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// 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)
}
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