1 // Copyright 2009 The Go Authors. All rights reserved.
2 // Use of this source code is governed by a BSD-style
3 // license that can be found in the LICENSE file.
4 5 package math
6 7 /*
8 Floating-point logarithm.
9 */
10 11 // The original C code, the long comment, and the constants
12 // below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
13 // and came with this notice. The go code is a simpler
14 // version of the original C.
15 //
16 // ====================================================
17 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
18 //
19 // Developed at SunPro, a Sun Microsystems, Inc. business.
20 // Permission to use, copy, modify, and distribute this
21 // software is freely granted, provided that this notice
22 // is preserved.
23 // ====================================================
24 //
25 // __ieee754_log(x)
26 // Return the logarithm of x
27 //
28 // Method :
29 // 1. Argument Reduction: find k and f such that
30 // x = 2**k * (1+f),
31 // where sqrt(2)/2 < 1+f < sqrt(2) .
32 //
33 // 2. Approximation of log(1+f).
34 // Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
35 // = 2s + 2/3 s**3 + 2/5 s**5 + .....,
36 // = 2s + s*R
37 // We use a special Reme algorithm on [0,0.1716] to generate
38 // a polynomial of degree 14 to approximate R. The maximum error
39 // of this polynomial approximation is bounded by 2**-58.45. In
40 // other words,
41 // 2 4 6 8 10 12 14
42 // R(z) ~ L1*s +L2*s +L3*s +L4*s +L5*s +L6*s +L7*s
43 // (the values of L1 to L7 are listed in the program) and
44 // | 2 14 | -58.45
45 // | L1*s +...+L7*s - R(z) | <= 2
46 // | |
47 // Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
48 // In order to guarantee error in log below 1ulp, we compute log by
49 // log(1+f) = f - s*(f - R) (if f is not too large)
50 // log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
51 //
52 // 3. Finally, log(x) = k*Ln2 + log(1+f).
53 // = k*Ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*Ln2_lo)))
54 // Here Ln2 is split into two floating point number:
55 // Ln2_hi + Ln2_lo,
56 // where n*Ln2_hi is always exact for |n| < 2000.
57 //
58 // Special cases:
59 // log(x) is NaN with signal if x < 0 (including -INF) ;
60 // log(+INF) is +INF; log(0) is -INF with signal;
61 // log(NaN) is that NaN with no signal.
62 //
63 // Accuracy:
64 // according to an error analysis, the error is always less than
65 // 1 ulp (unit in the last place).
66 //
67 // Constants:
68 // The hexadecimal values are the intended ones for the following
69 // constants. The decimal values may be used, provided that the
70 // compiler will convert from decimal to binary accurately enough
71 // to produce the hexadecimal values shown.
72 73 // Log returns the natural logarithm of x.
74 //
75 // Special cases are:
76 //
77 // Log(+Inf) = +Inf
78 // Log(0) = -Inf
79 // Log(x < 0) = NaN
80 // Log(NaN) = NaN
81 func Log(x float64) float64 {
82 if haveArchLog {
83 return archLog(x)
84 }
85 return log(x)
86 }
87 88 func log(x float64) float64 {
89 const (
90 Ln2Hi = 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
91 Ln2Lo = 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
92 L1 = 6.666666666666735130e-01 /* 3FE55555 55555593 */
93 L2 = 3.999999999940941908e-01 /* 3FD99999 9997FA04 */
94 L3 = 2.857142874366239149e-01 /* 3FD24924 94229359 */
95 L4 = 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
96 L5 = 1.818357216161805012e-01 /* 3FC74664 96CB03DE */
97 L6 = 1.531383769920937332e-01 /* 3FC39A09 D078C69F */
98 L7 = 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */
99 )
100 101 // special cases
102 switch {
103 case IsNaN(x) || IsInf(x, 1):
104 return x
105 case x < 0:
106 return NaN()
107 case x == 0:
108 return Inf(-1)
109 }
110 111 // reduce
112 f1, ki := Frexp(x)
113 if f1 < Sqrt2/2 {
114 f1 *= 2
115 ki--
116 }
117 f := f1 - 1
118 k := float64(ki)
119 120 // compute
121 s := f / (2 + f)
122 s2 := s * s
123 s4 := s2 * s2
124 t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
125 t2 := s4 * (L2 + s4*(L4+s4*L6))
126 R := t1 + t2
127 hfsq := 0.5 * f * f
128 return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f)
129 }
130