# Source file src/math/log.go

```     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  // ====================================================
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
```

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