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+// 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)
+}