775807)
+ int64(0), // Intn(1)
+ int64(2), // Intn(10)
+ int64(14), // Intn(32)
+ int64(515775), // Intn(1048576)
+ int64(839455), // Intn(1048577)
+ int64(690228450), // Intn(1000000000)
+ int64(642943070), // Intn(1073741824)
+ float64(-0.28158587086436215), // NormFloat64()
+ float64(0.570933095808067), // NormFloat64()
+ float64(-1.6920196326157044), // NormFloat64()
+ float64(0.1996229111693099), // NormFloat64()
+ float64(1.9195199291234621), // NormFloat64()
+ float64(0.8954838794918353), // NormFloat64()
+ float64(0.41457072128813166), // NormFloat64()
+ float64(-0.48700161491544713), // NormFloat64()
+ float64(-0.1684059662402393), // NormFloat64()
+ float64(0.37056410998929545), // NormFloat64()
+ float64(1.0156889027029008), // NormFloat64()
+ float64(-0.5174422210625114), // NormFloat64()
+ float64(-0.5565834214413804), // NormFloat64()
+ float64(0.778320596648391), // NormFloat64()
+ float64(-1.8970718197702225), // NormFloat64()
+ float64(0.5229525761688676), // NormFloat64()
+ float64(-1.5515595563231523), // NormFloat64()
+ float64(0.0182029289376123), // NormFloat64()
+ float64(-0.6820951356608795), // NormFloat64()
+ float64(-0.5987943422687668), // NormFloat64()
+ []int{}, // Perm(0)
+ []int{0}, // Perm(1)
+ []int{0, 4, 1, 3, 2}, // Perm(5)
+ []int{3, 1, 0, 4, 7, 5, 2, 6}, // Perm(8)
+ []int{5, 0, 3, 6, 7, 4, 2, 1, 8}, // Perm(9)
+ []int{4, 5, 0, 2, 6, 9, 3, 1, 8, 7}, // Perm(10)
+ []int{14, 2, 0, 8, 3, 5, 13, 12, 1, 4, 6, 7, 11, 9, 15, 10}, // Perm(16)
+ []int{}, // Perm(0)
+ []int{0}, // Perm(1)
+ []int{3, 0, 1, 2, 4}, // Perm(5)
+ []int{5, 1, 2, 0, 4, 7, 3, 6}, // Perm(8)
+ []int{4, 0, 6, 8, 1, 5, 2, 7, 3}, // Perm(9)
+ []int{8, 6, 1, 7, 5, 4, 3, 2, 9, 0}, // Perm(10)
+ []int{0, 3, 13, 2, 15, 4, 10, 1, 8, 14, 7, 6, 12, 9, 5, 11}, // Perm(16)
+ []int{}, // Perm(0)
+ []int{0}, // Perm(1)
+ []int{0, 4, 2, 1, 3}, // Perm(5)
+ []int{2, 1, 7, 0, 6, 3, 4, 5}, // Perm(8)
+ []int{8, 7, 5, 3, 4, 6, 0, 1, 2}, // Perm(9)
+ []int{1, 0, 2, 5, 7, 6, 9, 8, 3, 4}, // Perm(10)
+ []byte{0x1}, // Read([0])
+ []byte{0x94, 0xfd, 0xc2, 0xfa, 0x2f, 0xfc, 0xc0}, // Read([0 0 0 0 0 0 0])
+ []byte{0x41, 0xd3, 0xff, 0x12, 0x4, 0x5b, 0x73, 0xc8}, // Read([0 0 0 0 0 0 0 0])
+ []byte{0x6e, 0x4f, 0xf9, 0x5f, 0xf6, 0x62, 0xa5, 0xee, 0xe8}, // Read([0 0 0 0 0 0 0 0 0])
+ []byte{0x2a, 0xbd, 0xf4, 0x4a, 0x2d, 0xb, 0x75, 0xfb, 0x18, 0xd}, // Read([0 0 0 0 0 0 0 0 0 0])
+ []byte{0xaf}, // Read([0])
+ []byte{0x48, 0xa7, 0x9e, 0xe0, 0xb1, 0xd, 0x39}, // Read([0 0 0 0 0 0 0])
+ []byte{0x46, 0x51, 0x85, 0xf, 0xd4, 0xa1, 0x78, 0x89}, // Read([0 0 0 0 0 0 0 0])
+ []byte{0x2e, 0xe2, 0x85, 0xec, 0xe1, 0x51, 0x14, 0x55, 0x78}, // Read([0 0 0 0 0 0 0 0 0])
+ []byte{0x8, 0x75, 0xd6, 0x4e, 0xe2, 0xd3, 0xd0, 0xd0, 0xde, 0x6b}, // Read([0 0 0 0 0 0 0 0 0 0])
+ []byte{0xf8}, // Read([0])
+ []byte{0xf9, 0xb4, 0x4c, 0xe8, 0x5f, 0xf0, 0x44}, // Read([0 0 0 0 0 0 0])
+ []byte{0xc6, 0xb1, 0xf8, 0x3b, 0x8e, 0x88, 0x3b, 0xbf}, // Read([0 0 0 0 0 0 0 0])
+ []byte{0x85, 0x7a, 0xab, 0x99, 0xc5, 0xb2, 0x52, 0xc7, 0x42}, // Read([0 0 0 0 0 0 0 0 0])
+ []byte{0x9c, 0x32, 0xf3, 0xa8, 0xae, 0xb7, 0x9e, 0xf8, 0x56, 0xf6}, // Read([0 0 0 0 0 0 0 0 0 0])
+ []byte{0x59}, // Read([0])
+ []byte{0xc1, 0x8f, 0xd, 0xce, 0xcc, 0x77, 0xc7}, // Read([0 0 0 0 0 0 0])
+ []byte{0x5e, 0x7a, 0x81, 0xbf, 0xde, 0x27, 0x5f, 0x67}, // Read([0 0 0 0 0 0 0 0])
+ []byte{0xcf, 0xe2, 0x42, 0xcf, 0x3c, 0xc3, 0x54, 0xf3, 0xed}, // Read([0 0 0 0 0 0 0 0 0])
+ []byte{0xe2, 0xd6, 0xbe, 0xcc, 0x4e, 0xa3, 0xae, 0x5e, 0x88, 0x52}, // Read([0 0 0 0 0 0 0 0 0 0])
+ uint32(4059586549), // Uint32()
+ uint32(1052117029), // Uint32()
+ uint32(2817310706), // Uint32()
+ uint32(233405013), // Uint32()
+ uint32(1578775030), // Uint32()
+ uint32(1243308993), // Uint32()
+ uint32(826517535), // Uint32()
+ uint32(2814630155), // Uint32()
+ uint32(3853314576), // Uint32()
+ uint32(718781857), // Uint32()
+ uint32(1239465936), // Uint32()
+ uint32(3876658295), // Uint32()
+ uint32(3649778518), // Uint32()
+ uint32(1172727096), // Uint32()
+ uint32(2615979505), // Uint32()
+ uint32(1089444252), // Uint32()
+ uint32(3327114623), // Uint32()
+ uint32(75079301), // Uint32()
+ uint32(3380456901), // Uint32()
+ uint32(3433369789), // Uint32()
+ uint64(8717895732742165505), // Uint64()
+ uint64(2259404117704393152), // Uint64()
+ uint64(6050128673802995827), // Uint64()
+ uint64(9724605487393973602), // Uint64()
+ uint64(12613765599614152010), // Uint64()
+ uint64(11893357769247901871), // Uint64()
+ uint64(1774932891286980153), // Uint64()
+ uint64(15267744271532198264), // Uint64()
+ uint64(17498302081433670737), // Uint64()
+ uint64(1543572285742637646), // Uint64()
+ uint64(11885104867954719224), // Uint64()
+ uint64(17548432336275752516), // Uint64()
+ uint64(7837839688282259259), // Uint64()
+ uint64(2518412263346885298), // Uint64()
+ uint64(5617773211005988520), // Uint64()
+ uint64(11562935753659892057), // Uint64()
+ uint64(16368296284793757383), // Uint64()
+ uint64(161231572858529631), // Uint64()
+ uint64(16482847956365694147), // Uint64()
+ uint64(16596477517051940556), // Uint64()
+}
diff --git a/src/math/rand/rng.go b/src/math/rand/rng.go
new file mode 100644
index 0000000..f305df1
--- /dev/null
+++ b/src/math/rand/rng.go
@@ -0,0 +1,252 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand
+
+/*
+ * Uniform distribution
+ *
+ * algorithm by
+ * DP Mitchell and JA Reeds
+ */
+
+const (
+ rngLen = 607
+ rngTap = 273
+ rngMax = 1 << 63
+ rngMask = rngMax - 1
+ int32max = (1 << 31) - 1
+)
+
+var (
+ // rngCooked used for seeding. See gen_cooked.go for details.
+ rngCooked [rngLen]int64 = [...]int64{
+ -4181792142133755926, -4576982950128230565, 1395769623340756751, 5333664234075297259,
+ -6347679516498800754, 9033628115061424579, 7143218595135194537, 4812947590706362721,
+ 7937252194349799378, 5307299880338848416, 8209348851763925077, -7107630437535961764,
+ 4593015457530856296, 8140875735541888011, -5903942795589686782, -603556388664454774,
+ -7496297993371156308, 113108499721038619, 4569519971459345583, -4160538177779461077,
+ -6835753265595711384, -6507240692498089696, 6559392774825876886, 7650093201692370310,
+ 7684323884043752161, -8965504200858744418, -2629915517445760644, 271327514973697897,
+ -6433985589514657524, 1065192797246149621, 3344507881999356393, -4763574095074709175,
+ 7465081662728599889, 1014950805555097187, -4773931307508785033, -5742262670416273165,
+ 2418672789110888383, 5796562887576294778, 4484266064449540171, 3738982361971787048,
+ -4699774852342421385, 10530508058128498, -589538253572429690, -6598062107225984180,
+ 8660405965245884302, 10162832508971942, -2682657355892958417, 7031802312784620857,
+ 6240911277345944669, 831864355460801054, -1218937899312622917, 2116287251661052151,
+ 2202309800992166967, 9161020366945053561, 4069299552407763864, 4936383537992622449,
+ 457351505131524928, -8881176990926596454, -6375600354038175299, -7155351920868399290,
+ 4368649989588021065, 887231587095185257, -3659780529968199312, -2407146836602825512,
+ 5616972787034086048, -751562733459939242, 1686575021641186857, -5177887698780513806,
+ -4979215821652996885, -1375154703071198421, 5632136521049761902, -8390088894796940536,
+ -193645528485698615, -5979788902190688516, -4907000935050298721, -285522056888777828,
+ -2776431630044341707, 1679342092332374735, 6050638460742422078, -2229851317345194226,
+ -1582494184340482199, 5881353426285907985, 812786550756860885, 4541845584483343330,
+ -6497901820577766722, 4980675660146853729, -4012602956251539747, -329088717864244987,
+ -2896929232104691526, 1495812843684243920, -2153620458055647789, 7370257291860230865,
+ -2466442761497833547, 4706794511633873654, -1398851569026877145, 8549875090542453214,
+ -9189721207376179652, -7894453601103453165, 7297902601803624459, 1011190183918857495,
+ -6985347000036920864, 5147159997473910359, -8326859945294252826, 2659470849286379941,
+ 6097729358393448602, -7491646050550022124, -5117116194870963097, -896216826133240300,
+ -745860416168701406, 5803876044675762232, -787954255994554146, -3234519180203704564,
+ -4507534739750823898, -1657200065590290694, 505808562678895611, -4153273856159712438,
+ -8381261370078904295, 572156825025677802, 1791881013492340891, 3393267094866038768,
+ -5444650186382539299, 2352769483186201278, -7930912453007408350, -325464993179687389,
+ -3441562999710612272, -6489413242825283295, 5092019688680754699, -227247482082248967,
+ 4234737173186232084, 5027558287275472836, 4635198586344772304, -536033143587636457,
+ 5907508150730407386, -8438615781380831356, 972392927514829904, -3801314342046600696,
+ -4064951393885491917, -174840358296132583, 2407211146698877100, -1640089820333676239,
+ 3940796514530962282, -5882197405809569433, 3095313889586102949, -1818050141166537098,
+ 5832080132947175283, 7890064875145919662, 8184139210799583195, -8073512175445549678,
+ -7758774793014564506, -4581724029666783935, 3516491885471466898, -8267083515063118116,
+ 6657089965014657519, 5220884358887979358, 1796677326474620641, 5340761970648932916,
+ 1147977171614181568, 5066037465548252321, 2574765911837859848, 1085848279845204775,
+ -5873264506986385449, 6116438694366558490, 2107701075971293812, -7420077970933506541,
+ 2469478054175558874, -1855128755834809824, -5431463669011098282, -9038325065738319171,
+ -6966276280341336160, 7217693971077460129, -8314322083775271549, 7196649268545224266,
+ -3585711691453906209, -5267827091426810625, 8057528650917418961, -5084103596553648165,
+ -2601445448341207749, -7850010900052094367, 6527366231383600011, 3507654575162700890,
+ 9202058512774729859, 1954818376891585542, -2582991129724600103, 8299563319178235687,
+ -5321504681635821435, 7046310742295574065, -2376176645520785576, -7650733936335907755,
+ 8850422670118399721, 3631909142291992901, 5158881091950831288, -6340413719511654215,
+ 4763258931815816403, 6280052734341785344, -4979582628649810958, 2043464728020827976,
+ -2678071570832690343, 4562580375758598164, 5495451168795427352, -7485059175264624713,
+ 553004618757816492, 6895160632757959823, -989748114590090637, 7139506338801360852,
+ -672480814466784139, 5535668688139305547, 2430933853350256242, -3821430778991574732,
+ -1063731997747047009, -3065878205254005442, 7632066283658143750, 6308328381617103346,
+ 3681878764086140361, 3289686137190109749, 6587997200611086848, 244714774258135476,
+ -5143583659437639708, 8090302575944624335, 2945117363431356361, -8359047641006034763,
+ 3009039260312620700, -793344576772241777, 401084700045993341, -1968749590416080887,
+ 4707864159563588614, -3583123505891281857, -3240864324164777915, -5908273794572565703,
+ -3719524458082857382, -5281400669679581926, 8118566580304798074, 3839261274019871296,
+ 7062410411742090847, -8481991033874568140, 6027994129690250817, -6725542042704711878,
+ -2971981702428546974, -7854441788951256975, 8809096399316380241, 6492004350391900708,
+ 2462145737463489636, -8818543617934476634, -5070345602623085213, -8961586321599299868,
+ -3758656652254704451, -8630661632476012791, 6764129236657751224, -709716318315418359,
+ -3403028373052861600, -8838073512170985897, -3999237033416576341, -2920240395515973663,
+ -2073249475545404416, 368107899140673753, -6108185202296464250, -6307735683270494757,
+ 4782583894627718279, 6718292300699989587, 8387085186914375220, 3387513132024756289,
+ 4654329375432538231, -292704475491394206, -3848998599978456535, 7623042350483453954,
+ 7725442901813263321, 9186225467561587250, -5132344747257272453, -6865740430362196008,
+ 2530936820058611833, 1636551876240043639, -3658707362519810009, 1452244145334316253,
+ -7161729655835084979, -7943791770359481772, 9108481583171221009, -3200093350120725999,
+ 5007630032676973346, 2153168792952589781, 6720334534964750538, -3181825545719981703,
+ 3433922409283786309, 2285479922797300912, 3110614940896576130, -2856812446131932915,
+ -3804580617188639299, 7163298419643543757, 4891138053923696990, 580618510277907015,
+ 1684034065251686769, 4429514767357295841, -8893025458299325803, -8103734041042601133,
+ 7177515271653460134, 4589042248470800257, -1530083407795771245, 143607045258444228,
+ 246994305896273627, -8356954712051676521, 6473547110565816071, 3092379936208876896,
+ 2058427839513754051, -4089587328327907870, 8785882556301281247, -3074039370013608197,
+ -637529855400303673, 6137678347805511274, -7152924852417805802, 5708223427705576541,
+ -3223714144396531304, 4358391411789012426, 325123008708389849, 6837621693887290924,
+ 4843721905315627004, -3212720814705499393, -3825019837890901156, 4602025990114250980,
+ 1044646352569048800, 9106614159853161675, -8394115921626182539, -4304087667751778808,
+ 2681532557646850893, 3681559472488511871, -3915372517896561773, -2889241648411946534,
+ -6564663803938238204, -8060058171802589521, 581945337509520675, 3648778920718647903,
+ -4799698790548231394, -7602572252857820065, 220828013409515943, -1072987336855386047,
+ 4287360518296753003, -4633371852008891965, 5513660857261085186, -2258542936462001533,
+ -8744380348503999773, 8746140185685648781, 228500091334420247, 1356187007457302238,
+ 3019253992034194581, 3152601605678500003, -8793219284148773595, 5559581553696971176,
+ 4916432985369275664, -8559797105120221417, -5802598197927043732, 2868348622579915573,
+ -7224052902810357288, -5894682518218493085, 2587672709781371173, -7706116723325376475,
+ 3092343956317362483, -5561119517847711700, 972445599196498113, -1558506600978816441,
+ 1708913533482282562, -2305554874185907314, -6005743014309462908, -6653329009633068701,
+ -483583197311151195, 2488075924621352812, -4529369641467339140, -4663743555056261452,
+ 2997203966153298104, 1282559373026354493, 240113143146674385, 8665713329246516443,
+ 628141331766346752, -4651421219668005332, -7750560848702540400, 7596648026010355826,
+ -3132152619100351065, 7834161864828164065, 7103445518877254909, 4390861237357459201,
+ -4780718172614204074, -319889632007444440, 622261699494173647, -3186110786557562560,
+ -8718967088789066690, -1948156510637662747, -8212195255998774408, -7028621931231314745,
+ 2623071828615234808, -4066058308780939700, -5484966924888173764, -6683604512778046238,
+ -6756087640505506466, 5256026990536851868, 7841086888628396109, 6640857538655893162,
+ -8021284697816458310, -7109857044414059830, -1689021141511844405, -4298087301956291063,
