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+// 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
+}