Source file src/pkg/math/exp.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 // Exp returns e**x, the base-e exponential of x.
8 //
9 // Special cases are:
10 // Exp(+Inf) = +Inf
11 // Exp(NaN) = NaN
12 // Very large values overflow to 0 or +Inf.
13 // Very small values underflow to 1.
14 func Exp(x float64) float64
15
16 // The original C code, the long comment, and the constants
17 // below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
18 // and came with this notice. The go code is a simplified
19 // version of the original C.
20 //
21 // ====================================================
22 // Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
23 //
24 // Permission to use, copy, modify, and distribute this
25 // software is freely granted, provided that this notice
26 // is preserved.
27 // ====================================================
28 //
29 //
30 // exp(x)
31 // Returns the exponential of x.
32 //
33 // Method
34 // 1. Argument reduction:
35 // Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
36 // Given x, find r and integer k such that
37 //
38 // x = k*ln2 + r, |r| <= 0.5*ln2.
39 //
40 // Here r will be represented as r = hi-lo for better
41 // accuracy.
42 //
43 // 2. Approximation of exp(r) by a special rational function on
44 // the interval [0,0.34658]:
45 // Write
46 // R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
47 // We use a special Remes algorithm on [0,0.34658] to generate
48 // a polynomial of degree 5 to approximate R. The maximum error
49 // of this polynomial approximation is bounded by 2**-59. In
50 // other words,
51 // R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
52 // (where z=r*r, and the values of P1 to P5 are listed below)
53 // and
54 // | 5 | -59
55 // | 2.0+P1*z+...+P5*z - R(z) | <= 2
56 // | |
57 // The computation of exp(r) thus becomes
58 // 2*r
59 // exp(r) = 1 + -------
60 // R - r
61 // r*R1(r)
62 // = 1 + r + ----------- (for better accuracy)
63 // 2 - R1(r)
64 // where
65 // 2 4 10
66 // R1(r) = r - (P1*r + P2*r + ... + P5*r ).
67 //
68 // 3. Scale back to obtain exp(x):
69 // From step 1, we have
70 // exp(x) = 2**k * exp(r)
71 //
72 // Special cases:
73 // exp(INF) is INF, exp(NaN) is NaN;
74 // exp(-INF) is 0, and
75 // for finite argument, only exp(0)=1 is exact.
76 //
77 // Accuracy:
78 // according to an error analysis, the error is always less than
79 // 1 ulp (unit in the last place).
80 //
81 // Misc. info.
82 // For IEEE double
83 // if x > 7.09782712893383973096e+02 then exp(x) overflow
84 // if x < -7.45133219101941108420e+02 then exp(x) underflow
85 //
86 // Constants:
87 // The hexadecimal values are the intended ones for the following
88 // constants. The decimal values may be used, provided that the
89 // compiler will convert from decimal to binary accurately enough
90 // to produce the hexadecimal values shown.
91
92 func exp(x float64) float64 {
93 const (
94 Ln2Hi = 6.93147180369123816490e-01
95 Ln2Lo = 1.90821492927058770002e-10
96 Log2e = 1.44269504088896338700e+00
97
98 Overflow = 7.09782712893383973096e+02
99 Underflow = -7.45133219101941108420e+02
100 NearZero = 1.0 / (1 << 28) // 2**-28
101 )
102
103 // special cases
104 switch {
105 case IsNaN(x) || IsInf(x, 1):
106 return x
107 case IsInf(x, -1):
108 return 0
109 case x > Overflow:
110 return Inf(1)
111 case x < Underflow:
112 return 0
113 case -NearZero < x && x < NearZero:
114 return 1 + x
115 }
116
117 // reduce; computed as r = hi - lo for extra precision.
118 var k int
119 switch {
120 case x < 0:
121 k = int(Log2e*x - 0.5)
122 case x > 0:
123 k = int(Log2e*x + 0.5)
124 }
125 hi := x - float64(k)*Ln2Hi
126 lo := float64(k) * Ln2Lo
127
128 // compute
129 return expmulti(hi, lo, k)
130 }
131
132 // Exp2 returns 2**x, the base-2 exponential of x.
133 //
134 // Special cases are the same as Exp.
135 func Exp2(x float64) float64
136
137 func exp2(x float64) float64 {
138 const (
139 Ln2Hi = 6.93147180369123816490e-01
140 Ln2Lo = 1.90821492927058770002e-10
141
142 Overflow = 1.0239999999999999e+03
143 Underflow = -1.0740e+03
144 )
145
146 // special cases
147 switch {
148 case IsNaN(x) || IsInf(x, 1):
149 return x
150 case IsInf(x, -1):
151 return 0
152 case x > Overflow:
153 return Inf(1)
154 case x < Underflow:
155 return 0
156 }
157
158 // argument reduction; x = r×lg(e) + k with |r| ≤ ln(2)/2.
159 // computed as r = hi - lo for extra precision.
160 var k int
161 switch {
162 case x > 0:
163 k = int(x + 0.5)
164 case x < 0:
165 k = int(x - 0.5)
166 }
167 t := x - float64(k)
168 hi := t * Ln2Hi
169 lo := -t * Ln2Lo
170
171 // compute
172 return expmulti(hi, lo, k)
173 }
174
175 // exp1 returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2.
176 func expmulti(hi, lo float64, k int) float64 {
177 const (
178 P1 = 1.66666666666666019037e-01 /* 0x3FC55555; 0x5555553E */
179 P2 = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */
180 P3 = 6.61375632143793436117e-05 /* 0x3F11566A; 0xAF25DE2C */
181 P4 = -1.65339022054652515390e-06 /* 0xBEBBBD41; 0xC5D26BF1 */
182 P5 = 4.13813679705723846039e-08 /* 0x3E663769; 0x72BEA4D0 */
183 )
184
185 r := hi - lo
186 t := r * r
187 c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))))
188 y := 1 - ((lo - (r*c)/(2-c)) - hi)
189 // TODO(rsc): make sure Ldexp can handle boundary k
190 return Ldexp(y, k)
191 }