+ -4077748265377282003, -998231156719803476, 2719520354384050532, 9132346697815513771,
+ 4332154495710163773, -2085582442760428892, 6994721091344268833, -2556143461985726874,
+ -8567931991128098309, 59934747298466858, -3098398008776739403, -265597256199410390,
+ 2332206071942466437, -7522315324568406181, 3154897383618636503, -7585605855467168281,
+ -6762850759087199275, 197309393502684135, -8579694182469508493, 2543179307861934850,
+ 4350769010207485119, -4468719947444108136, -7207776534213261296, -1224312577878317200,
+ 4287946071480840813, 8362686366770308971, 6486469209321732151, -5605644191012979782,
+ -1669018511020473564, 4450022655153542367, -7618176296641240059, -3896357471549267421,
+ -4596796223304447488, -6531150016257070659, -8982326463137525940, -4125325062227681798,
+ -1306489741394045544, -8338554946557245229, 5329160409530630596, 7790979528857726136,
+ 4955070238059373407, -4304834761432101506, -6215295852904371179, 3007769226071157901,
+ -6753025801236972788, 8928702772696731736, 7856187920214445904, -4748497451462800923,
+ 7900176660600710914, -7082800908938549136, -6797926979589575837, -6737316883512927978,
+ 4186670094382025798, 1883939007446035042, -414705992779907823, 3734134241178479257,
+ 4065968871360089196, 6953124200385847784, -7917685222115876751, -7585632937840318161,
+ -5567246375906782599, -5256612402221608788, 3106378204088556331, -2894472214076325998,
+ 4565385105440252958, 1979884289539493806, -6891578849933910383, 3783206694208922581,
+ 8464961209802336085, 2843963751609577687, 3030678195484896323, -4429654462759003204,
+ 4459239494808162889, 402587895800087237, 8057891408711167515, 4541888170938985079,
+ 1042662272908816815, -3666068979732206850, 2647678726283249984, 2144477441549833761,
+ -3417019821499388721, -2105601033380872185, 5916597177708541638, -8760774321402454447,
+ 8833658097025758785, 5970273481425315300, 563813119381731307, -6455022486202078793,
+ 1598828206250873866, -4016978389451217698, -2988328551145513985, -6071154634840136312,
+ 8469693267274066490, 125672920241807416, -3912292412830714870, -2559617104544284221,
+ -486523741806024092, -4735332261862713930, 5923302823487327109, -9082480245771672572,
+ -1808429243461201518, 7990420780896957397, 4317817392807076702, 3625184369705367340,
+ -6482649271566653105, -3480272027152017464, -3225473396345736649, -368878695502291645,
+ -3981164001421868007, -8522033136963788610, 7609280429197514109, 3020985755112334161,
+ -2572049329799262942, 2635195723621160615, 5144520864246028816, -8188285521126945980,
+ 1567242097116389047, 8172389260191636581, -2885551685425483535, -7060359469858316883,
+ -6480181133964513127, -7317004403633452381, 6011544915663598137, 5932255307352610768,
+ 2241128460406315459, -8327867140638080220, 3094483003111372717, 4583857460292963101,
+ 9079887171656594975, -384082854924064405, -3460631649611717935, 4225072055348026230,
+ -7385151438465742745, 3801620336801580414, -399845416774701952, -7446754431269675473,
+ 7899055018877642622, 5421679761463003041, 5521102963086275121, -4975092593295409910,
+ 8735487530905098534, -7462844945281082830, -2080886987197029914, -1000715163927557685,
+ -4253840471931071485, -5828896094657903328, 6424174453260338141, 359248545074932887,
+ -5949720754023045210, -2426265837057637212, 3030918217665093212, -9077771202237461772,
+ -3186796180789149575, 740416251634527158, -2142944401404840226, 6951781370868335478,
+ 399922722363687927, -8928469722407522623, -1378421100515597285, -8343051178220066766,
+ -3030716356046100229, -8811767350470065420, 9026808440365124461, 6440783557497587732,
+ 4615674634722404292, 539897290441580544, 2096238225866883852, 8751955639408182687,
+ -7316147128802486205, 7381039757301768559, 6157238513393239656, -1473377804940618233,
+ 8629571604380892756, 5280433031239081479, 7101611890139813254, 2479018537985767835,
+ 7169176924412769570, -1281305539061572506, -7865612307799218120, 2278447439451174845,
+ 3625338785743880657, 6477479539006708521, 8976185375579272206, -3712000482142939688,
+ 1326024180520890843, 7537449876596048829, 5464680203499696154, 3189671183162196045,
+ 6346751753565857109, -8982212049534145501, -6127578587196093755, -245039190118465649,
+ -6320577374581628592, 7208698530190629697, 7276901792339343736, -7490986807540332668,
+ 4133292154170828382, 2918308698224194548, -7703910638917631350, -3929437324238184044,
+ -4300543082831323144, -6344160503358350167, 5896236396443472108, -758328221503023383,
+ -1894351639983151068, -307900319840287220, -6278469401177312761, -2171292963361310674,
+ 8382142935188824023, 9103922860780351547, 4152330101494654406,
+ }
+)
+
+type rngSource struct {
+ tap int // index into vec
+ feed int // index into vec
+ vec [rngLen]int64 // current feedback register
+}
+
+// seed rng x[n+1] = 48271 * x[n] mod (2**31 - 1)
+func seedrand(x int32) int32 {
+ const (
+ A = 48271
+ Q = 44488
+ R = 3399
+ )
+
+ hi := x / Q
+ lo := x % Q
+ x = A*lo - R*hi
+ if x < 0 {
+ x += int32max
+ }
+ return x
+}
+
+// Seed uses the provided seed value to initialize the generator to a deterministic state.
+func (rng *rngSource) Seed(seed int64) {
+ rng.tap = 0
+ rng.feed = rngLen - rngTap
+
+ seed = seed % int32max
+ if seed < 0 {
+ seed += int32max
+ }
+ if seed == 0 {
+ seed = 89482311
+ }
+
+ x := int32(seed)
+ for i := -20; i < rngLen; i++ {
+ x = seedrand(x)
+ if i >= 0 {
+ var u int64
+ u = int64(x) << 40
+ x = seedrand(x)
+ u ^= int64(x) << 20
+ x = seedrand(x)
+ u ^= int64(x)
+ u ^= rngCooked[i]
+ rng.vec[i] = u
+ }
+ }
+}
+
+// Int63 returns a non-negative pseudo-random 63-bit integer as an int64.
+func (rng *rngSource) Int63() int64 {
+ return int64(rng.Uint64() & rngMask)
+}
+
+// Uint64 returns a non-negative pseudo-random 64-bit integer as an uint64.
+func (rng *rngSource) Uint64() uint64 {
+ rng.tap--
+ if rng.tap < 0 {
+ rng.tap += rngLen
+ }
+
+ rng.feed--
+ if rng.feed < 0 {
+ rng.feed += rngLen
+ }
+
+ x := rng.vec[rng.feed] + rng.vec[rng.tap]
+ rng.vec[rng.feed] = x
+ return uint64(x)
+}
diff --git a/src/math/rand/zipf.go b/src/math/rand/zipf.go
new file mode 100644
index 0000000..f04c814
--- /dev/null
+++ b/src/math/rand/zipf.go
@@ -0,0 +1,77 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// W.Hormann, G.Derflinger:
+// "Rejection-Inversion to Generate Variates
+// from Monotone Discrete Distributions"
+// http://eeyore.wu-wien.ac.at/papers/96-04-04.wh-der.ps.gz
+
+package rand
+
+import "math"
+
+// A Zipf generates Zipf distributed variates.
+type Zipf struct {
+ r *Rand
+ imax float64
+ v float64
+ q float64
+ s float64
+ oneminusQ float64
+ oneminusQinv float64
+ hxm float64
+ hx0minusHxm float64
+}
+
+func (z *Zipf) h(x float64) float64 {
+ return math.Exp(z.oneminusQ*math.Log(z.v+x)) * z.oneminusQinv
+}
+
+func (z *Zipf) hinv(x float64) float64 {
+ return math.Exp(z.oneminusQinv*math.Log(z.oneminusQ*x)) - z.v
+}
+
+// NewZipf returns a Zipf variate generator.
+// The generator generates values k ∈ [0, imax]
+// such that P(k) is proportional to (v + k) ** (-s).
+// Requirements: s > 1 and v >= 1.
+func NewZipf(r *Rand, s float64, v float64, imax uint64) *Zipf {
+ z := new(Zipf)
+ if s <= 1.0 || v < 1 {
+ return nil
+ }
+ z.r = r
+ z.imax = float64(imax)
+ z.v = v
+ z.q = s
+ z.oneminusQ = 1.0 - z.q
+ z.oneminusQinv = 1.0 / z.oneminusQ
+ z.hxm = z.h(z.imax + 0.5)
+ z.hx0minusHxm = z.h(0.5) - math.Exp(math.Log(z.v)*(-z.q)) - z.hxm
+ z.s = 1 - z.hinv(z.h(1.5)-math.Exp(-z.q*math.Log(z.v+1.0)))
+ return z
+}
+
+// Uint64 returns a value drawn from the Zipf distribution described
+// by the Zipf object.
+func (z *Zipf) Uint64() uint64 {
+ if z == nil {
+ panic("rand: nil Zipf")
+ }
+ k := 0.0
+
+ for {
+ r := z.r.Float64() // r on [0,1]
+ ur := z.hxm + r*z.hx0minusHxm
+ x := z.hinv(ur)
+ k = math.Floor(x + 0.5)
+ if k-x <= z.s {
+ break
+ }
+ if ur >= z.h(k+0.5)-math.Exp(-math.Log(k+z.v)*z.q) {
+ break
+ }
+ }
+ return uint64(k)
+}
diff --git a/src/math/remainder.go b/src/math/remainder.go
new file mode 100644
index 0000000..7c77d6e
--- /dev/null
+++ b/src/math/remainder.go
@@ -0,0 +1,89 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code and the comment below are from
+// FreeBSD's /usr/src/lib/msun/src/e_remainder.c and came
+// with this notice. The go code is a simplified version of
+// the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_remainder(x,y)
+// Return :
+// returns x REM y = x - [x/y]*y as if in infinite
+// precision arithmetic, where [x/y] is the (infinite bit)
+// integer nearest x/y (in half way cases, choose the even one).
+// Method :
+// Based on Mod() returning x - [x/y]chopped * y exactly.
+
+// Remainder returns the IEEE 754 floating-point remainder of x/y.
+//
+// Special cases are:
+// Remainder(±Inf, y) = NaN
+// Remainder(NaN, y) = NaN
+// Remainder(x, 0) = NaN
+// Remainder(x, ±Inf) = x
+// Remainder(x, NaN) = NaN
+func Remainder(x, y float64) float64
+
+func remainder(x, y float64) float64 {
+ const (
+ Tiny = 4.45014771701440276618e-308 // 0x0020000000000000
+ HalfMax = MaxFloat64 / 2
+ )
+ // special cases
+ switch {
+ case IsNaN(x) || IsNaN(y) || IsInf(x, 0) || y == 0:
+ return NaN()
+ case IsInf(y, 0):
+ return x
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ if y < 0 {
+ y = -y
+ }
+ if x == y {
+ if sign {
+ zero := 0.0
+ return -zero
+ }
+ return 0
+ }
+ if y <= HalfMax {
+ x = Mod(x, y+y) // now x < 2y
+ }
+ if y < Tiny {
+ if x+x > y {
+ x -= y
+ if x+x >= y {
+ x -= y
+ }
+ }
+ } else {
+ yHalf := 0.5 * y
+ if x > yHalf {
+ x -= y
+ if x >= yHalf {
+ x -= y
+ }
+ }
+ }
+ if sign {
+ x = -x
+ }
+ return x
+}
diff --git a/src/math/remainder_386.s b/src/math/remainder_386.s
new file mode 100644
index 0000000..318fa2c
--- /dev/null
+++ b/src/math/remainder_386.s
@@ -0,0 +1,17 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Remainder(x, y float64) float64
+TEXT ·Remainder(SB),NOSPLIT,$0
+ FMOVD y+8(FP), F0 // F0=y
+ FMOVD x+0(FP), F0 // F0=x, F1=y
+ FPREM1 // F0=reduced_x, F1=y
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE -3(PC) // jump if reduction incomplete
+ FMOVDP F0, F1 // F0=x-q*y
+ FMOVDP F0, ret+16(FP)
+ RET
diff --git a/src/math/signbit.go b/src/math/signbit.go
new file mode 100644
index 0000000..f6e61d6
--- /dev/null
+++ b/src/math/signbit.go
@@ -0,0 +1,10 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Signbit reports whether x is negative or negative zero.
+func Signbit(x float64) bool {
+ return Float64bits(x)&(1<<63) != 0
+}
diff --git a/src/math/sin.go b/src/math/sin.go
new file mode 100644
index 0000000..3b6dbe3
--- /dev/null
+++ b/src/math/sin.go
@@ -0,0 +1,232 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point sine and cosine.
+*/
+
+// The original C code, the long comment, and the constants
+// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
+// available from http://www.netlib.org/cephes/cmath.tgz.
+// The go code is a simplified version of the original C.
+//
+// sin.c
+//
+// Circular sine
+//
+// SYNOPSIS:
+//
+// double x, y, sin();
+// y = sin( x );
+//
+// DESCRIPTION:
+//
+// Range reduction is into intervals of pi/4. The reduction error is nearly
+// eliminated by contriving an extended precision modular arithmetic.
+//
+// Two polynomial approximating functions are employed.
+// Between 0 and pi/4 the sine is approximated by
+// x + x**3 P(x**2).
+// Between pi/4 and pi/2 the cosine is represented as
+// 1 - x**2 Q(x**2).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC 0, 10 150000 3.0e-17 7.8e-18
+// IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
+//
+// Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9. The loss
+// is not gradual, but jumps suddenly to about 1 part in 10e7. Results may
+// be meaningless for x > 2**49 = 5.6e14.
+//
+// cos.c
+//
+// Circular cosine
+//
+// SYNOPSIS:
+//
+// double x, y, cos();
+// y = cos( x );
+//
+// DESCRIPTION:
+//
+// Range reduction is into intervals of pi/4. The reduction error is nearly
+// eliminated by contriving an extended precision modular arithmetic.
+//
+// Two polynomial approximating functions are employed.
+// Between 0 and pi/4 the cosine is approximated by
+// 1 - x**2 Q(x**2).
+// Between pi/4 and pi/2 the sine is represented as
+// x + x**3 P(x**2).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
+// DEC 0,+1.07e9 17000 3.0e-17 7.2e-18
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// sin coefficients
+var _sin = [...]float64{
+ 1.58962301576546568060e-10, // 0x3de5d8fd1fd19ccd
+ -2.50507477628578072866e-8, // 0xbe5ae5e5a9291f5d
+ 2.75573136213857245213e-6, // 0x3ec71de3567d48a1
+ -1.98412698295895385996e-4, // 0xbf2a01a019bfdf03
+ 8.33333333332211858878e-3, // 0x3f8111111110f7d0
+ -1.66666666666666307295e-1, // 0xbfc5555555555548
+}
+
+// cos coefficients
+var _cos = [...]float64{
+ -1.13585365213876817300e-11, // 0xbda8fa49a0861a9b
+ 2.08757008419747316778e-9, // 0x3e21ee9d7b4e3f05
+ -2.75573141792967388112e-7, // 0xbe927e4f7eac4bc6
+ 2.48015872888517045348e-5, // 0x3efa01a019c844f5
+ -1.38888888888730564116e-3, // 0xbf56c16c16c14f91
+ 4.16666666666665929218e-2, // 0x3fa555555555554b
+}
+
+// Cos returns the cosine of the radian argument x.
+//
+// Special cases are:
+// Cos(±Inf) = NaN
+// Cos(NaN) = NaN
+func Cos(x float64) float64
+
+func cos(x float64) float64 {
+ const (
+ PI4A = 7.85398125648498535156e-1 // 0x3fe921fb40000000, Pi/4 split into three parts
+ PI4B = 3.77489470793079817668e-8 // 0x3e64442d00000000,
+ PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170,
+ )
+ // special cases
+ switch {
+ case IsNaN(x) || IsInf(x, 0):
+ return NaN()
+ }
+
+ // make argument positive
+ sign := false
+ x = Abs(x)
+
+ var j uint64
+ var y, z float64
+ if x >= reduceThreshold {
+ j, z = trigReduce(x)
+ } else {
+ j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
+ y = float64(j) // integer part of x/(Pi/4), as float
+
+ // map zeros to origin
+ if j&1 == 1 {
+ j++
+ y++
+ }
+ j &= 7 // octant modulo 2Pi radians (360 degrees)
+ z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
+ }
+
+ if j > 3 {
+ j -= 4
+ sign = !sign
+ }
+ if j > 1 {
+ sign = !sign
+ }
+
+ zz := z * z
+ if j == 1 || j == 2 {
+ y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
+ } else {
+ y = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
+ }
+ if sign {
+ y = -y
+ }
+ return y
+}
+
+// Sin returns the sine of the radian argument x.
