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
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
|
// Copyright 2010 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
// The original C code, the long comment, and the constants
// below are from FreeBSD's /usr/src/lib/msun/src/s_log1p.c
// and came with this notice. The go code is a simplified
// 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.
// ====================================================
//
//
// double log1p(double x)
//
// Method :
// 1. Argument Reduction: find k and f such that
// 1+x = 2**k * (1+f),
// where sqrt(2)/2 < 1+f < sqrt(2) .
//
// Note. If k=0, then f=x is exact. However, if k!=0, then f
// may not be representable exactly. In that case, a correction
// term is need. Let u=1+x rounded. Let c = (1+x)-u, then
// log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
// and add back the correction term c/u.
// (Note: when x > 2**53, one can simply return log(x))
//
// 2. Approximation of log1p(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) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
// (the values of Lp1 to Lp7 are listed in the program)
// a-0.2929nd
// | 2 14 | -58.45
// | Lp1*s +...+Lp7*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
// log1p(f) = f - (hfsq - s*(hfsq+R)).
//
// 3. Finally, log1p(x) = k*ln2 + log1p(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:
// log1p(x) is NaN with signal if x < -1 (including -INF) ;
// log1p(+INF) is +INF; log1p(-1) is -INF with signal;
// log1p(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.
//
// Note: Assuming log() return accurate answer, the following
// algorithm can be used to compute log1p(x) to within a few ULP:
//
// u = 1+x;
// if(u==1.0) return x ; else
// return log(u)*(x/(u-1.0));
//
// See HP-15C Advanced Functions Handbook, p.193.
// Log1p returns the natural logarithm of 1 plus its argument x.
// It is more accurate than Log(1 + x) when x is near zero.
//
// Special cases are:
// Log1p(+Inf) = +Inf
// Log1p(-1) = -Inf
// Log1p(x < -1) = NaN
// Log1p(NaN) = NaN
func Log1p(x float64) float64 {
const (
Sqrt2M1 = 4.142135623730950488017e-01 // Sqrt(2)-1 = 0x3fda827999fcef34
Sqrt2HalfM1 = -2.928932188134524755992e-01 // Sqrt(2)/2-1 = 0xbfd2bec333018866
Small = 1.0 / (1 << 29) // 2**-29 = 0x3e20000000000000
Tiny = 1.0 / (1 << 54) // 2**-54
Two53 = 1 << 53 // 2**53
Ln2Hi = 6.93147180369123816490e-01 // 3fe62e42fee00000
Ln2Lo = 1.90821492927058770002e-10 // 3dea39ef35793c76
Lp1 = 6.666666666666735130e-01 // 3FE5555555555593
Lp2 = 3.999999999940941908e-01 // 3FD999999997FA04
Lp3 = 2.857142874366239149e-01 // 3FD2492494229359
Lp4 = 2.222219843214978396e-01 // 3FCC71C51D8E78AF
Lp5 = 1.818357216161805012e-01 // 3FC7466496CB03DE
Lp6 = 1.531383769920937332e-01 // 3FC39A09D078C69F
Lp7 = 1.479819860511658591e-01 // 3FC2F112DF3E5244
)
// special cases
// TODO(rsc): Remove manual inlining of IsNaN, IsInf
// when compiler does it for us
switch {
case x < -1 || x != x: // x < -1 || IsNaN(x): // includes -Inf
return NaN()
case x == -1:
return Inf(-1)
case x > MaxFloat64: // IsInf(x, 1):
return Inf(1)
}
absx := x
if absx < 0 {
absx = -absx
}
var f float64
var iu uint64
k := 1
if absx < Sqrt2M1 { // |x| < Sqrt(2)-1
if absx < Small { // |x| < 2**-29
if absx < Tiny { // |x| < 2**-54
return x
}
return x - x*x*0.5
}
if x > Sqrt2HalfM1 { // Sqrt(2)/2-1 < x
// (Sqrt(2)/2-1) < x < (Sqrt(2)-1)
k = 0
f = x
iu = 1
}
}
var c float64
if k != 0 {
var u float64
if absx < Two53 { // 1<<53
u = 1.0 + x
iu = Float64bits(u)
k = int((iu >> 52) - 1023)
if k > 0 {
c = 1.0 - (u - x)
} else {
c = x - (u - 1.0) // correction term
c /= u
}
} else {
u = x
iu = Float64bits(u)
k = int((iu >> 52) - 1023)
c = 0
}
iu &= 0x000fffffffffffff
if iu < 0x0006a09e667f3bcd { // mantissa of Sqrt(2)
u = Float64frombits(iu | 0x3ff0000000000000) // normalize u
} else {
k += 1
u = Float64frombits(iu | 0x3fe0000000000000) // normalize u/2
iu = (0x0010000000000000 - iu) >> 2
}
f = u - 1.0 // Sqrt(2)/2 < u < Sqrt(2)
}
hfsq := 0.5 * f * f
var s, R, z float64
if iu == 0 { // |f| < 2**-20
if f == 0 {
if k == 0 {
return 0
} else {
c += float64(k) * Ln2Lo
return float64(k)*Ln2Hi + c
}
}
R = hfsq * (1.0 - 0.66666666666666666*f) // avoid division
if k == 0 {
return f - R
}
return float64(k)*Ln2Hi - ((R - (float64(k)*Ln2Lo + c)) - f)
}
s = f / (2.0 + f)
z = s * s
R = z * (Lp1 + z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))))
if k == 0 {
return f - (hfsq - s*(hfsq+R))
}
return float64(k)*Ln2Hi - ((hfsq - (s*(hfsq+R) + (float64(k)*Ln2Lo + c))) - f)
}
|
