1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
|
// 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
import "math"
// 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 logrithm 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 {
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);
}
// Log10 returns the decimal logarthm of x.
// The special cases are the same as for Log.
func Log10(x float64) float64 {
if x <= 0 {
return NaN();
}
return Log(x) * (1/Ln10);
}
|