+//
+// Special cases are:
+// Sin(±0) = ±0
+// Sin(±Inf) = NaN
+// Sin(NaN) = NaN
+func Sin(x float64) float64
+
+func sin(x float64) float64 {
+ const (
+ PI4A = 7.85398125648498535156e-1 // 0x3fe921fb40000000, Pi/4 split into three parts
+ PI4B = 3.77489470793079817668e-8 // 0x3e64442d00000000,
+ PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170,
+ )
+ // special cases
+ switch {
+ case x == 0 || IsNaN(x):
+ return x // return ±0 || NaN()
+ case IsInf(x, 0):
+ return NaN()
+ }
+
+ // make argument positive but save the sign
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+
+ var j uint64
+ var y, z float64
+ if x >= reduceThreshold {
+ j, z = trigReduce(x)
+ } else {
+ j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
+ y = float64(j) // integer part of x/(Pi/4), as float
+
+ // map zeros to origin
+ if j&1 == 1 {
+ j++
+ y++
+ }
+ j &= 7 // octant modulo 2Pi radians (360 degrees)
+ z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
+ }
+ // reflect in x axis
+ if j > 3 {
+ sign = !sign
+ j -= 4
+ }
+ zz := z * z
+ if j == 1 || j == 2 {
+ y = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
+ } else {
+ y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
+ }
+ if sign {
+ y = -y
+ }
+ return y
+}
diff --git a/src/math/sin_s390x.s b/src/math/sin_s390x.s
new file mode 100644
index 0000000..7eb2206
--- /dev/null
+++ b/src/math/sin_s390x.s
@@ -0,0 +1,356 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Various constants
+DATA sincosxnan<>+0(SB)/8, $0x7ff8000000000000
+GLOBL sincosxnan<>+0(SB), RODATA, $8
+DATA sincosxlim<>+0(SB)/8, $0x432921fb54442d19
+GLOBL sincosxlim<>+0(SB), RODATA, $8
+DATA sincosxadd<>+0(SB)/8, $0xc338000000000000
+GLOBL sincosxadd<>+0(SB), RODATA, $8
+DATA sincosxpi2l<>+0(SB)/8, $0.108285667392191389e-31
+GLOBL sincosxpi2l<>+0(SB), RODATA, $8
+DATA sincosxpi2m<>+0(SB)/8, $0.612323399573676480e-16
+GLOBL sincosxpi2m<>+0(SB), RODATA, $8
+DATA sincosxpi2h<>+0(SB)/8, $0.157079632679489656e+01
+GLOBL sincosxpi2h<>+0(SB), RODATA, $8
+DATA sincosrpi2<>+0(SB)/8, $0.636619772367581341e+00
+GLOBL sincosrpi2<>+0(SB), RODATA, $8
+
+// Minimax polynomial approximations
+DATA sincosc0<>+0(SB)/8, $0.100000000000000000E+01
+GLOBL sincosc0<>+0(SB), RODATA, $8
+DATA sincosc1<>+0(SB)/8, $-.499999999999999833E+00
+GLOBL sincosc1<>+0(SB), RODATA, $8
+DATA sincosc2<>+0(SB)/8, $0.416666666666625843E-01
+GLOBL sincosc2<>+0(SB), RODATA, $8
+DATA sincosc3<>+0(SB)/8, $-.138888888885498984E-02
+GLOBL sincosc3<>+0(SB), RODATA, $8
+DATA sincosc4<>+0(SB)/8, $0.248015871681607202E-04
+GLOBL sincosc4<>+0(SB), RODATA, $8
+DATA sincosc5<>+0(SB)/8, $-.275572911309937875E-06
+GLOBL sincosc5<>+0(SB), RODATA, $8
+DATA sincosc6<>+0(SB)/8, $0.208735047247632818E-08
+GLOBL sincosc6<>+0(SB), RODATA, $8
+DATA sincosc7<>+0(SB)/8, $-.112753632738365317E-10
+GLOBL sincosc7<>+0(SB), RODATA, $8
+DATA sincoss0<>+0(SB)/8, $0.100000000000000000E+01
+GLOBL sincoss0<>+0(SB), RODATA, $8
+DATA sincoss1<>+0(SB)/8, $-.166666666666666657E+00
+GLOBL sincoss1<>+0(SB), RODATA, $8
+DATA sincoss2<>+0(SB)/8, $0.833333333333309209E-02
+GLOBL sincoss2<>+0(SB), RODATA, $8
+DATA sincoss3<>+0(SB)/8, $-.198412698410701448E-03
+GLOBL sincoss3<>+0(SB), RODATA, $8
+DATA sincoss4<>+0(SB)/8, $0.275573191453906794E-05
+GLOBL sincoss4<>+0(SB), RODATA, $8
+DATA sincoss5<>+0(SB)/8, $-.250520918387633290E-07
+GLOBL sincoss5<>+0(SB), RODATA, $8
+DATA sincoss6<>+0(SB)/8, $0.160571285514715856E-09
+GLOBL sincoss6<>+0(SB), RODATA, $8
+DATA sincoss7<>+0(SB)/8, $-.753213484933210972E-12
+GLOBL sincoss7<>+0(SB), RODATA, $8
+
+// Sin returns the sine of the radian argument x.
+//
+// Special cases are:
+// Sin(±0) = ±0
+// Sin(±Inf) = NaN
+// Sin(NaN) = NaN
+// The algorithm used is minimax polynomial approximation.
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·sinAsm(SB),NOSPLIT,$0-16
+ FMOVD x+0(FP), F0
+ //special case Sin(±0) = ±0
+ FMOVD $(0.0), F1
+ FCMPU F0, F1
+ BEQ sinIsZero
+ LTDBR F0, F0
+ BLTU L17
+ FMOVD F0, F5
+L2:
+ MOVD $sincoss7<>+0(SB), R1
+ FMOVD 0(R1), F4
+ MOVD $sincoss6<>+0(SB), R1
+ FMOVD 0(R1), F1
+ MOVD $sincoss5<>+0(SB), R1
+ VLEG $0, 0(R1), V18
+ MOVD $sincoss4<>+0(SB), R1
+ FMOVD 0(R1), F6
+ MOVD $sincoss2<>+0(SB), R1
+ VLEG $0, 0(R1), V16
+ MOVD $sincoss3<>+0(SB), R1
+ FMOVD 0(R1), F7
+ MOVD $sincoss1<>+0(SB), R1
+ FMOVD 0(R1), F3
+ MOVD $sincoss0<>+0(SB), R1
+ FMOVD 0(R1), F2
+ WFCHDBS V2, V5, V2
+ BEQ L18
+ MOVD $sincosrpi2<>+0(SB), R1
+ FMOVD 0(R1), F3
+ MOVD $sincosxadd<>+0(SB), R1
+ FMOVD 0(R1), F2
+ WFMSDB V0, V3, V2, V3
+ FMOVD 0(R1), F6
+ FADD F3, F6
+ MOVD $sincosxpi2h<>+0(SB), R1
+ FMOVD 0(R1), F2
+ FMSUB F2, F6, F0
+ MOVD $sincosxpi2m<>+0(SB), R1
+ FMOVD 0(R1), F4
+ FMADD F4, F6, F0
+ MOVD $sincosxpi2l<>+0(SB), R1
+ WFMDB V0, V0, V1
+ FMOVD 0(R1), F7
+ WFMDB V1, V1, V2
+ LGDR F3, R1
+ MOVD $sincosxlim<>+0(SB), R2
+ TMLL R1, $1
+ BEQ L6
+ FMOVD 0(R2), F0
+ WFCHDBS V0, V5, V0
+ BNE L14
+ MOVD $sincosc7<>+0(SB), R2
+ FMOVD 0(R2), F0
+ MOVD $sincosc6<>+0(SB), R2
+ FMOVD 0(R2), F4
+ MOVD $sincosc5<>+0(SB), R2
+ WFMADB V1, V0, V4, V0
+ FMOVD 0(R2), F6
+ MOVD $sincosc4<>+0(SB), R2
+ WFMADB V1, V0, V6, V0
+ FMOVD 0(R2), F4
+ MOVD $sincosc2<>+0(SB), R2
+ FMOVD 0(R2), F6
+ WFMADB V2, V4, V6, V4
+ MOVD $sincosc3<>+0(SB), R2
+ FMOVD 0(R2), F3
+ MOVD $sincosc1<>+0(SB), R2
+ WFMADB V2, V0, V3, V0
+ FMOVD 0(R2), F6
+ WFMADB V1, V4, V6, V4
+ TMLL R1, $2
+ WFMADB V2, V0, V4, V0
+ MOVD $sincosc0<>+0(SB), R1
+ FMOVD 0(R1), F2
+ WFMADB V1, V0, V2, V0
+ BNE L15
+ FMOVD F0, ret+8(FP)
+ RET
+
+L6:
+ FMOVD 0(R2), F4
+ WFCHDBS V4, V5, V4
+ BNE L14
+ MOVD $sincoss7<>+0(SB), R2
+ FMOVD 0(R2), F4
+ MOVD $sincoss6<>+0(SB), R2
+ FMOVD 0(R2), F3
+ MOVD $sincoss5<>+0(SB), R2
+ WFMADB V1, V4, V3, V4
+ WFMADB V6, V7, V0, V6
+ FMOVD 0(R2), F0
+ MOVD $sincoss4<>+0(SB), R2
+ FMADD F4, F1, F0
+ FMOVD 0(R2), F3
+ MOVD $sincoss2<>+0(SB), R2
+ FMOVD 0(R2), F4
+ MOVD $sincoss3<>+0(SB), R2
+ WFMADB V2, V3, V4, V3
+ FMOVD 0(R2), F4
+ MOVD $sincoss1<>+0(SB), R2
+ WFMADB V2, V0, V4, V0
+ FMOVD 0(R2), F4
+ WFMADB V1, V3, V4, V3
+ FNEG F6, F4
+ WFMADB V2, V0, V3, V2
+ WFMDB V4, V1, V0
+ TMLL R1, $2
+ WFMSDB V0, V2, V6, V0
+ BNE L15
+ FMOVD F0, ret+8(FP)
+ RET
+
+L14:
+ MOVD $sincosxnan<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L18:
+ WFMDB V0, V0, V2
+ WFMADB V2, V4, V1, V4
+ WFMDB V2, V2, V1
+ WFMADB V2, V4, V18, V4
+ WFMADB V1, V6, V16, V6
+ WFMADB V1, V4, V7, V4
+ WFMADB V2, V6, V3, V6
+ FMUL F0, F2
+ WFMADB V1, V4, V6, V4
+ FMADD F4, F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L17:
+ FNEG F0, F5
+ BR L2
+L15:
+ FNEG F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+
+sinIsZero:
+ FMOVD F0, ret+8(FP)
+ RET
+
+// Cos returns the cosine of the radian argument.
+//
+// Special cases are:
+// Cos(±Inf) = NaN
+// Cos(NaN) = NaN
+// The algorithm used is minimax polynomial approximation.
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·cosAsm(SB),NOSPLIT,$0-16
+ FMOVD x+0(FP), F0
+ LTDBR F0, F0
+ BLTU L35
+ FMOVD F0, F1
+L21:
+ MOVD $sincosc7<>+0(SB), R1
+ FMOVD 0(R1), F4
+ MOVD $sincosc6<>+0(SB), R1
+ VLEG $0, 0(R1), V20
+ MOVD $sincosc5<>+0(SB), R1
+ VLEG $0, 0(R1), V18
+ MOVD $sincosc4<>+0(SB), R1
+ FMOVD 0(R1), F6
+ MOVD $sincosc2<>+0(SB), R1
+ VLEG $0, 0(R1), V16
+ MOVD $sincosc3<>+0(SB), R1
+ FMOVD 0(R1), F7
+ MOVD $sincosc1<>+0(SB), R1
+ FMOVD 0(R1), F5
+ MOVD $sincosrpi2<>+0(SB), R1
+ FMOVD 0(R1), F2
+ MOVD $sincosxadd<>+0(SB), R1
+ FMOVD 0(R1), F3
+ MOVD $sincoss0<>+0(SB), R1
+ WFMSDB V0, V2, V3, V2
+ FMOVD 0(R1), F3
+ WFCHDBS V3, V1, V3
+ LGDR F2, R1
+ BEQ L36
+ MOVD $sincosxadd<>+0(SB), R2
+ FMOVD 0(R2), F4
+ FADD F2, F4
+ MOVD $sincosxpi2h<>+0(SB), R2
+ FMOVD 0(R2), F2
+ WFMSDB V4, V2, V0, V2
+ MOVD $sincosxpi2m<>+0(SB), R2
+ FMOVD 0(R2), F0
+ WFMADB V4, V0, V2, V0
+ MOVD $sincosxpi2l<>+0(SB), R2
+ WFMDB V0, V0, V2
+ FMOVD 0(R2), F5
+ WFMDB V2, V2, V6
+ MOVD $sincosxlim<>+0(SB), R2
+ TMLL R1, $1
+ BNE L25
+ FMOVD 0(R2), F0
+ WFCHDBS V0, V1, V0
+ BNE L33
+ MOVD $sincosc7<>+0(SB), R2
+ FMOVD 0(R2), F0
+ MOVD $sincosc6<>+0(SB), R2
+ FMOVD 0(R2), F4
+ MOVD $sincosc5<>+0(SB), R2
+ WFMADB V2, V0, V4, V0
+ FMOVD 0(R2), F1
+ MOVD $sincosc4<>+0(SB), R2
+ WFMADB V2, V0, V1, V0
+ FMOVD 0(R2), F4
+ MOVD $sincosc2<>+0(SB), R2
+ FMOVD 0(R2), F1
+ WFMADB V6, V4, V1, V4
+ MOVD $sincosc3<>+0(SB), R2
+ FMOVD 0(R2), F3
+ MOVD $sincosc1<>+0(SB), R2
+ WFMADB V6, V0, V3, V0
+ FMOVD 0(R2), F1
+ WFMADB V2, V4, V1, V4
+ TMLL R1, $2
+ WFMADB V6, V0, V4, V0
+ MOVD $sincosc0<>+0(SB), R1
+ FMOVD 0(R1), F4
+ WFMADB V2, V0, V4, V0
+ BNE L34
+ FMOVD F0, ret+8(FP)
+ RET
+
+L25:
+ FMOVD 0(R2), F3
+ WFCHDBS V3, V1, V1
+ BNE L33
+ MOVD $sincoss7<>+0(SB), R2
+ FMOVD 0(R2), F1
+ MOVD $sincoss6<>+0(SB), R2
+ FMOVD 0(R2), F3
+ MOVD $sincoss5<>+0(SB), R2
+ WFMADB V2, V1, V3, V1
+ FMOVD 0(R2), F3
+ MOVD $sincoss4<>+0(SB), R2
+ WFMADB V2, V1, V3, V1
+ FMOVD 0(R2), F3
+ MOVD $sincoss2<>+0(SB), R2
+ FMOVD 0(R2), F7
+ WFMADB V6, V3, V7, V3
+ MOVD $sincoss3<>+0(SB), R2
+ FMADD F5, F4, F0
+ FMOVD 0(R2), F4
+ MOVD $sincoss1<>+0(SB), R2
+ FMADD F1, F6, F4
+ FMOVD 0(R2), F1
+ FMADD F3, F2, F1
+ FMUL F0, F2
+ WFMADB V6, V4, V1, V6
+ TMLL R1, $2
+ FMADD F6, F2, F0
+ BNE L34
+ FMOVD F0, ret+8(FP)
+ RET
+
+L33:
+ MOVD $sincosxnan<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L36:
+ FMUL F0, F0
+ MOVD $sincosc0<>+0(SB), R1
+ WFMDB V0, V0, V1
+ WFMADB V0, V4, V20, V4
+ WFMADB V1, V6, V16, V6
+ WFMADB V0, V4, V18, V4
+ WFMADB V0, V6, V5, V6
+ WFMADB V1, V4, V7, V4
+ FMOVD 0(R1), F2
+ WFMADB V1, V4, V6, V4
+ WFMADB V0, V4, V2, V0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L35:
+ FNEG F0, F1
+ BR L21
+L34:
+ FNEG F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/sincos.go b/src/math/sincos.go
new file mode 100644
index 0000000..5c5726f
--- /dev/null
+++ b/src/math/sincos.go
@@ -0,0 +1,72 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Coefficients _sin[] and _cos[] are found in pkg/math/sin.go.
+
+// Sincos returns Sin(x), Cos(x).
+//
+// Special cases are:
+// Sincos(±0) = ±0, 1
+// Sincos(±Inf) = NaN, NaN
+// Sincos(NaN) = NaN, NaN
+func Sincos(x float64) (sin, cos float64) {
+ const (
+ PI4A = 7.85398125648498535156e-1 // 0x3fe921fb40000000, Pi/4 split into three parts
+ PI4B = 3.77489470793079817668e-8 // 0x3e64442d00000000,
+ PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170,
+ )
+ // special cases
+ switch {
+ case x == 0:
+ return x, 1 // return ±0.0, 1.0
+ case IsNaN(x) || IsInf(x, 0):
+ return NaN(), NaN()
+ }
+
+ // make argument positive
+ sinSign, cosSign := false, false
+ if x < 0 {
+ x = -x
+ sinSign = true
+ }
+
+ var j uint64
+ var y, z float64
+ if x >= reduceThreshold {
+ j, z = trigReduce(x)
+ } else {
+ j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
+ y = float64(j) // integer part of x/(Pi/4), as float
+
+ if j&1 == 1 { // map zeros to origin
+ j++
+ y++
+ }
+ j &= 7 // octant modulo 2Pi radians (360 degrees)
+ z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
+ }
+ if j > 3 { // reflect in x axis
+ j -= 4
+ sinSign, cosSign = !sinSign, !cosSign
+ }
+ if j > 1 {
+ cosSign = !cosSign
+ }
+
+ zz := z * z
+ cos = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
+ sin = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
+ if j == 1 || j == 2 {
+ sin, cos = cos, sin
+ }
+ if cosSign {
+ cos = -cos
+ }
+ if sinSign {
+ sin = -sin
+ }
+ return
+}
diff --git a/src/math/sinh.go b/src/math/sinh.go
new file mode 100644
index 0000000..573a37e
--- /dev/null
+++ b/src/math/sinh.go
@@ -0,0 +1,81 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point hyperbolic sine and cosine.
+
+ The exponential func is called for arguments
+ greater in magnitude than 0.5.
+
+ A series is used for arguments smaller in magnitude than 0.5.
+
+ Cosh(x) is computed from the exponential func for
+ all arguments.
+*/
+
+// Sinh returns the hyperbolic sine of x.
