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
|
// 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
// The original C code, the long comment, and the constants
// below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
// and came with this notice. The go code is a simplified
// version of the original C.
//
// ====================================================
// Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
//
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
//
// exp(x)
// Returns the exponential of x.
//
// Method
// 1. Argument reduction:
// Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
// Given x, find r and integer k such that
//
// x = k*ln2 + r, |r| <= 0.5*ln2.
//
// Here r will be represented as r = hi-lo for better
// accuracy.
//
// 2. Approximation of exp(r) by a special rational function on
// the interval [0,0.34658]:
// Write
// R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
// We use a special Remes algorithm on [0,0.34658] to generate
// a polynomial of degree 5 to approximate R. The maximum error
// of this polynomial approximation is bounded by 2**-59. In
// other words,
// R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
// (where z=r*r, and the values of P1 to P5 are listed below)
// and
// | 5 | -59
// | 2.0+P1*z+...+P5*z - R(z) | <= 2
// | |
// The computation of exp(r) thus becomes
// 2*r
// exp(r) = 1 + -------
// R - r
// r*R1(r)
// = 1 + r + ----------- (for better accuracy)
// 2 - R1(r)
// where
// 2 4 10
// R1(r) = r - (P1*r + P2*r + ... + P5*r ).
//
// 3. Scale back to obtain exp(x):
// From step 1, we have
// exp(x) = 2**k * exp(r)
//
// Special cases:
// exp(INF) is INF, exp(NaN) is NaN;
// exp(-INF) is 0, and
// for finite argument, only exp(0)=1 is exact.
//
// Accuracy:
// according to an error analysis, the error is always less than
// 1 ulp (unit in the last place).
//
// Misc. info.
// For IEEE double
// if x > 7.09782712893383973096e+02 then exp(x) overflow
// if x < -7.45133219101941108420e+02 then exp(x) underflow
//
// 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.
// Exp returns e**x, the base-e exponential of x.
//
// Special cases are:
// Exp(+Inf) = +Inf
// Exp(NaN) = NaN
// Very large values overflow to 0 or +Inf.
// Very small values underflow to 1.
func expGo(x float64) float64 {
const (
Ln2Hi = 6.93147180369123816490e-01
Ln2Lo = 1.90821492927058770002e-10
Log2e = 1.44269504088896338700e+00
Overflow = 7.09782712893383973096e+02
Underflow = -7.45133219101941108420e+02
NearZero = 1.0 / (1 << 28) // 2**-28
)
// TODO(rsc): Remove manual inlining of IsNaN, IsInf
// when compiler does it for us
// special cases
switch {
case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1):
return x
case x < -MaxFloat64: // IsInf(x, -1):
return 0
case x > Overflow:
return Inf(1)
case x < Underflow:
return 0
case -NearZero < x && x < NearZero:
return 1 + x
}
// reduce; computed as r = hi - lo for extra precision.
var k int
switch {
case x < 0:
k = int(Log2e*x - 0.5)
case x > 0:
k = int(Log2e*x + 0.5)
}
hi := x - float64(k)*Ln2Hi
lo := float64(k) * Ln2Lo
// compute
return exp(hi, lo, k)
}
// Exp2 returns 2**x, the base-2 exponential of x.
//
// Special cases are the same as Exp.
func exp2Go(x float64) float64 {
const (
Ln2Hi = 6.93147180369123816490e-01
Ln2Lo = 1.90821492927058770002e-10
Overflow = 1.0239999999999999e+03
Underflow = -1.0740e+03
)
// TODO: remove manual inlining of IsNaN and IsInf
// when compiler does it for us
// special cases
switch {
case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1):
return x
case x < -MaxFloat64: // IsInf(x, -1):
return 0
case x > Overflow:
return Inf(1)
case x < Underflow:
return 0
}
// argument reduction; x = r×lg(e) + k with |r| ≤ ln(2)/2.
// computed as r = hi - lo for extra precision.
var k int
switch {
case x > 0:
k = int(x + 0.5)
case x < 0:
k = int(x - 0.5)
}
t := x - float64(k)
hi := t * Ln2Hi
lo := -t * Ln2Lo
// compute
return exp(hi, lo, k)
}
// exp returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2.
func exp(hi, lo float64, k int) float64 {
const (
P1 = 1.66666666666666019037e-01 /* 0x3FC55555; 0x5555553E */
P2 = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */
P3 = 6.61375632143793436117e-05 /* 0x3F11566A; 0xAF25DE2C */
P4 = -1.65339022054652515390e-06 /* 0xBEBBBD41; 0xC5D26BF1 */
P5 = 4.13813679705723846039e-08 /* 0x3E663769; 0x72BEA4D0 */
)
r := hi - lo
t := r * r
c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))))
y := 1 - ((lo - (r*c)/(2-c)) - hi)
// TODO(rsc): make sure Ldexp can handle boundary k
return Ldexp(y, k)
}
|
