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Diffstat (limited to 'src/math/log.go')
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diff --git a/src/math/log.go b/src/math/log.go new file mode 100644 index 0000000..e328348 --- /dev/null +++ b/src/math/log.go @@ -0,0 +1,123 @@ +// 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) +} |