+//
+// Special cases are:
+// Sinh(±0) = ±0
+// Sinh(±Inf) = ±Inf
+// Sinh(NaN) = NaN
+func Sinh(x float64) float64
+
+func sinh(x float64) float64 {
+ // The coefficients are #2029 from Hart & Cheney. (20.36D)
+ const (
+ P0 = -0.6307673640497716991184787251e+6
+ P1 = -0.8991272022039509355398013511e+5
+ P2 = -0.2894211355989563807284660366e+4
+ P3 = -0.2630563213397497062819489e+2
+ Q0 = -0.6307673640497716991212077277e+6
+ Q1 = 0.1521517378790019070696485176e+5
+ Q2 = -0.173678953558233699533450911e+3
+ )
+
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+
+ var temp float64
+ switch {
+ case x > 21:
+ temp = Exp(x) * 0.5
+
+ case x > 0.5:
+ ex := Exp(x)
+ temp = (ex - 1/ex) * 0.5
+
+ default:
+ sq := x * x
+ temp = (((P3*sq+P2)*sq+P1)*sq + P0) * x
+ temp = temp / (((sq+Q2)*sq+Q1)*sq + Q0)
+ }
+
+ if sign {
+ temp = -temp
+ }
+ return temp
+}
+
+// Cosh returns the hyperbolic cosine of x.
+//
+// Special cases are:
+// Cosh(±0) = 1
+// Cosh(±Inf) = +Inf
+// Cosh(NaN) = NaN
+func Cosh(x float64) float64
+
+func cosh(x float64) float64 {
+ x = Abs(x)
+ if x > 21 {
+ return Exp(x) * 0.5
+ }
+ ex := Exp(x)
+ return (ex + 1/ex) * 0.5
+}
diff --git a/src/math/sinh_s390x.s b/src/math/sinh_s390x.s
new file mode 100644
index 0000000..73701f2
--- /dev/null
+++ b/src/math/sinh_s390x.s
@@ -0,0 +1,251 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+
+#include "textflag.h"
+
+// Constants
+DATA sinhrodataL21<>+0(SB)/8, $0.231904681384629956E-16
+DATA sinhrodataL21<>+8(SB)/8, $0.693147180559945286E+00
+DATA sinhrodataL21<>+16(SB)/8, $704.E0
+GLOBL sinhrodataL21<>+0(SB), RODATA, $24
+DATA sinhrlog2<>+0(SB)/8, $0x3ff7154760000000
+GLOBL sinhrlog2<>+0(SB), RODATA, $8
+DATA sinhxinf<>+0(SB)/8, $0x7ff0000000000000
+GLOBL sinhxinf<>+0(SB), RODATA, $8
+DATA sinhxinit<>+0(SB)/8, $0x3ffb504f333f9de6
+GLOBL sinhxinit<>+0(SB), RODATA, $8
+DATA sinhxlim1<>+0(SB)/8, $800.E0
+GLOBL sinhxlim1<>+0(SB), RODATA, $8
+DATA sinhxadd<>+0(SB)/8, $0xc3200001610007fb
+GLOBL sinhxadd<>+0(SB), RODATA, $8
+DATA sinhx4ff<>+0(SB)/8, $0x4ff0000000000000
+GLOBL sinhx4ff<>+0(SB), RODATA, $8
+
+// Minimax polynomial approximations
+DATA sinhe0<>+0(SB)/8, $0.11715728752538099300E+01
+GLOBL sinhe0<>+0(SB), RODATA, $8
+DATA sinhe1<>+0(SB)/8, $0.11715728752538099300E+01
+GLOBL sinhe1<>+0(SB), RODATA, $8
+DATA sinhe2<>+0(SB)/8, $0.58578643762688526692E+00
+GLOBL sinhe2<>+0(SB), RODATA, $8
+DATA sinhe3<>+0(SB)/8, $0.19526214587563004497E+00
+GLOBL sinhe3<>+0(SB), RODATA, $8
+DATA sinhe4<>+0(SB)/8, $0.48815536475176217404E-01
+GLOBL sinhe4<>+0(SB), RODATA, $8
+DATA sinhe5<>+0(SB)/8, $0.97631072948627397816E-02
+GLOBL sinhe5<>+0(SB), RODATA, $8
+DATA sinhe6<>+0(SB)/8, $0.16271839297756073153E-02
+GLOBL sinhe6<>+0(SB), RODATA, $8
+DATA sinhe7<>+0(SB)/8, $0.23245485387271142509E-03
+GLOBL sinhe7<>+0(SB), RODATA, $8
+DATA sinhe8<>+0(SB)/8, $0.29080955860869629131E-04
+GLOBL sinhe8<>+0(SB), RODATA, $8
+DATA sinhe9<>+0(SB)/8, $0.32311267157667725278E-05
+GLOBL sinhe9<>+0(SB), RODATA, $8
+
+// Sinh returns the hyperbolic sine of the argument.
+//
+// Special cases are:
+// Sinh(±0) = ±0
+// Sinh(±Inf) = ±Inf
+// Sinh(NaN) = NaN
+// The algorithm used is minimax polynomial approximation
+// with coefficients determined with a Remez exchange algorithm.
+
+TEXT ·sinhAsm(SB),NOSPLIT,$0-16
+ FMOVD x+0(FP), F0
+ //specail case Sinh(±0) = ±0
+ FMOVD $(0.0), F1
+ FCMPU F0, F1
+ BEQ sinhIsZero
+ //specail case Sinh(±Inf = ±Inf
+ FMOVD $1.797693134862315708145274237317043567981e+308, F1
+ FCMPU F1, F0
+ BLEU sinhIsInf
+ FMOVD $-1.797693134862315708145274237317043567981e+308, F1
+ FCMPU F1, F0
+ BGT sinhIsInf
+
+ MOVD $sinhrodataL21<>+0(SB), R5
+ LTDBR F0, F0
+ MOVD sinhxinit<>+0(SB), R1
+ FMOVD F0, F4
+ MOVD R1, R3
+ BLTU L19
+ FMOVD F0, F2
+L2:
+ WORD $0xED205010 //cdb %f2,.L22-.L21(%r5)
+ BYTE $0x00
+ BYTE $0x19
+ BGE L15 //jnl .L15
+ BVS L15
+ WFCEDBS V2, V2, V0
+ BEQ L20
+L12:
+ FMOVD F4, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L15:
+ WFCEDBS V2, V2, V0
+ BVS L12
+ MOVD $sinhxlim1<>+0(SB), R2
+ FMOVD 0(R2), F0
+ WFCHDBS V0, V2, V0
+ BEQ L6
+ WFCHEDBS V4, V2, V6
+ MOVD $sinhxinf<>+0(SB), R1
+ FMOVD 0(R1), F0
+ BNE LEXITTAGsinh
+ WFCHDBS V2, V4, V2
+ BNE L16
+ FNEG F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L19:
+ FNEG F0, F2
+ BR L2
+L6:
+ MOVD $sinhxadd<>+0(SB), R2
+ FMOVD 0(R2), F0
+ MOVD sinhrlog2<>+0(SB), R2
+ LDGR R2, F6
+ WFMSDB V4, V6, V0, V16
+ FMOVD sinhrodataL21<>+8(SB), F6
+ WFADB V0, V16, V0
+ FMOVD sinhrodataL21<>+0(SB), F3
+ WFMSDB V0, V6, V4, V6
+ MOVD $sinhe9<>+0(SB), R2
+ WFMADB V0, V3, V6, V0
+ FMOVD 0(R2), F1
+ MOVD $sinhe7<>+0(SB), R2
+ WFMDB V0, V0, V6
+ FMOVD 0(R2), F5
+ MOVD $sinhe8<>+0(SB), R2
+ FMOVD 0(R2), F3
+ MOVD $sinhe6<>+0(SB), R2
+ WFMADB V6, V1, V5, V1
+ FMOVD 0(R2), F5
+ MOVD $sinhe5<>+0(SB), R2
+ FMOVD 0(R2), F7
+ MOVD $sinhe3<>+0(SB), R2
+ WFMADB V6, V3, V5, V3
+ FMOVD 0(R2), F5
+ MOVD $sinhe4<>+0(SB), R2
+ WFMADB V6, V7, V5, V7
+ FMOVD 0(R2), F5
+ MOVD $sinhe2<>+0(SB), R2
+ VLEG $0, 0(R2), V20
+ WFMDB V6, V6, V18
+ WFMADB V6, V5, V20, V5
+ WFMADB V1, V18, V7, V1
+ FNEG F0, F0
+ WFMADB V3, V18, V5, V3
+ MOVD $sinhe1<>+0(SB), R3
+ WFCEDBS V2, V4, V2
+ FMOVD 0(R3), F5
+ MOVD $sinhe0<>+0(SB), R3
+ WFMADB V6, V1, V5, V1
+ FMOVD 0(R3), F5
+ VLGVG $0, V16, R2
+ WFMADB V6, V3, V5, V6
+ RLL $3, R2, R2
+ RISBGN $0, $15, $48, R2, R1
+ BEQ L9
+ WFMSDB V0, V1, V6, V0
+ MOVD $sinhx4ff<>+0(SB), R3
+ FNEG F0, F0
+ FMOVD 0(R3), F2
+ FMUL F2, F0
+ ANDW $0xFFFF, R2
+ WORD $0xA53FEFB6 //llill %r3,61366
+ SUBW R2, R3, R2
+ RISBGN $0, $15, $48, R2, R1
+ LDGR R1, F2
+ FMUL F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L20:
+ MOVD $sinhxadd<>+0(SB), R2
+ FMOVD 0(R2), F2
+ MOVD sinhrlog2<>+0(SB), R2
+ LDGR R2, F0
+ WFMSDB V4, V0, V2, V6
+ FMOVD sinhrodataL21<>+8(SB), F0
+ FADD F6, F2
+ MOVD $sinhe9<>+0(SB), R2
+ FMSUB F0, F2, F4
+ FMOVD 0(R2), F1
+ FMOVD sinhrodataL21<>+0(SB), F3
+ MOVD $sinhe7<>+0(SB), R2
+ FMADD F3, F2, F4
+ FMOVD 0(R2), F0
+ MOVD $sinhe8<>+0(SB), R2
+ WFMDB V4, V4, V2
+ FMOVD 0(R2), F3
+ MOVD $sinhe6<>+0(SB), R2
+ FMOVD 0(R2), F5
+ LGDR F6, R2
+ RLL $3, R2, R2
+ RISBGN $0, $15, $48, R2, R1
+ WFMADB V2, V1, V0, V1
+ LDGR R1, F0
+ MOVD $sinhe5<>+0(SB), R1
+ WFMADB V2, V3, V5, V3
+ FMOVD 0(R1), F5
+ MOVD $sinhe3<>+0(SB), R1
+ FMOVD 0(R1), F6
+ WFMDB V2, V2, V7
+ WFMADB V2, V5, V6, V5
+ WORD $0xA7487FB6 //lhi %r4,32694
+ FNEG F4, F4
+ ANDW $0xFFFF, R2
+ SUBW R2, R4, R2
+ RISBGN $0, $15, $48, R2, R3
+ LDGR R3, F6
+ WFADB V0, V6, V16
+ MOVD $sinhe4<>+0(SB), R1
+ WFMADB V1, V7, V5, V1
+ WFMDB V4, V16, V4
+ FMOVD 0(R1), F5
+ MOVD $sinhe2<>+0(SB), R1
+ VLEG $0, 0(R1), V16
+ MOVD $sinhe1<>+0(SB), R1
+ WFMADB V2, V5, V16, V5
+ VLEG $0, 0(R1), V16
+ WFMADB V3, V7, V5, V3
+ WFMADB V2, V1, V16, V1
+ FSUB F6, F0
+ FMUL F1, F4
+ MOVD $sinhe0<>+0(SB), R1
+ FMOVD 0(R1), F6
+ WFMADB V2, V3, V6, V2
+ WFMADB V0, V2, V4, V0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L9:
+ WFMADB V0, V1, V6, V0
+ MOVD $sinhx4ff<>+0(SB), R3
+ FMOVD 0(R3), F2
+ FMUL F2, F0
+ WORD $0xA72AF000 //ahi %r2,-4096
+ RISBGN $0, $15, $48, R2, R1
+ LDGR R1, F2
+ FMUL F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L16:
+ FMOVD F0, ret+8(FP)
+ RET
+
+LEXITTAGsinh:
+sinhIsInf:
+sinhIsZero:
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/sqrt.go b/src/math/sqrt.go
new file mode 100644
index 0000000..1077a62
--- /dev/null
+++ b/src/math/sqrt.go
@@ -0,0 +1,144 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_sqrt.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_sqrt(x)
+// Return correctly rounded sqrt.
+// -----------------------------------------
+// | Use the hardware sqrt if you have one |
+// -----------------------------------------
+// Method:
+// Bit by bit method using integer arithmetic. (Slow, but portable)
+// 1. Normalization
+// Scale x to y in [1,4) with even powers of 2:
+// find an integer k such that 1 <= (y=x*2**(2k)) < 4, then
+// sqrt(x) = 2**k * sqrt(y)
+// 2. Bit by bit computation
+// Let q = sqrt(y) truncated to i bit after binary point (q = 1),
+// i 0
+// i+1 2
+// s = 2*q , and y = 2 * ( y - q ). (1)
+// i i i i
+//
+// To compute q from q , one checks whether
+// i+1 i
+//
+// -(i+1) 2
+// (q + 2 ) <= y. (2)
+// i
+// -(i+1)
+// If (2) is false, then q = q ; otherwise q = q + 2 .
+// i+1 i i+1 i
+//
+// With some algebraic manipulation, it is not difficult to see
+// that (2) is equivalent to
+// -(i+1)
+// s + 2 <= y (3)
+// i i
+//
+// The advantage of (3) is that s and y can be computed by
+// i i
+// the following recurrence formula:
+// if (3) is false
+//
+// s = s , y = y ; (4)
+// i+1 i i+1 i
+//
+// otherwise,
+// -i -(i+1)
+// s = s + 2 , y = y - s - 2 (5)
+// i+1 i i+1 i i
+//
+// One may easily use induction to prove (4) and (5).
+// Note. Since the left hand side of (3) contain only i+2 bits,
+// it is not necessary to do a full (53-bit) comparison
+// in (3).
+// 3. Final rounding
+// After generating the 53 bits result, we compute one more bit.
+// Together with the remainder, we can decide whether the
+// result is exact, bigger than 1/2ulp, or less than 1/2ulp
+// (it will never equal to 1/2ulp).
+// The rounding mode can be detected by checking whether
+// huge + tiny is equal to huge, and whether huge - tiny is
+// equal to huge for some floating point number "huge" and "tiny".
+//
+//
+// Notes: Rounding mode detection omitted. The constants "mask", "shift",
+// and "bias" are found in src/math/bits.go
+
+// Sqrt returns the square root of x.
+//
+// Special cases are:
+// Sqrt(+Inf) = +Inf
+// Sqrt(±0) = ±0
+// Sqrt(x < 0) = NaN
+// Sqrt(NaN) = NaN
+func Sqrt(x float64) float64
+
+// Note: Sqrt is implemented in assembly on some systems.
+// Others have assembly stubs that jump to func sqrt below.
+// On systems where Sqrt is a single instruction, the compiler
+// may turn a direct call into a direct use of that instruction instead.
+
+func sqrt(x float64) float64 {
+ // special cases
+ switch {
+ case x == 0 || IsNaN(x) || IsInf(x, 1):
+ return x
+ case x < 0:
+ return NaN()
+ }
+ ix := Float64bits(x)
+ // normalize x
+ exp := int((ix >> shift) & mask)
+ if exp == 0 { // subnormal x
+ for ix&(1<<shift) == 0 {
+ ix <<= 1
+ exp--
+ }
+ exp++
+ }
+ exp -= bias // unbias exponent
+ ix &^= mask << shift
+ ix |= 1 << shift
+ if exp&1 == 1 { // odd exp, double x to make it even
+ ix <<= 1
+ }
+ exp >>= 1 // exp = exp/2, exponent of square root
+ // generate sqrt(x) bit by bit
+ ix <<= 1
+ var q, s uint64 // q = sqrt(x)
+ r := uint64(1 << (shift + 1)) // r = moving bit from MSB to LSB
+ for r != 0 {
+ t := s + r
+ if t <= ix {
+ s = t + r
+ ix -= t
+ q += r
+ }
+ ix <<= 1
+ r >>= 1
+ }
+ // final rounding
+ if ix != 0 { // remainder, result not exact
+ q += q & 1 // round according to extra bit
+ }
+ ix = q>>1 + uint64(exp-1+bias)<<shift // significand + biased exponent
+ return Float64frombits(ix)
+}
diff --git a/src/math/sqrt_386.s b/src/math/sqrt_386.s
new file mode 100644
index 0000000..5a5c33a
--- /dev/null
+++ b/src/math/sqrt_386.s
@@ -0,0 +1,12 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB),NOSPLIT,$0
+ FMOVD x+0(FP),F0
+ FSQRT
+ FMOVDP F0,ret+8(FP)
+ RET
diff --git a/src/math/sqrt_amd64.s b/src/math/sqrt_amd64.s
new file mode 100644
index 0000000..1102903
--- /dev/null
+++ b/src/math/sqrt_amd64.s
@@ -0,0 +1,12 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB), NOSPLIT, $0
+ XORPS X0, X0 // break dependency
+ SQRTSD x+0(FP), X0
+ MOVSD X0, ret+8(FP)
+ RET
diff --git a/src/math/sqrt_arm.s b/src/math/sqrt_arm.s
new file mode 100644
index 0000000..ffc7d10
--- /dev/null
+++ b/src/math/sqrt_arm.s
@@ -0,0 +1,20 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB),NOSPLIT,$0
+ MOVB runtime·goarm(SB), R11
+ CMP $5, R11
+ BEQ arm5
+ MOVD x+0(FP),F0
+ SQRTD F0,F0
+ MOVD F0,ret+8(FP)
+ RET
+arm5:
+ // Tail call to Go implementation.
