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Diffstat (limited to 'src/math/log1p.go')
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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) +} |