mirror of git://gcc.gnu.org/git/gcc.git
130 lines
3.9 KiB
Go
130 lines
3.9 KiB
Go
// 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
|
|
|
|
//extern log
|
|
func libc_log(float64) float64
|
|
|
|
func Log(x float64) float64 {
|
|
return libc_log(x)
|
|
}
|
|
|
|
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)
|
|
}
|