+ // Can't use JMP, as in softfloat mode SQRTD is rewritten
+ // to a CALL, which makes this function have a frame.
+ RET ·sqrt(SB)
diff --git a/src/math/sqrt_arm64.s b/src/math/sqrt_arm64.s
new file mode 100644
index 0000000..3041d25
--- /dev/null
+++ b/src/math/sqrt_arm64.s
@@ -0,0 +1,12 @@
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FSQRTD F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/sqrt_mipsx.s b/src/math/sqrt_mipsx.s
new file mode 100644
index 0000000..a63ea9e
--- /dev/null
+++ b/src/math/sqrt_mipsx.s
@@ -0,0 +1,18 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build mips mipsle
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB),NOSPLIT,$0
+#ifdef GOMIPS_softfloat
+ JMP ·sqrt(SB)
+#else
+ MOVD x+0(FP), F0
+ SQRTD F0, F0
+ MOVD F0, ret+8(FP)
+#endif
+ RET
diff --git a/src/math/sqrt_ppc64x.s b/src/math/sqrt_ppc64x.s
new file mode 100644
index 0000000..0469f4d
--- /dev/null
+++ b/src/math/sqrt_ppc64x.s
@@ -0,0 +1,14 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build ppc64 ppc64le
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0
+ FSQRT F0, F0
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/sqrt_riscv64.s b/src/math/sqrt_riscv64.s
new file mode 100644
index 0000000..048171b
--- /dev/null
+++ b/src/math/sqrt_riscv64.s
@@ -0,0 +1,14 @@
+// Copyright 2020 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build riscv64
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB),NOSPLIT,$0
+ MOVD x+0(FP), F0
+ FSQRTD F0, F0
+ MOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/sqrt_s390x.s b/src/math/sqrt_s390x.s
new file mode 100644
index 0000000..37ca0be
--- /dev/null
+++ b/src/math/sqrt_s390x.s
@@ -0,0 +1,12 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F1
+ FSQRT F1, F1
+ FMOVD F1, ret+8(FP)
+ RET
diff --git a/src/math/sqrt_wasm.s b/src/math/sqrt_wasm.s
new file mode 100644
index 0000000..cbfe598
--- /dev/null
+++ b/src/math/sqrt_wasm.s
@@ -0,0 +1,12 @@
+// Copyright 2018 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Sqrt(SB),NOSPLIT,$0
+ Get SP
+ F64Load x+0(FP)
+ F64Sqrt
+ F64Store ret+8(FP)
+ RET
diff --git a/src/math/stubs_386.s b/src/math/stubs_386.s
new file mode 100644
index 0000000..92c8621
--- /dev/null
+++ b/src/math/stubs_386.s
@@ -0,0 +1,53 @@
+// Copyright 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Acosh(SB), NOSPLIT, $0
+ JMP ·acosh(SB)
+
+TEXT ·Asinh(SB), NOSPLIT, $0
+ JMP ·asinh(SB)
+
+TEXT ·Atanh(SB), NOSPLIT, $0
+ JMP ·atanh(SB)
+
+TEXT ·Cbrt(SB), NOSPLIT, $0
+ JMP ·cbrt(SB)
+
+TEXT ·Cos(SB), NOSPLIT, $0
+ JMP ·cos(SB)
+
+TEXT ·Cosh(SB), NOSPLIT, $0
+ JMP ·cosh(SB)
+
+TEXT ·Erf(SB), NOSPLIT, $0
+ JMP ·erf(SB)
+
+TEXT ·Erfc(SB), NOSPLIT, $0
+ JMP ·erfc(SB)
+
+TEXT ·Exp(SB), NOSPLIT, $0
+ JMP ·exp(SB)
+
+TEXT ·Max(SB), NOSPLIT, $0
+ JMP ·max(SB)
+
+TEXT ·Min(SB), NOSPLIT, $0
+ JMP ·min(SB)
+
+TEXT ·Pow(SB), NOSPLIT, $0
+ JMP ·pow(SB)
+
+TEXT ·Sin(SB), NOSPLIT, $0
+ JMP ·sin(SB)
+
+TEXT ·Sinh(SB), NOSPLIT, $0
+ JMP ·sinh(SB)
+
+TEXT ·Tan(SB), NOSPLIT, $0
+ JMP ·tan(SB)
+
+TEXT ·Tanh(SB), NOSPLIT, $0
+ JMP ·tanh(SB)
diff --git a/src/math/stubs_amd64.s b/src/math/stubs_amd64.s
new file mode 100644
index 0000000..31e01fd
--- /dev/null
+++ b/src/math/stubs_amd64.s
@@ -0,0 +1,86 @@
+// Copyright 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Acos(SB), NOSPLIT, $0
+ JMP ·acos(SB)
+
+TEXT ·Acosh(SB), NOSPLIT, $0
+ JMP ·acosh(SB)
+
+TEXT ·Asin(SB), NOSPLIT, $0
+ JMP ·asin(SB)
+
+TEXT ·Asinh(SB), NOSPLIT, $0
+ JMP ·asinh(SB)
+
+TEXT ·Atan(SB), NOSPLIT, $0
+ JMP ·atan(SB)
+
+TEXT ·Atan2(SB), NOSPLIT, $0
+ JMP ·atan2(SB)
+
+TEXT ·Atanh(SB), NOSPLIT, $0
+ JMP ·atanh(SB)
+
+TEXT ·Cbrt(SB), NOSPLIT, $0
+ JMP ·cbrt(SB)
+
+TEXT ·Cos(SB), NOSPLIT, $0
+ JMP ·cos(SB)
+
+TEXT ·Cosh(SB), NOSPLIT, $0
+ JMP ·cosh(SB)
+
+TEXT ·Erf(SB), NOSPLIT, $0
+ JMP ·erf(SB)
+
+TEXT ·Erfc(SB), NOSPLIT, $0
+ JMP ·erfc(SB)
+
+TEXT ·Exp2(SB), NOSPLIT, $0
+ JMP ·exp2(SB)
+
+TEXT ·Expm1(SB), NOSPLIT, $0
+ JMP ·expm1(SB)
+
+TEXT ·Frexp(SB), NOSPLIT, $0
+ JMP ·frexp(SB)
+
+TEXT ·Ldexp(SB), NOSPLIT, $0
+ JMP ·ldexp(SB)
+
+TEXT ·Log10(SB), NOSPLIT, $0
+ JMP ·log10(SB)
+
+TEXT ·Log2(SB), NOSPLIT, $0
+ JMP ·log2(SB)
+
+TEXT ·Log1p(SB), NOSPLIT, $0
+ JMP ·log1p(SB)
+
+TEXT ·Mod(SB), NOSPLIT, $0
+ JMP ·mod(SB)
+
+TEXT ·Modf(SB), NOSPLIT, $0
+ JMP ·modf(SB)
+
+TEXT ·Pow(SB), NOSPLIT, $0
+ JMP ·pow(SB)
+
+TEXT ·Remainder(SB), NOSPLIT, $0
+ JMP ·remainder(SB)
+
+TEXT ·Sin(SB), NOSPLIT, $0
+ JMP ·sin(SB)
+
+TEXT ·Sinh(SB), NOSPLIT, $0
+ JMP ·sinh(SB)
+
+TEXT ·Tan(SB), NOSPLIT, $0
+ JMP ·tan(SB)
+
+TEXT ·Tanh(SB), NOSPLIT, $0
+ JMP ·tanh(SB)
diff --git a/src/math/stubs_arm.s b/src/math/stubs_arm.s
new file mode 100644
index 0000000..31bf872
--- /dev/null
+++ b/src/math/stubs_arm.s
@@ -0,0 +1,110 @@
+// Copyright 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Acos(SB), NOSPLIT, $0
+ B ·acos(SB)
+
+TEXT ·Acosh(SB), NOSPLIT, $0
+ B ·acosh(SB)
+
+TEXT ·Asin(SB), NOSPLIT, $0
+ B ·asin(SB)
+
+TEXT ·Asinh(SB), NOSPLIT, $0
+ B ·asinh(SB)
+
+TEXT ·Atan(SB), NOSPLIT, $0
+ B ·atan(SB)
+
+TEXT ·Atan2(SB), NOSPLIT, $0
+ B ·atan2(SB)
+
+TEXT ·Atanh(SB), NOSPLIT, $0
+ B ·atanh(SB)
+
+TEXT ·Cbrt(SB), NOSPLIT, $0
+ B ·cbrt(SB)
+
+TEXT ·Cos(SB), NOSPLIT, $0
+ B ·cos(SB)
+
+TEXT ·Cosh(SB), NOSPLIT, $0
+ B ·cosh(SB)
+
+TEXT ·Erf(SB), NOSPLIT, $0
+ B ·erf(SB)
+
+TEXT ·Erfc(SB), NOSPLIT, $0
+ B ·erfc(SB)
+
+TEXT ·Exp2(SB), NOSPLIT, $0
+ B ·exp2(SB)
+
+TEXT ·Exp(SB), NOSPLIT, $0
+ B ·exp(SB)
+
+TEXT ·Expm1(SB), NOSPLIT, $0
+ B ·expm1(SB)
+
+TEXT ·Floor(SB), NOSPLIT, $0
+ B ·floor(SB)
+
+TEXT ·Ceil(SB), NOSPLIT, $0
+ B ·ceil(SB)
+
+TEXT ·Trunc(SB), NOSPLIT, $0
+ B ·trunc(SB)
+
+TEXT ·Frexp(SB), NOSPLIT, $0
+ B ·frexp(SB)
+
+TEXT ·Hypot(SB), NOSPLIT, $0
+ B ·hypot(SB)
+
+TEXT ·Ldexp(SB), NOSPLIT, $0
+ B ·ldexp(SB)
+
+TEXT ·Log10(SB), NOSPLIT, $0
+ B ·log10(SB)
+
+TEXT ·Log2(SB), NOSPLIT, $0
+ B ·log2(SB)
+
+TEXT ·Log1p(SB), NOSPLIT, $0
+ B ·log1p(SB)
+
+TEXT ·Log(SB), NOSPLIT, $0
+ B ·log(SB)
+
+TEXT ·Max(SB), NOSPLIT, $0
+ B ·max(SB)
+
+TEXT ·Min(SB), NOSPLIT, $0
+ B ·min(SB)
+
+TEXT ·Mod(SB), NOSPLIT, $0
+ B ·mod(SB)
+
+TEXT ·Modf(SB), NOSPLIT, $0
+ B ·modf(SB)
+
+TEXT ·Pow(SB), NOSPLIT, $0
+ JMP ·pow(SB)
+
+TEXT ·Remainder(SB), NOSPLIT, $0
+ B ·remainder(SB)
+
+TEXT ·Sin(SB), NOSPLIT, $0
+ B ·sin(SB)
+
+TEXT ·Sinh(SB), NOSPLIT, $0
+ B ·sinh(SB)
+
+TEXT ·Tan(SB), NOSPLIT, $0
+ B ·tan(SB)
+
+TEXT ·Tanh(SB), NOSPLIT, $0
+ B ·tanh(SB)
diff --git a/src/math/stubs_arm64.s b/src/math/stubs_arm64.s
new file mode 100644
index 0000000..24564d0
--- /dev/null
+++ b/src/math/stubs_arm64.s
@@ -0,0 +1,88 @@
+// Copyright 2014 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build arm64
+
+#include "textflag.h"
+
+TEXT ·Asin(SB), NOSPLIT, $0
+ B ·asin(SB)
+
+TEXT ·Acos(SB), NOSPLIT, $0
+ B ·acos(SB)
+
+TEXT ·Asinh(SB), NOSPLIT, $0
+ B ·asinh(SB)
+
+TEXT ·Acosh(SB), NOSPLIT, $0
+ B ·acosh(SB)
+
+TEXT ·Atan2(SB), NOSPLIT, $0
+ B ·atan2(SB)
+
+TEXT ·Atan(SB), NOSPLIT, $0
+ B ·atan(SB)
+
+TEXT ·Atanh(SB), NOSPLIT, $0
+ B ·atanh(SB)
+
+TEXT ·Erf(SB), NOSPLIT, $0
+ B ·erf(SB)
+
+TEXT ·Erfc(SB), NOSPLIT, $0
+ B ·erfc(SB)
+
+TEXT ·Cbrt(SB), NOSPLIT, $0
+ B ·cbrt(SB)
+
+TEXT ·Cosh(SB), NOSPLIT, $0
+ B ·cosh(SB)
+
+TEXT ·Expm1(SB), NOSPLIT, $0
+ B ·expm1(SB)
+
+TEXT ·Frexp(SB), NOSPLIT, $0
+ B ·frexp(SB)
+
+TEXT ·Hypot(SB), NOSPLIT, $0
+ B ·hypot(SB)
+
+TEXT ·Ldexp(SB), NOSPLIT, $0
+ B ·ldexp(SB)
+
+TEXT ·Log10(SB), NOSPLIT, $0
+ B ·log10(SB)
+
+TEXT ·Log2(SB), NOSPLIT, $0
+ B ·log2(SB)
+
+TEXT ·Log1p(SB), NOSPLIT, $0
+ B ·log1p(SB)
+
+TEXT ·Log(SB), NOSPLIT, $0
+ B ·log(SB)
+
+TEXT ·Mod(SB), NOSPLIT, $0
+ B ·mod(SB)
+
+TEXT ·Remainder(SB), NOSPLIT, $0
+ B ·remainder(SB)
+
+TEXT ·Sin(SB), NOSPLIT, $0
+ B ·sin(SB)
+
+TEXT ·Sinh(SB), NOSPLIT, $0
+ B ·sinh(SB)
+
+TEXT ·Cos(SB), NOSPLIT, $0
+ B ·cos(SB)
+
+TEXT ·Tan(SB), NOSPLIT, $0
+ B ·tan(SB)
+
+TEXT ·Tanh(SB), NOSPLIT, $0
+ B ·tanh(SB)
+
+TEXT ·Pow(SB), NOSPLIT, $0
+ B ·pow(SB)
diff --git a/src/math/stubs_mips64x.s b/src/math/stubs_mips64x.s
new file mode 100644
index 0000000..187b069
--- /dev/null
+++ b/src/math/stubs_mips64x.s
@@ -0,0 +1,115 @@
+// Copyright 2014 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build mips64 mips64le
+
+#include "textflag.h"
+
+TEXT ·Asin(SB), NOSPLIT, $0
+ JMP ·asin(SB)
+
+TEXT ·Acos(SB), NOSPLIT, $0
+ JMP ·acos(SB)
+
+TEXT ·Asinh(SB), NOSPLIT, $0
+ JMP ·asinh(SB)
+
+TEXT ·Acosh(SB), NOSPLIT, $0
+ JMP ·acosh(SB)
+
+TEXT ·Atan2(SB), NOSPLIT, $0
+ JMP ·atan2(SB)
+
+TEXT ·Atan(SB), NOSPLIT, $0
+ JMP ·atan(SB)
+
+TEXT ·Atanh(SB), NOSPLIT, $0
+ JMP ·atanh(SB)
+
+TEXT ·Min(SB), NOSPLIT, $0
+ JMP ·min(SB)
+
+TEXT ·Max(SB), NOSPLIT, $0
+ JMP ·max(SB)
+
+TEXT ·Erf(SB), NOSPLIT, $0
+ JMP ·erf(SB)
+
+TEXT ·Erfc(SB), NOSPLIT, $0
+ JMP ·erfc(SB)
+
+TEXT ·Exp2(SB), NOSPLIT, $0
+ JMP ·exp2(SB)
+
+TEXT ·Expm1(SB), NOSPLIT, $0
+ JMP ·expm1(SB)
+
+TEXT ·Exp(SB), NOSPLIT, $0
+ JMP ·exp(SB)
+
+TEXT ·Floor(SB), NOSPLIT, $0
+ JMP ·floor(SB)
+
+TEXT ·Ceil(SB), NOSPLIT, $0
+ JMP ·ceil(SB)
+
+TEXT ·Trunc(SB), NOSPLIT, $0
+ JMP ·trunc(SB)
+
+TEXT ·Frexp(SB), NOSPLIT, $0
+ JMP ·frexp(SB)
+
+TEXT ·Hypot(SB), NOSPLIT, $0
+ JMP ·hypot(SB)
+
+TEXT ·Ldexp(SB), NOSPLIT, $0
+ JMP ·ldexp(SB)
+
+TEXT ·Log10(SB), NOSPLIT, $0
+ JMP ·log10(SB)
+
+TEXT ·Log2(SB), NOSPLIT, $0
+ JMP ·log2(SB)
+
+TEXT ·Log1p(SB), NOSPLIT, $0
+ JMP ·log1p(SB)
+
+TEXT ·Log(SB), NOSPLIT, $0
+ JMP ·log(SB)
+
+TEXT ·Modf(SB), NOSPLIT, $0
+ JMP ·modf(SB)
+
+TEXT ·Mod(SB), NOSPLIT, $0
+ JMP ·mod(SB)
+
+TEXT ·Remainder(SB), NOSPLIT, $0
+ JMP ·remainder(SB)
+
+TEXT ·Sin(SB), NOSPLIT, $0
+ JMP ·sin(SB)
+
+TEXT ·Sinh(SB), NOSPLIT, $0
+ JMP ·sinh(SB)
+
+TEXT ·Cos(SB), NOSPLIT, $0
+ JMP ·cos(SB)
+
+TEXT ·Cosh(SB), NOSPLIT, $0
+ JMP ·cosh(SB)
+
+TEXT ·Sqrt(SB), NOSPLIT, $0
+ JMP ·sqrt(SB)
+
+TEXT ·Tan(SB), NOSPLIT, $0
+ JMP ·tan(SB)
+
+TEXT ·Tanh(SB), NOSPLIT, $0
+ JMP ·tanh(SB)
+
+TEXT ·Cbrt(SB), NOSPLIT, $0
+ JMP ·cbrt(SB)
+
+TEXT ·Pow(SB), NOSPLIT, $0
+ JMP ·pow(SB)
diff --git a/src/math/stubs_mipsx.s b/src/math/stubs_mipsx.s
new file mode 100644
index 0000000..4b82449
--- /dev/null
+++ b/src/math/stubs_mipsx.s
@@ -0,0 +1,113 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build mips mipsle
+
+#include "textflag.h"
+
+TEXT ·Asin(SB), NOSPLIT, $0
+ JMP ·asin(SB)
+
+TEXT ·Acos(SB), NOSPLIT, $0
+ JMP ·acos(SB)
+
+TEXT ·Asinh(SB), NOSPLIT, $0
+ JMP ·asinh(SB)
+
+TEXT ·Acosh(SB), NOSPLIT, $0
+ JMP ·acosh(SB)
+
+TEXT ·Atan2(SB), NOSPLIT, $0
+ JMP ·atan2(SB)
+
+TEXT ·Atan(SB), NOSPLIT, $0
+ JMP ·atan(SB)
+
+TEXT ·Atanh(SB), NOSPLIT, $0
+ JMP ·atanh(SB)
+
+TEXT ·Min(SB), NOSPLIT, $0
+ JMP ·min(SB)
+
+TEXT ·Max(SB), NOSPLIT, $0
+ JMP ·max(SB)
+
+TEXT ·Erf(SB), NOSPLIT, $0
+ JMP ·erf(SB)
+
+TEXT ·Erfc(SB), NOSPLIT, $0
+ JMP ·erfc(SB)
+
+TEXT ·Exp2(SB), NOSPLIT, $0
+ JMP ·exp2(SB)
+
+TEXT ·Expm1(SB), NOSPLIT, $0
+ JMP ·expm1(SB)
+
+TEXT ·Exp(SB), NOSPLIT, $0
+ JMP ·exp(SB)
+
+TEXT ·Floor(SB), NOSPLIT, $0
+ JMP ·floor(SB)
+
+TEXT ·Ceil(SB), NOSPLIT, $0
+ JMP ·ceil(SB)
+
+TEXT ·Trunc(SB), NOSPLIT, $0
+ JMP ·trunc(SB)
+
+TEXT ·Frexp(SB), NOSPLIT, $0
