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Diffstat (limited to 'src/pkg/math/log1p.go')
| -rw-r--r-- | src/pkg/math/log1p.go | 200 |
1 files changed, 0 insertions, 200 deletions
diff --git a/src/pkg/math/log1p.go b/src/pkg/math/log1p.go deleted file mode 100644 index 12b98684c..000000000 --- a/src/pkg/math/log1p.go +++ /dev/null @@ -1,200 +0,0 @@ -// 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 := x - if absx < 0 { - absx = -absx - } - - 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) - if k > 0 { - c = 1.0 - (u - x) - } else { - c = x - (u - 1.0) // correction term - 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 += 1 - 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 - } else { - 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) -} |