// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package math
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/*
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Floating-point logarithm.
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*/
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// The original C code, the long comment, and the constants
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// below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
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// and came with this notice. The go code is a simpler
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// version of the original C.
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//
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// ====================================================
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// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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//
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// Developed at SunPro, a Sun Microsystems, Inc. business.
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// Permission to use, copy, modify, and distribute this
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// software is freely granted, provided that this notice
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// is preserved.
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// ====================================================
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//
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// __ieee754_log(x)
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// Return the logarithm of x
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//
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// Method :
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// 1. Argument Reduction: find k and f such that
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// x = 2**k * (1+f),
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// where sqrt(2)/2 < 1+f < sqrt(2) .
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//
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// 2. Approximation of log(1+f).
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// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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// = 2s + s*R
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// We use a special Reme algorithm on [0,0.1716] to generate
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// a polynomial of degree 14 to approximate R. The maximum error
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// of this polynomial approximation is bounded by 2**-58.45. In
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// other words,
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// 2 4 6 8 10 12 14
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// R(z) ~ L1*s +L2*s +L3*s +L4*s +L5*s +L6*s +L7*s
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// (the values of L1 to L7 are listed in the program) and
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// | 2 14 | -58.45
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// | L1*s +...+L7*s - R(z) | <= 2
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// | |
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// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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// In order to guarantee error in log below 1ulp, we compute log by
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// log(1+f) = f - s*(f - R) (if f is not too large)
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// log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
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//
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// 3. Finally, log(x) = k*Ln2 + log(1+f).
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// = k*Ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*Ln2_lo)))
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// Here Ln2 is split into two floating point number:
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// Ln2_hi + Ln2_lo,
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// where n*Ln2_hi is always exact for |n| < 2000.
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//
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// Special cases:
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// log(x) is NaN with signal if x < 0 (including -INF) ;
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// log(+INF) is +INF; log(0) is -INF with signal;
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// log(NaN) is that NaN with no signal.
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//
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// Accuracy:
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// according to an error analysis, the error is always less than
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// 1 ulp (unit in the last place).
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//
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// Constants:
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// The hexadecimal values are the intended ones for the following
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// constants. The decimal values may be used, provided that the
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// compiler will convert from decimal to binary accurately enough
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// to produce the hexadecimal values shown.
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// Log returns the natural logarithm of x.
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//
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// Special cases are:
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// Log(+Inf) = +Inf
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// Log(0) = -Inf
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// Log(x < 0) = NaN
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// Log(NaN) = NaN
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func Log(x float64) float64
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func log(x float64) float64 {
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const (
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Ln2Hi = 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
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Ln2Lo = 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
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L1 = 6.666666666666735130e-01 /* 3FE55555 55555593 */
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L2 = 3.999999999940941908e-01 /* 3FD99999 9997FA04 */
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L3 = 2.857142874366239149e-01 /* 3FD24924 94229359 */
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L4 = 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
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L5 = 1.818357216161805012e-01 /* 3FC74664 96CB03DE */
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L6 = 1.531383769920937332e-01 /* 3FC39A09 D078C69F */
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L7 = 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */
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)
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// special cases
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switch {
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case IsNaN(x) || IsInf(x, 1):
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return x
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case x < 0:
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return NaN()
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case x == 0:
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return Inf(-1)
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}
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// reduce
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f1, ki := Frexp(x)
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if f1 < Sqrt2/2 {
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f1 *= 2
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ki--
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}
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f := f1 - 1
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k := float64(ki)
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// compute
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s := f / (2 + f)
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s2 := s * s
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s4 := s2 * s2
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t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
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t2 := s4 * (L2 + s4*(L4+s4*L6))
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R := t1 + t2
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hfsq := 0.5 * f * f
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return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f)
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}
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