+ JMP ·frexp(SB)
+
+TEXT ·Hypot(SB), NOSPLIT, $0
+ JMP ·hypot(SB)
+
+TEXT ·Ldexp(SB), NOSPLIT, $0
+ JMP ·ldexp(SB)
+
+TEXT ·Log10(SB), NOSPLIT, $0
+ JMP ·log10(SB)
+
+TEXT ·Log2(SB), NOSPLIT, $0
+ JMP ·log2(SB)
+
+TEXT ·Log1p(SB), NOSPLIT, $0
+ JMP ·log1p(SB)
+
+TEXT ·Log(SB), NOSPLIT, $0
+ JMP ·log(SB)
+
+TEXT ·Modf(SB), NOSPLIT, $0
+ JMP ·modf(SB)
+
+TEXT ·Mod(SB), NOSPLIT, $0
+ JMP ·mod(SB)
+
+TEXT ·Remainder(SB), NOSPLIT, $0
+ JMP ·remainder(SB)
+
+TEXT ·Sin(SB), NOSPLIT, $0
+ JMP ·sin(SB)
+
+TEXT ·Sinh(SB), NOSPLIT, $0
+ JMP ·sinh(SB)
+
+TEXT ·Cos(SB), NOSPLIT, $0
+ JMP ·cos(SB)
+
+TEXT ·Cosh(SB), NOSPLIT, $0
+ JMP ·cosh(SB)
+
+TEXT ·Tan(SB), NOSPLIT, $0
+ JMP ·tan(SB)
+
+TEXT ·Tanh(SB), NOSPLIT, $0
+ JMP ·tanh(SB)
+
+TEXT ·Cbrt(SB), NOSPLIT, $0
+ JMP ·cbrt(SB)
+
+TEXT ·Pow(SB), NOSPLIT, $0
+ JMP ·pow(SB)
+
diff --git a/src/math/stubs_ppc64x.s b/src/math/stubs_ppc64x.s
new file mode 100644
index 0000000..e7be07e
--- /dev/null
+++ b/src/math/stubs_ppc64x.s
@@ -0,0 +1,101 @@
+// Copyright 2014 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build ppc64 ppc64le
+
+#include "textflag.h"
+
+TEXT ·Asin(SB), NOSPLIT, $0
+ BR ·asin(SB)
+
+TEXT ·Acos(SB), NOSPLIT, $0
+ BR ·acos(SB)
+
+TEXT ·Asinh(SB), NOSPLIT, $0
+ BR ·asinh(SB)
+
+TEXT ·Acosh(SB), NOSPLIT, $0
+ BR ·acosh(SB)
+
+TEXT ·Atan2(SB), NOSPLIT, $0
+ BR ·atan2(SB)
+
+TEXT ·Atan(SB), NOSPLIT, $0
+ BR ·atan(SB)
+
+TEXT ·Atanh(SB), NOSPLIT, $0
+ BR ·atanh(SB)
+
+TEXT ·Min(SB), NOSPLIT, $0
+ BR ·min(SB)
+
+TEXT ·Max(SB), NOSPLIT, $0
+ BR ·max(SB)
+
+TEXT ·Erf(SB), NOSPLIT, $0
+ BR ·erf(SB)
+
+TEXT ·Erfc(SB), NOSPLIT, $0
+ BR ·erfc(SB)
+
+TEXT ·Exp2(SB), NOSPLIT, $0
+ BR ·exp2(SB)
+
+TEXT ·Expm1(SB), NOSPLIT, $0
+ BR ·expm1(SB)
+
+TEXT ·Exp(SB), NOSPLIT, $0
+ BR ·exp(SB)
+
+TEXT ·Frexp(SB), NOSPLIT, $0
+ BR ·frexp(SB)
+
+TEXT ·Hypot(SB), NOSPLIT, $0
+ BR ·hypot(SB)
+
+TEXT ·Ldexp(SB), NOSPLIT, $0
+ BR ·ldexp(SB)
+
+TEXT ·Log10(SB), NOSPLIT, $0
+ BR ·log10(SB)
+
+TEXT ·Log2(SB), NOSPLIT, $0
+ BR ·log2(SB)
+
+TEXT ·Log1p(SB), NOSPLIT, $0
+ BR ·log1p(SB)
+
+TEXT ·Log(SB), NOSPLIT, $0
+ BR ·log(SB)
+
+TEXT ·Mod(SB), NOSPLIT, $0
+ BR ·mod(SB)
+
+TEXT ·Remainder(SB), NOSPLIT, $0
+ BR ·remainder(SB)
+
+TEXT ·Sin(SB), NOSPLIT, $0
+ BR ·sin(SB)
+
+TEXT ·Sinh(SB), NOSPLIT, $0
+ BR ·sinh(SB)
+
+TEXT ·Cos(SB), NOSPLIT, $0
+ BR ·cos(SB)
+
+TEXT ·Cosh(SB), NOSPLIT, $0
+ BR ·cosh(SB)
+
+TEXT ·Tan(SB), NOSPLIT, $0
+ BR ·tan(SB)
+
+TEXT ·Tanh(SB), NOSPLIT, $0
+ BR ·tanh(SB)
+
+TEXT ·Cbrt(SB), NOSPLIT, $0
+ BR ·cbrt(SB)
+
+TEXT ·Pow(SB), NOSPLIT, $0
+ BR ·pow(SB)
+
diff --git a/src/math/stubs_riscv64.s b/src/math/stubs_riscv64.s
new file mode 100644
index 0000000..b36efb8
--- /dev/null
+++ b/src/math/stubs_riscv64.s
@@ -0,0 +1,104 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Asin(SB),NOSPLIT,$0
+ JMP ·asin(SB)
+
+TEXT ·Acos(SB),NOSPLIT,$0
+ JMP ·acos(SB)
+
+TEXT ·Asinh(SB),NOSPLIT,$0
+ JMP ·asinh(SB)
+
+TEXT ·Acosh(SB),NOSPLIT,$0
+ JMP ·acosh(SB)
+
+TEXT ·Atan2(SB),NOSPLIT,$0
+ JMP ·atan2(SB)
+
+TEXT ·Atan(SB),NOSPLIT,$0
+ JMP ·atan(SB)
+
+TEXT ·Atanh(SB),NOSPLIT,$0
+ JMP ·atanh(SB)
+
+TEXT ·Erf(SB),NOSPLIT,$0
+ JMP ·erf(SB)
+
+TEXT ·Erfc(SB),NOSPLIT,$0
+ JMP ·erfc(SB)
+
+TEXT ·Exp2(SB),NOSPLIT,$0
+ JMP ·exp2(SB)
+
+TEXT ·Expm1(SB),NOSPLIT,$0
+ JMP ·expm1(SB)
+
+TEXT ·Exp(SB),NOSPLIT,$0
+ JMP ·exp(SB)
+
+TEXT ·Floor(SB),NOSPLIT,$0
+ JMP ·floor(SB)
+
+TEXT ·Ceil(SB),NOSPLIT,$0
+ JMP ·ceil(SB)
+
+TEXT ·Trunc(SB),NOSPLIT,$0
+ JMP ·trunc(SB)
+
+TEXT ·Frexp(SB),NOSPLIT,$0
+ JMP ·frexp(SB)
+
+TEXT ·Hypot(SB),NOSPLIT,$0
+ JMP ·hypot(SB)
+
+TEXT ·Ldexp(SB),NOSPLIT,$0
+ JMP ·ldexp(SB)
+
+TEXT ·Log10(SB),NOSPLIT,$0
+ JMP ·log10(SB)
+
+TEXT ·Log2(SB),NOSPLIT,$0
+ JMP ·log2(SB)
+
+TEXT ·Log1p(SB),NOSPLIT,$0
+ JMP ·log1p(SB)
+
+TEXT ·Log(SB),NOSPLIT,$0
+ JMP ·log(SB)
+
+TEXT ·Modf(SB),NOSPLIT,$0
+ JMP ·modf(SB)
+
+TEXT ·Mod(SB),NOSPLIT,$0
+ JMP ·mod(SB)
+
+TEXT ·Remainder(SB),NOSPLIT,$0
+ JMP ·remainder(SB)
+
+TEXT ·Sin(SB),NOSPLIT,$0
+ JMP ·sin(SB)
+
+TEXT ·Sinh(SB),NOSPLIT,$0
+ JMP ·sinh(SB)
+
+TEXT ·Cos(SB),NOSPLIT,$0
+ JMP ·cos(SB)
+
+TEXT ·Cosh(SB),NOSPLIT,$0
+ JMP ·cosh(SB)
+
+TEXT ·Tan(SB),NOSPLIT,$0
+ JMP ·tan(SB)
+
+TEXT ·Tanh(SB),NOSPLIT,$0
+ JMP ·tanh(SB)
+
+TEXT ·Cbrt(SB),NOSPLIT,$0
+ JMP ·cbrt(SB)
+
+TEXT ·Pow(SB),NOSPLIT,$0
+ JMP ·pow(SB)
diff --git a/src/math/stubs_s390x.s b/src/math/stubs_s390x.s
new file mode 100644
index 0000000..d0087ab
--- /dev/null
+++ b/src/math/stubs_s390x.s
@@ -0,0 +1,492 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Exp2(SB), NOSPLIT, $0
+ BR ·exp2(SB)
+
+TEXT ·Frexp(SB), NOSPLIT, $0
+ BR ·frexp(SB)
+
+TEXT ·Hypot(SB), NOSPLIT, $0
+ BR ·hypot(SB)
+
+TEXT ·Ldexp(SB), NOSPLIT, $0
+ BR ·ldexp(SB)
+
+TEXT ·Log2(SB), NOSPLIT, $0
+ BR ·log2(SB)
+
+TEXT ·Modf(SB), NOSPLIT, $0
+ BR ·modf(SB)
+
+TEXT ·Mod(SB), NOSPLIT, $0
+ BR ·mod(SB)
+
+TEXT ·Remainder(SB), NOSPLIT, $0
+ BR ·remainder(SB)
+
+TEXT ·Log10(SB), NOSPLIT, $0
+ MOVD ·log10vectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·log10TrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·log10vectorfacility+0x00(SB), R1
+ MOVD $·log10(SB), R2
+ MOVD R2, 0(R1)
+ BR ·log10(SB)
+
+vectorimpl:
+ MOVD $·log10vectorfacility+0x00(SB), R1
+ MOVD $·log10Asm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·log10Asm(SB)
+
+GLOBL ·log10vectorfacility+0x00(SB), NOPTR, $8
+DATA ·log10vectorfacility+0x00(SB)/8, $·log10TrampolineSetup(SB)
+
+TEXT ·Cos(SB), NOSPLIT, $0
+ MOVD ·cosvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·cosTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·cosvectorfacility+0x00(SB), R1
+ MOVD $·cos(SB), R2
+ MOVD R2, 0(R1)
+ BR ·cos(SB)
+
+vectorimpl:
+ MOVD $·cosvectorfacility+0x00(SB), R1
+ MOVD $·cosAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·cosAsm(SB)
+
+GLOBL ·cosvectorfacility+0x00(SB), NOPTR, $8
+DATA ·cosvectorfacility+0x00(SB)/8, $·cosTrampolineSetup(SB)
+
+TEXT ·Cosh(SB), NOSPLIT, $0
+ MOVD ·coshvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·coshTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·coshvectorfacility+0x00(SB), R1
+ MOVD $·cosh(SB), R2
+ MOVD R2, 0(R1)
+ BR ·cosh(SB)
+
+vectorimpl:
+ MOVD $·coshvectorfacility+0x00(SB), R1
+ MOVD $·coshAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·coshAsm(SB)
+
+GLOBL ·coshvectorfacility+0x00(SB), NOPTR, $8
+DATA ·coshvectorfacility+0x00(SB)/8, $·coshTrampolineSetup(SB)
+
+TEXT ·Sin(SB), NOSPLIT, $0
+ MOVD ·sinvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·sinTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·sinvectorfacility+0x00(SB), R1
+ MOVD $·sin(SB), R2
+ MOVD R2, 0(R1)
+ BR ·sin(SB)
+
+vectorimpl:
+ MOVD $·sinvectorfacility+0x00(SB), R1
+ MOVD $·sinAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·sinAsm(SB)
+
+GLOBL ·sinvectorfacility+0x00(SB), NOPTR, $8
+DATA ·sinvectorfacility+0x00(SB)/8, $·sinTrampolineSetup(SB)
+
+TEXT ·Sinh(SB), NOSPLIT, $0
+ MOVD ·sinhvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·sinhTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·sinhvectorfacility+0x00(SB), R1
+ MOVD $·sinh(SB), R2
+ MOVD R2, 0(R1)
+ BR ·sinh(SB)
+
+vectorimpl:
+ MOVD $·sinhvectorfacility+0x00(SB), R1
+ MOVD $·sinhAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·sinhAsm(SB)
+
+GLOBL ·sinhvectorfacility+0x00(SB), NOPTR, $8
+DATA ·sinhvectorfacility+0x00(SB)/8, $·sinhTrampolineSetup(SB)
+
+TEXT ·Tanh(SB), NOSPLIT, $0
+ MOVD ·tanhvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·tanhTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·tanhvectorfacility+0x00(SB), R1
+ MOVD $·tanh(SB), R2
+ MOVD R2, 0(R1)
+ BR ·tanh(SB)
+
+vectorimpl:
+ MOVD $·tanhvectorfacility+0x00(SB), R1
+ MOVD $·tanhAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·tanhAsm(SB)
+
+GLOBL ·tanhvectorfacility+0x00(SB), NOPTR, $8
+DATA ·tanhvectorfacility+0x00(SB)/8, $·tanhTrampolineSetup(SB)
+
+TEXT ·Log1p(SB), NOSPLIT, $0
+ MOVD ·log1pvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·log1pTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·log1pvectorfacility+0x00(SB), R1
+ MOVD $·log1p(SB), R2
+ MOVD R2, 0(R1)
+ BR ·log1p(SB)
+
+vectorimpl:
+ MOVD $·log1pvectorfacility+0x00(SB), R1
+ MOVD $·log1pAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·log1pAsm(SB)
+
+GLOBL ·log1pvectorfacility+0x00(SB), NOPTR, $8
+DATA ·log1pvectorfacility+0x00(SB)/8, $·log1pTrampolineSetup(SB)
+
+TEXT ·Atanh(SB), NOSPLIT, $0
+ MOVD ·atanhvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·atanhTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·atanhvectorfacility+0x00(SB), R1
+ MOVD $·atanh(SB), R2
+ MOVD R2, 0(R1)
+ BR ·atanh(SB)
+
+vectorimpl:
+ MOVD $·atanhvectorfacility+0x00(SB), R1
+ MOVD $·atanhAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·atanhAsm(SB)
+
+GLOBL ·atanhvectorfacility+0x00(SB), NOPTR, $8
+DATA ·atanhvectorfacility+0x00(SB)/8, $·atanhTrampolineSetup(SB)
+
+TEXT ·Acos(SB), NOSPLIT, $0
+ MOVD ·acosvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·acosTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·acosvectorfacility+0x00(SB), R1
+ MOVD $·acos(SB), R2
+ MOVD R2, 0(R1)
+ BR ·acos(SB)
+
+vectorimpl:
+ MOVD $·acosvectorfacility+0x00(SB), R1
+ MOVD $·acosAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·acosAsm(SB)
+
+GLOBL ·acosvectorfacility+0x00(SB), NOPTR, $8
+DATA ·acosvectorfacility+0x00(SB)/8, $·acosTrampolineSetup(SB)
+
+TEXT ·Asin(SB), NOSPLIT, $0
+ MOVD ·asinvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·asinTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·asinvectorfacility+0x00(SB), R1
+ MOVD $·asin(SB), R2
+ MOVD R2, 0(R1)
+ BR ·asin(SB)
+
+vectorimpl:
+ MOVD $·asinvectorfacility+0x00(SB), R1
+ MOVD $·asinAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·asinAsm(SB)
+
+GLOBL ·asinvectorfacility+0x00(SB), NOPTR, $8
+DATA ·asinvectorfacility+0x00(SB)/8, $·asinTrampolineSetup(SB)
+
+TEXT ·Asinh(SB), NOSPLIT, $0
+ MOVD ·asinhvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·asinhTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·asinhvectorfacility+0x00(SB), R1
+ MOVD $·asinh(SB), R2
+ MOVD R2, 0(R1)
+ BR ·asinh(SB)
+
+vectorimpl:
+ MOVD $·asinhvectorfacility+0x00(SB), R1
+ MOVD $·asinhAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·asinhAsm(SB)
+
+GLOBL ·asinhvectorfacility+0x00(SB), NOPTR, $8
+DATA ·asinhvectorfacility+0x00(SB)/8, $·asinhTrampolineSetup(SB)
+
+TEXT ·Acosh(SB), NOSPLIT, $0
+ MOVD ·acoshvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·acoshTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·acoshvectorfacility+0x00(SB), R1
+ MOVD $·acosh(SB), R2
+ MOVD R2, 0(R1)
+ BR ·acosh(SB)
+
+vectorimpl:
+ MOVD $·acoshvectorfacility+0x00(SB), R1
+ MOVD $·acoshAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·acoshAsm(SB)
+
+GLOBL ·acoshvectorfacility+0x00(SB), NOPTR, $8
+DATA ·acoshvectorfacility+0x00(SB)/8, $·acoshTrampolineSetup(SB)
+
+TEXT ·Erf(SB), NOSPLIT, $0
+ MOVD ·erfvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·erfTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·erfvectorfacility+0x00(SB), R1
