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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
+
+// 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)
+}