+ MOVD $·erf(SB), R2
+ MOVD R2, 0(R1)
+ BR ·erf(SB)
+
+vectorimpl:
+ MOVD $·erfvectorfacility+0x00(SB), R1
+ MOVD $·erfAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·erfAsm(SB)
+
+GLOBL ·erfvectorfacility+0x00(SB), NOPTR, $8
+DATA ·erfvectorfacility+0x00(SB)/8, $·erfTrampolineSetup(SB)
+
+TEXT ·Erfc(SB), NOSPLIT, $0
+ MOVD ·erfcvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·erfcTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·erfcvectorfacility+0x00(SB), R1
+ MOVD $·erfc(SB), R2
+ MOVD R2, 0(R1)
+ BR ·erfc(SB)
+
+vectorimpl:
+ MOVD $·erfcvectorfacility+0x00(SB), R1
+ MOVD $·erfcAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·erfcAsm(SB)
+
+GLOBL ·erfcvectorfacility+0x00(SB), NOPTR, $8
+DATA ·erfcvectorfacility+0x00(SB)/8, $·erfcTrampolineSetup(SB)
+
+TEXT ·Atan(SB), NOSPLIT, $0
+ MOVD ·atanvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·atanTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·atanvectorfacility+0x00(SB), R1
+ MOVD $·atan(SB), R2
+ MOVD R2, 0(R1)
+ BR ·atan(SB)
+
+vectorimpl:
+ MOVD $·atanvectorfacility+0x00(SB), R1
+ MOVD $·atanAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·atanAsm(SB)
+
+GLOBL ·atanvectorfacility+0x00(SB), NOPTR, $8
+DATA ·atanvectorfacility+0x00(SB)/8, $·atanTrampolineSetup(SB)
+
+TEXT ·Atan2(SB), NOSPLIT, $0
+ MOVD ·atan2vectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·atan2TrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·atan2vectorfacility+0x00(SB), R1
+ MOVD $·atan2(SB), R2
+ MOVD R2, 0(R1)
+ BR ·atan2(SB)
+
+vectorimpl:
+ MOVD $·atan2vectorfacility+0x00(SB), R1
+ MOVD $·atan2Asm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·atan2Asm(SB)
+
+GLOBL ·atan2vectorfacility+0x00(SB), NOPTR, $8
+DATA ·atan2vectorfacility+0x00(SB)/8, $·atan2TrampolineSetup(SB)
+
+TEXT ·Cbrt(SB), NOSPLIT, $0
+ MOVD ·cbrtvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·cbrtTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·cbrtvectorfacility+0x00(SB), R1
+ MOVD $·cbrt(SB), R2
+ MOVD R2, 0(R1)
+ BR ·cbrt(SB)
+
+vectorimpl:
+ MOVD $·cbrtvectorfacility+0x00(SB), R1
+ MOVD $·cbrtAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·cbrtAsm(SB)
+
+GLOBL ·cbrtvectorfacility+0x00(SB), NOPTR, $8
+DATA ·cbrtvectorfacility+0x00(SB)/8, $·cbrtTrampolineSetup(SB)
+
+TEXT ·Log(SB), NOSPLIT, $0
+ MOVD ·logvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·logTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·logvectorfacility+0x00(SB), R1
+ MOVD $·log(SB), R2
+ MOVD R2, 0(R1)
+ BR ·log(SB)
+
+vectorimpl:
+ MOVD $·logvectorfacility+0x00(SB), R1
+ MOVD $·logAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·logAsm(SB)
+
+GLOBL ·logvectorfacility+0x00(SB), NOPTR, $8
+DATA ·logvectorfacility+0x00(SB)/8, $·logTrampolineSetup(SB)
+
+TEXT ·Tan(SB), NOSPLIT, $0
+ MOVD ·tanvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·tanTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·tanvectorfacility+0x00(SB), R1
+ MOVD $·tan(SB), R2
+ MOVD R2, 0(R1)
+ BR ·tan(SB)
+
+vectorimpl:
+ MOVD $·tanvectorfacility+0x00(SB), R1
+ MOVD $·tanAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·tanAsm(SB)
+
+GLOBL ·tanvectorfacility+0x00(SB), NOPTR, $8
+DATA ·tanvectorfacility+0x00(SB)/8, $·tanTrampolineSetup(SB)
+
+TEXT ·Exp(SB), NOSPLIT, $0
+ MOVD ·expvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·expTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·expvectorfacility+0x00(SB), R1
+ MOVD $·exp(SB), R2
+ MOVD R2, 0(R1)
+ BR ·exp(SB)
+
+vectorimpl:
+ MOVD $·expvectorfacility+0x00(SB), R1
+ MOVD $·expAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·expAsm(SB)
+
+GLOBL ·expvectorfacility+0x00(SB), NOPTR, $8
+DATA ·expvectorfacility+0x00(SB)/8, $·expTrampolineSetup(SB)
+
+TEXT ·Expm1(SB), NOSPLIT, $0
+ MOVD ·expm1vectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·expm1TrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·expm1vectorfacility+0x00(SB), R1
+ MOVD $·expm1(SB), R2
+ MOVD R2, 0(R1)
+ BR ·expm1(SB)
+
+vectorimpl:
+ MOVD $·expm1vectorfacility+0x00(SB), R1
+ MOVD $·expm1Asm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·expm1Asm(SB)
+
+GLOBL ·expm1vectorfacility+0x00(SB), NOPTR, $8
+DATA ·expm1vectorfacility+0x00(SB)/8, $·expm1TrampolineSetup(SB)
+
+TEXT ·Pow(SB), NOSPLIT, $0
+ MOVD ·powvectorfacility+0x00(SB), R1
+ BR (R1)
+
+TEXT ·powTrampolineSetup(SB), NOSPLIT, $0
+ MOVB ·hasVX(SB), R1
+ CMPBEQ R1, $1, vectorimpl // vectorfacility = 1, vector supported
+ MOVD $·powvectorfacility+0x00(SB), R1
+ MOVD $·pow(SB), R2
+ MOVD R2, 0(R1)
+ BR ·pow(SB)
+
+vectorimpl:
+ MOVD $·powvectorfacility+0x00(SB), R1
+ MOVD $·powAsm(SB), R2
+ MOVD R2, 0(R1)
+ BR ·powAsm(SB)
+
+GLOBL ·powvectorfacility+0x00(SB), NOPTR, $8
+DATA ·powvectorfacility+0x00(SB)/8, $·powTrampolineSetup(SB)
+
diff --git a/src/math/stubs_wasm.s b/src/math/stubs_wasm.s
new file mode 100644
index 0000000..c97a2d7
--- /dev/null
+++ b/src/math/stubs_wasm.s
@@ -0,0 +1,101 @@
+// Copyright 2018 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Asin(SB), NOSPLIT, $0
+ JMP ·asin(SB)
+
+TEXT ·Asinh(SB), NOSPLIT, $0
+ JMP ·asinh(SB)
+
+TEXT ·Acos(SB), NOSPLIT, $0
+ JMP ·acos(SB)
+
+TEXT ·Acosh(SB), NOSPLIT, $0
+ JMP ·acosh(SB)
+
+TEXT ·Atan(SB), NOSPLIT, $0
+ JMP ·atan(SB)
+
+TEXT ·Atanh(SB), NOSPLIT, $0
+ JMP ·atanh(SB)
+
+TEXT ·Atan2(SB), NOSPLIT, $0
+ JMP ·atan2(SB)
+
+TEXT ·Cbrt(SB), NOSPLIT, $0
+ JMP ·cbrt(SB)
+
+TEXT ·Cos(SB), NOSPLIT, $0
+ JMP ·cos(SB)
+
+TEXT ·Cosh(SB), NOSPLIT, $0
+ JMP ·cosh(SB)
+
+TEXT ·Erf(SB), NOSPLIT, $0
+ JMP ·erf(SB)
+
+TEXT ·Erfc(SB), NOSPLIT, $0
+ JMP ·erfc(SB)
+
+TEXT ·Exp(SB), NOSPLIT, $0
+ JMP ·exp(SB)
+
+TEXT ·Expm1(SB), NOSPLIT, $0
+ JMP ·expm1(SB)
+
+TEXT ·Exp2(SB), NOSPLIT, $0
+ JMP ·exp2(SB)
+
+TEXT ·Frexp(SB), NOSPLIT, $0
+ JMP ·frexp(SB)
+
+TEXT ·Hypot(SB), NOSPLIT, $0
+ JMP ·hypot(SB)
+
+TEXT ·Ldexp(SB), NOSPLIT, $0
+ JMP ·ldexp(SB)
+
+TEXT ·Log(SB), NOSPLIT, $0
+ JMP ·log(SB)
+
+TEXT ·Log1p(SB), NOSPLIT, $0
+ JMP ·log1p(SB)
+
+TEXT ·Log10(SB), NOSPLIT, $0
+ JMP ·log10(SB)
+
+TEXT ·Log2(SB), NOSPLIT, $0
+ JMP ·log2(SB)
+
+TEXT ·Max(SB), NOSPLIT, $0
+ JMP ·max(SB)
+
+TEXT ·Min(SB), NOSPLIT, $0
+ JMP ·min(SB)
+
+TEXT ·Mod(SB), NOSPLIT, $0
+ JMP ·mod(SB)
+
+TEXT ·Modf(SB), NOSPLIT, $0
+ JMP ·modf(SB)
+
+TEXT ·Pow(SB), NOSPLIT, $0
+ JMP ·pow(SB)
+
+TEXT ·Remainder(SB), NOSPLIT, $0
+ JMP ·remainder(SB)
+
+TEXT ·Sin(SB), NOSPLIT, $0
+ JMP ·sin(SB)
+
+TEXT ·Sinh(SB), NOSPLIT, $0
+ JMP ·sinh(SB)
+
+TEXT ·Tan(SB), NOSPLIT, $0
+ JMP ·tan(SB)
+
+TEXT ·Tanh(SB), NOSPLIT, $0
+ JMP ·tanh(SB)
diff --git a/src/math/tan.go b/src/math/tan.go
new file mode 100644
index 0000000..49b1239
--- /dev/null
+++ b/src/math/tan.go
@@ -0,0 +1,134 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point tangent.
+*/
+
+// The original C code, the long comment, and the constants
+// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
+// available from http://www.netlib.org/cephes/cmath.tgz.
+// The go code is a simplified version of the original C.
+//
+// tan.c
+//
+// Circular tangent
+//
+// SYNOPSIS:
+//
+// double x, y, tan();
+// y = tan( x );
+//
+// DESCRIPTION:
+//
+// Returns the circular tangent of the radian argument x.
+//
+// Range reduction is modulo pi/4. A rational function
+// x + x**3 P(x**2)/Q(x**2)
+// is employed in the basic interval [0, pi/4].
+//
+// ACCURACY:
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC +-1.07e9 44000 4.1e-17 1.0e-17
+// IEEE +-1.07e9 30000 2.9e-16 8.1e-17
+//
+// Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9. The loss
+// is not gradual, but jumps suddenly to about 1 part in 10e7. Results may
+// be meaningless for x > 2**49 = 5.6e14.
+// [Accuracy loss statement from sin.go comments.]
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// tan coefficients
+var _tanP = [...]float64{
+ -1.30936939181383777646e4, // 0xc0c992d8d24f3f38
+ 1.15351664838587416140e6, // 0x413199eca5fc9ddd
+ -1.79565251976484877988e7, // 0xc1711fead3299176
+}
+var _tanQ = [...]float64{
+ 1.00000000000000000000e0,
+ 1.36812963470692954678e4, //0x40cab8a5eeb36572
+ -1.32089234440210967447e6, //0xc13427bc582abc96
+ 2.50083801823357915839e7, //0x4177d98fc2ead8ef
+ -5.38695755929454629881e7, //0xc189afe03cbe5a31
+}
+
+// Tan returns the tangent of the radian argument x.
+//
+// Special cases are:
+// Tan(±0) = ±0
+// Tan(±Inf) = NaN
+// Tan(NaN) = NaN
+func Tan(x float64) float64
+
+func tan(x float64) float64 {
+ const (
+ PI4A = 7.85398125648498535156e-1 // 0x3fe921fb40000000, Pi/4 split into three parts
+ PI4B = 3.77489470793079817668e-8 // 0x3e64442d00000000,
+ PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170,
+ )
+ // special cases
+ switch {
+ case x == 0 || IsNaN(x):
+ return x // return ±0 || NaN()
+ case IsInf(x, 0):
+ return NaN()
+ }
+
+ // make argument positive but save the sign
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ var j uint64
+ var y, z float64
+ if x >= reduceThreshold {
+ j, z = trigReduce(x)
+ } else {
+ j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
+ y = float64(j) // integer part of x/(Pi/4), as float
+
+ /* map zeros and singularities to origin */
+ if j&1 == 1 {
+ j++
+ y++
+ }
+
+ z = ((x - y*PI4A) - y*PI4B) - y*PI4C
+ }
+ zz := z * z
+
+ if zz > 1e-14 {
+ y = z + z*(zz*(((_tanP[0]*zz)+_tanP[1])*zz+_tanP[2])/((((zz+_tanQ[1])*zz+_tanQ[2])*zz+_tanQ[3])*zz+_tanQ[4]))
+ } else {
+ y = z
+ }
+ if j&2 == 2 {
+ y = -1 / y
+ }
+ if sign {
+ y = -y
+ }
+ return y
+}
diff --git a/src/math/tan_s390x.s b/src/math/tan_s390x.s
new file mode 100644
index 0000000..b6e2295
--- /dev/null
+++ b/src/math/tan_s390x.s
@@ -0,0 +1,110 @@
+// Copyright 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial approximations
+DATA ·tanrodataL13<> + 0(SB)/8, $0.181017336383229927e-07
+DATA ·tanrodataL13<> + 8(SB)/8, $-.256590857271311164e-03
+DATA ·tanrodataL13<> + 16(SB)/8, $-.464359274328689195e+00
+DATA ·tanrodataL13<> + 24(SB)/8, $1.0
+DATA ·tanrodataL13<> + 32(SB)/8, $-.333333333333333464e+00
+DATA ·tanrodataL13<> + 40(SB)/8, $0.245751217306830032e-01
+DATA ·tanrodataL13<> + 48(SB)/8, $-.245391301343844510e-03
+DATA ·tanrodataL13<> + 56(SB)/8, $0.214530914428992319e-01
+DATA ·tanrodataL13<> + 64(SB)/8, $0.108285667160535624e-31
+DATA ·tanrodataL13<> + 72(SB)/8, $0.612323399573676480e-16
+DATA ·tanrodataL13<> + 80(SB)/8, $0.157079632679489656e+01
+DATA ·tanrodataL13<> + 88(SB)/8, $0.636619772367581341e+00
+GLOBL ·tanrodataL13<> + 0(SB), RODATA, $96
+
+// Constants
+DATA ·tanxnan<> + 0(SB)/8, $0x7ff8000000000000
+GLOBL ·tanxnan<> + 0(SB), RODATA, $8
+DATA ·tanxlim<> + 0(SB)/8, $0x432921fb54442d19
+GLOBL ·tanxlim<> + 0(SB), RODATA, $8
+DATA ·tanxadd<> + 0(SB)/8, $0xc338000000000000
+GLOBL ·tanxadd<> + 0(SB), RODATA, $8
+
+// Tan returns the tangent of the radian argument.
+//
+// Special cases are:
+// Tan(±0) = ±0
+// Tan(±Inf) = NaN
+// Tan(NaN) = NaN
+// The algorithm used is minimax polynomial approximation using a table of
+// polynomial coefficients determined with a Remez exchange algorithm.
+
+TEXT ·tanAsm(SB), NOSPLIT, $0-16
+ FMOVD x+0(FP), F0
+ //specail case Tan(±0) = ±0
+ FMOVD $(0.0), F1
+ FCMPU F0, F1
+ BEQ atanIsZero
+
+ MOVD $·tanrodataL13<>+0(SB), R5
+ LTDBR F0, F0
+ BLTU L10
+ FMOVD F0, F2
+L2:
+ MOVD $·tanxlim<>+0(SB), R1
+ WORD $0xED201000 //cdb %f2,0(%r1)
+ BYTE $0x00
+ BYTE $0x19
+ BGE L11
+ BVS L11
+ MOVD $·tanxadd<>+0(SB), R1
+ FMOVD 88(R5), F6
+ FMOVD 0(R1), F4
+ WFMSDB V0, V6, V4, V6
+ FMOVD 80(R5), F1
+ FADD F6, F4
+ FMOVD 72(R5), F2
+ FMSUB F1, F4, F0
+ FMOVD 64(R5), F3
+ WFMADB V4, V2, V0, V2
+ FMOVD 56(R5), F1
+ WFMADB V4, V3, V2, V4
+ FMUL F2, F2
+ VLEG $0, 48(R5), V18
+ LGDR F6, R1
+ FMOVD 40(R5), F5
+ FMOVD 32(R5), F3
+ FMADD F1, F2, F3
+ FMOVD 24(R5), F1
+ FMOVD 16(R5), F7
+ FMOVD 8(R5), F0
+ WFMADB V2, V7, V1, V7
+ WFMADB V2, V0, V5, V0
+ WFMDB V2, V2, V1
+ FMOVD 0(R5), F5
+ WFLCDB V4, V16
+ WFMADB V2, V5, V18, V5
+ WFMADB V1, V0, V7, V0
+ TMLL R1, $1
+ WFMADB V1, V5, V3, V1
+ BNE L12
+ WFDDB V0, V1, V0
+ WFMDB V2, V16, V2
+ WFMADB V2, V0, V4, V0
+ WORD $0xB3130000 //lcdbr %f0,%f0
+ FMOVD F0, ret+8(FP)
+ RET
+L12:
+ WFMSDB V2, V1, V0, V2
+ WFMDB V16, V2, V2
+ FDIV F2, F0
+ FMOVD F0, ret+8(FP)
+ RET
+L11:
+ MOVD $·tanxnan<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+L10:
+ WORD $0xB3130020 //lcdbr %f2,%f0
+ BR L2
+atanIsZero:
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/tanh.go b/src/math/tanh.go
new file mode 100644
index 0000000..0b7fb7f
--- /dev/null
+++ b/src/math/tanh.go
@@ -0,0 +1,99 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
+// available from http://www.netlib.org/cephes/cmath.tgz.
+// The go code is a simplified version of the original C.
+// tanh.c
+//
+// Hyperbolic tangent
+//
+// SYNOPSIS:
+//
+// double x, y, tanh();
+//
+// y = tanh( x );
+//
+// DESCRIPTION:
+//
+// Returns hyperbolic tangent of argument in the range MINLOG to MAXLOG.
+// MAXLOG = 8.8029691931113054295988e+01 = log(2**127)
+// MINLOG = -8.872283911167299960540e+01 = log(2**-128)
+//
+// A rational function is used for |x| < 0.625. The form
+// x + x**3 P(x)/Q(x) of Cody & Waite is employed.
+// Otherwise,
+// tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -2,2 30000 2.5e-16 5.8e-17
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+//
+
+var tanhP = [...]float64{
+ -9.64399179425052238628e-1,
+ -9.92877231001918586564e1,
+ -1.61468768441708447952e3,
+}
+var tanhQ = [...]float64{
+ 1.12811678491632931402e2,
+ 2.23548839060100448583e3,
+ 4.84406305325125486048e3,
+}
+
+// Tanh returns the hyperbolic tangent of x.
+//
+// Special cases are:
+// Tanh(±0) = ±0
+// Tanh(±Inf) = ±1
+// Tanh(NaN) = NaN
+func Tanh(x float64) float64
+
+func tanh(x float64) float64 {
+ const MAXLOG = 8.8029691931113054295988e+01 // log(2**127)
+ z := Abs(x)
+ switch {
+ case z > 0.5*MAXLOG:
+ if x < 0 {
+ return -1
+ }
+ return 1
+ case z >= 0.625:
+ s := Exp(2 * z)
+ z = 1 - 2/(s+1)
+ if x < 0 {
+ z = -z
+ }
+ default:
+ if x == 0 {
+ return x
+ }
+ s := x * x
+ z = x + x*s*((tanhP[0]*s+tanhP[1])*s+tanhP[2])/(((s+tanhQ[0])*s+tanhQ[1])*s+tanhQ[2])
+ }
+ return z
+}
diff --git a/src/math/tanh_s390x.s b/src/math/tanh_s390x.s
new file mode 100644
index 0000000..7e2d4dd
--- /dev/null
+++ b/src/math/tanh_s390x.s
@@ -0,0 +1,169 @@
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// Minimax polynomial approximations
+DATA tanhrodataL18<>+0(SB)/8, $-1.0
+DATA tanhrodataL18<>+8(SB)/8, $-2.0
+DATA tanhrodataL18<>+16(SB)/8, $1.0
+DATA tanhrodataL18<>+24(SB)/8, $2.0
+DATA tanhrodataL18<>+32(SB)/8, $0.20000000000000011868E+01
+DATA tanhrodataL18<>+40(SB)/8, $0.13333333333333341256E+01
+DATA tanhrodataL18<>+48(SB)/8, $0.26666666663549111502E+00
+DATA tanhrodataL18<>+56(SB)/8, $0.66666666658721844678E+00
+DATA tanhrodataL18<>+64(SB)/8, $0.88890217768964374821E-01
+DATA tanhrodataL18<>+72(SB)/8, $0.25397199429103821138E-01
+DATA tanhrodataL18<>+80(SB)/8, $-.346573590279972643E+00
+DATA tanhrodataL18<>+88(SB)/8, $20.E0
+GLOBL tanhrodataL18<>+0(SB), RODATA, $96
+
+// Constants
+DATA tanhrlog2<>+0(SB)/8, $0x4007154760000000
+GLOBL tanhrlog2<>+0(SB), RODATA, $8
+DATA tanhxadd<>+0(SB)/8, $0xc2f0000100003ff0
+GLOBL tanhxadd<>+0(SB), RODATA, $8
+DATA tanhxmone<>+0(SB)/8, $-1.0
+GLOBL tanhxmone<>+0(SB), RODATA, $8
+DATA tanhxzero<>+0(SB)/8, $0
+GLOBL tanhxzero<>+0(SB), RODATA, $8
+
+// Polynomial coefficients
+DATA tanhtab<>+0(SB)/8, $0.000000000000000000E+00
+DATA tanhtab<>+8(SB)/8, $-.171540871271399150E-01
+DATA tanhtab<>+16(SB)/8, $-.306597931864376363E-01
+DATA tanhtab<>+24(SB)/8, $-.410200970469965021E-01
+DATA tanhtab<>+32(SB)/8, $-.486343079978231466E-01
+DATA tanhtab<>+40(SB)/8, $-.538226193725835820E-01
+DATA tanhtab<>+48(SB)/8, $-.568439602538111520E-01
+DATA tanhtab<>+56(SB)/8, $-.579091847395528847E-01
+DATA tanhtab<>+64(SB)/8, $-.571909584179366341E-01
+DATA tanhtab<>+72(SB)/8, $-.548312665987204407E-01
+DATA tanhtab<>+80(SB)/8, $-.509471843643441085E-01
+DATA tanhtab<>+88(SB)/8, $-.456353588448863359E-01
+DATA tanhtab<>+96(SB)/8, $-.389755254243262365E-01
+DATA tanhtab<>+104(SB)/8, $-.310332908285244231E-01
+DATA tanhtab<>+112(SB)/8, $-.218623539150173528E-01
+DATA tanhtab<>+120(SB)/8, $-.115062908917949451E-01
+GLOBL tanhtab<>+0(SB), RODATA, $128
+
+// Tanh returns the hyperbolic tangent of the argument.
+//
+// Special cases are:
+// Tanh(±0) = ±0
+// Tanh(±Inf) = ±1
+// Tanh(NaN) = NaN
+// The algorithm used is minimax polynomial approximation using a table of
+// polynomial coefficients determined with a Remez exchange algorithm.
+
+TEXT ·tanhAsm(SB),NOSPLIT,$0-16
+ FMOVD x+0(FP), F0
+ // special case Tanh(±0) = ±0
+ FMOVD $(0.0), F1
+ FCMPU F0, F1
+ BEQ tanhIsZero
+ MOVD $tanhrodataL18<>+0(SB), R5
+ LTDBR F0, F0
+ MOVD $0x4034000000000000, R1
+ BLTU L15
+ FMOVD F0, F1
+L2:
+ MOVD $tanhxadd<>+0(SB), R2
+ FMOVD 0(R2), F2
+ MOVD tanhrlog2<>+0(SB), R2
+ LDGR R2, F4
+ WFMSDB V0, V4, V2, V4
+ MOVD $tanhtab<>+0(SB), R3
+ LGDR F4, R2
+ RISBGZ $57, $60, $3, R2, R4
+ WORD $0xED105058 //cdb %f1,.L19-.L18(%r5)
+ BYTE $0x00
+ BYTE $0x19
+ RISBGN $0, $15, $48, R2, R1
+ WORD $0x68543000 //ld %f5,0(%r4,%r3)
+ LDGR R1, F6
+ BLT L3
+ MOVD $tanhxzero<>+0(SB), R1
+ FMOVD 0(R1), F2
+ WFCHDBS V0, V2, V4
+ BEQ L9
+ WFCHDBS V2, V0, V2
+ BNE L1
+ MOVD $tanhxmone<>+0(SB), R1
+ FMOVD 0(R1), F0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L3:
+ FADD F4, F2
+ FMOVD tanhrodataL18<>+80(SB), F4
+ FMADD F4, F2, F0
+ FMOVD tanhrodataL18<>+72(SB), F1
+ WFMDB V0, V0, V3
+ FMOVD tanhrodataL18<>+64(SB), F2
+ WFMADB V0, V1, V2, V1
+ FMOVD tanhrodataL18<>+56(SB), F4
+ FMOVD tanhrodataL18<>+48(SB), F2
+ WFMADB V1, V3, V4, V1
+ FMOVD tanhrodataL18<>+40(SB), F4
+ WFMADB V3, V2, V4, V2
+ FMOVD tanhrodataL18<>+32(SB), F4
+ WORD $0xB9270022 //lhr %r2,%r2
+ WFMADB V3, V1, V4, V1
+ FMOVD tanhrodataL18<>+24(SB), F4
+ WFMADB V3, V2, V4, V3
+ WFMADB V0, V5, V0, V2
+ WFMADB V0, V1, V3, V0
+ WORD $0xA7183ECF //lhi %r1,16079
+ WFMADB V0, V2, V5, V2
+ FMUL F6, F2
+ MOVW R2, R10
+ MOVW R1, R11
+ CMPBLE R10, R11, L16
+ FMOVD F6, F0
+ WORD $0xED005010 //adb %f0,.L28-.L18(%r5)
+ BYTE $0x00
+ BYTE $0x1A
+ WORD $0xA7184330 //lhi %r1,17200
+ FADD F2, F0
+ MOVW R2, R10
+ MOVW R1, R11
+ CMPBGT R10, R11, L17
+ WORD $0xED605010 //sdb %f6,.L28-.L18(%r5)
+ BYTE $0x00
+ BYTE $0x1B
+ FADD F6, F2
+ WFDDB V0, V2, V0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L9:
+ FMOVD tanhrodataL18<>+16(SB), F0
+L1:
+ FMOVD F0, ret+8(FP)
+ RET
+
+L15:
+ FNEG F0, F1
+ BR L2
+L16:
+ FADD F6, F2
+ FMOVD tanhrodataL18<>+8(SB), F0
+ FMADD F4, F2, F0
+ FMOVD tanhrodataL18<>+0(SB), F4
+ FNEG F0, F0
+ WFMADB V0, V2, V4, V0
+ FMOVD F0, ret+8(FP)
+ RET
+
+L17:
+ WFDDB V0, V4, V0
+ FMOVD tanhrodataL18<>+16(SB), F2
+ WFSDB V0, V2, V0
+ FMOVD F0, ret+8(FP)
+ RET
+
+tanhIsZero: //return ±0
+ FMOVD F0, ret+8(FP)
+ RET
diff --git a/src/math/trig_reduce.go b/src/math/trig_reduce.go
new file mode 100644
index 0000000..5cdf4fa
--- /dev/null
+++ b/src/math/trig_reduce.go
@@ -0,0 +1,100 @@
+// Copyright 2018 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+import (
+ "math/bits"
+)
+
+// reduceThreshold is the maximum value of x where the reduction using Pi/4
+// in 3 float64 parts still gives accurate results. This threshold
+// is set by y*C being representable as a float64 without error
+// where y is given by y = floor(x * (4 / Pi)) and C is the leading partial
+// terms of 4/Pi. Since the leading terms (PI4A and PI4B in sin.go) have 30
+// and 32 trailing zero bits, y should have less than 30 significant bits.
+// y < 1<<30 -> floor(x*4/Pi) < 1<<30 -> x < (1<<30 - 1) * Pi/4
+// So, conservatively we can take x < 1<<29.
+// Above this threshold Payne-Hanek range reduction must be used.
+const reduceThreshold = 1 << 29
+
+// trigReduce implements Payne-Hanek range reduction by Pi/4
+// for x > 0. It returns the integer part mod 8 (j) and
+// the fractional part (z) of x / (Pi/4).
+// The implementation is based on:
+// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
+// K. C. Ng et al, March 24, 1992
+// The simulated multi-precision calculation of x*B uses 64-bit integer arithmetic.
+func trigReduce(x float64) (j uint64, z float64) {
+ const PI4 = Pi / 4
+ if x < PI4 {
+ return 0, x
+ }
+ // Extract out the integer and exponent such that,
+ // x = ix * 2 ** exp.
+ ix := Float64bits(x)
+ exp := int(ix>>shift&mask) - bias - shift
+ ix &^= mask << shift
+ ix |= 1 << shift
+ // Use the exponent to extract the 3 appropriate uint64 digits from mPi4,
+ // B ~ (z0, z1, z2), such that the product leading digit has the exponent -61.
+ // Note, exp >= -53 since x >= PI4 and exp < 971 for maximum float64.
+ digit, bitshift := uint(exp+61)/64, uint(exp+61)%64
+ z0 := (mPi4[digit] << bitshift) | (mPi4[digit+1] >> (64 - bitshift))
+ z1 := (mPi4[digit+1] << bitshift) | (mPi4[digit+2] >> (64 - bitshift))
+ z2 := (mPi4[digit+2] << bitshift) | (mPi4[digit+3] >> (64 - bitshift))
+ // Multiply mantissa by the digits and extract the upper two digits (hi, lo).
+ z2hi, _ := bits.Mul64(z2, ix)
+ z1hi, z1lo := bits.Mul64(z1, ix)
+ z0lo := z0 * ix
+ lo, c := bits.Add64(z1lo, z2hi, 0)
+ hi, _ := bits.Add64(z0lo, z1hi, c)
+ // The top 3 bits are j.
+ j = hi >> 61
+ // Extract the fraction and find its magnitude.
+ hi = hi<<3 | lo>>61
+ lz := uint(bits.LeadingZeros64(hi))
+ e := uint64(bias - (lz + 1))
+ // Clear implicit mantissa bit and shift into place.
+ hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1)))
+ hi >>= 64 - shift
+ // Include the exponent and convert to a float.
+ hi |= e << shift
+ z = Float64frombits(hi)
+ // Map zeros to origin.
+ if j&1 == 1 {
+ j++
+ j &= 7
+ z--
+ }
+ // Multiply the fractional part by pi/4.
+ return j, z * PI4
+}
+
+// mPi4 is the binary digits of 4/pi as a uint64 array,
+// that is, 4/pi = Sum mPi4[i]*2^(-64*i)
+// 19 64-bit digits and the leading one bit give 1217 bits
+// of precision to handle the largest possible float64 exponent.
+var mPi4 = [...]uint64{
+ 0x0000000000000001,
+ 0x45f306dc9c882a53,
+ 0xf84eafa3ea69bb81,
+ 0xb6c52b3278872083,
+ 0xfca2c757bd778ac3,
+ 0x6e48dc74849ba5c0,
+ 0x0c925dd413a32439,
+ 0xfc3bd63962534e7d,
+ 0xd1046bea5d768909,
+ 0xd338e04d68befc82,
+ 0x7323ac7306a673e9,
+ 0x3908bf177bf25076,
+ 0x3ff12fffbc0b301f,
+ 0xde5e2316b414da3e,
+ 0xda6cfd9e4f96136e,
+ 0x9e8c7ecd3cbfd45a,
+ 0xea4f758fd7cbe2f6,
+ 0x7a0e73ef14a525d4,
+ 0xd7f6bf623f1aba10,
+ 0xac06608df8f6d757,
+}
diff --git a/src/math/unsafe.go b/src/math/unsafe.go
new file mode 100644
index 0000000..e59f50c
--- /dev/null
+++ b/src/math/unsafe.go
@@ -0,0 +1,29 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+import "unsafe"
+
+// Float32bits returns the IEEE 754 binary representation of f,
+// with the sign bit of f and the result in the same bit position.
+// Float32bits(Float32frombits(x)) == x.
+func Float32bits(f float32) uint32 { return *(*uint32)(unsafe.Pointer(&f)) }
+
+// Float32frombits returns the floating-point number corresponding
+// to the IEEE 754 binary representation b, with the sign bit of b
+// and the result in the same bit position.
+// Float32frombits(Float32bits(x)) == x.
+func Float32frombits(b uint32) float32 { return *(*float32)(unsafe.Pointer(&b)) }
+
+// Float64bits returns the IEEE 754 binary representation of f,
+// with the sign bit of f and the result in the same bit position,
+// and Float64bits(Float64frombits(x)) == x.
+func Float64bits(f float64) uint64 { return *(*uint64)(unsafe.Pointer(&f)) }
+
+// Float64frombits returns the floating-point number corresponding
+// to the IEEE 754 binary representation b, with the sign bit of b
+// and the result in the same bit position.
+// Float64frombits(Float64bits(x)) == x.
+func Float64frombits(b uint64) float64 { return *(*float64)(unsafe.Pointer(&b)) }