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
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
|
// 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_expm1.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.
// ====================================================
//
// expm1(x)
// Returns exp(x)-1, the exponential of x minus 1.
//
// Method
// 1. Argument reduction:
// Given x, find r and integer k such that
//
// x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
//
// Here a correction term c will be computed to compensate
// the error in r when rounded to a floating-point number.
//
// 2. Approximating expm1(r) by a special rational function on
// the interval [0,0.34658]:
// Since
// r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 - r**4/360 + ...
// we define R1(r*r) by
// r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 * R1(r*r)
// That is,
// R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
// = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
// = 1 - r**2/60 + r**4/2520 - r**6/100800 + ...
// We use a special Reme algorithm on [0,0.347] to generate
// a polynomial of degree 5 in r*r to approximate R1. The
// maximum error of this polynomial approximation is bounded
// by 2**-61. In other words,
// R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
// where Q1 = -1.6666666666666567384E-2,
// Q2 = 3.9682539681370365873E-4,
// Q3 = -9.9206344733435987357E-6,
// Q4 = 2.5051361420808517002E-7,
// Q5 = -6.2843505682382617102E-9;
// (where z=r*r, and the values of Q1 to Q5 are listed below)
// with error bounded by
// | 5 | -61
// | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
// | |
//
// expm1(r) = exp(r)-1 is then computed by the following
// specific way which minimize the accumulation rounding error:
// 2 3
// r r [ 3 - (R1 + R1*r/2) ]
// expm1(r) = r + --- + --- * [--------------------]
// 2 2 [ 6 - r*(3 - R1*r/2) ]
//
// To compensate the error in the argument reduction, we use
// expm1(r+c) = expm1(r) + c + expm1(r)*c
// ~ expm1(r) + c + r*c
// Thus c+r*c will be added in as the correction terms for
// expm1(r+c). Now rearrange the term to avoid optimization
// screw up:
// ( 2 2 )
// ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
// expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
// ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
// ( )
//
// = r - E
// 3. Scale back to obtain expm1(x):
// From step 1, we have
// expm1(x) = either 2**k*[expm1(r)+1] - 1
// = or 2**k*[expm1(r) + (1-2**-k)]
// 4. Implementation notes:
// (A). To save one multiplication, we scale the coefficient Qi
// to Qi*2**i, and replace z by (x**2)/2.
// (B). To achieve maximum accuracy, we compute expm1(x) by
// (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
// (ii) if k=0, return r-E
// (iii) if k=-1, return 0.5*(r-E)-0.5
// (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
// else return 1.0+2.0*(r-E);
// (v) if (k<-2||k>56) return 2**k(1-(E-r)) - 1 (or exp(x)-1)
// (vi) if k <= 20, return 2**k((1-2**-k)-(E-r)), else
// (vii) return 2**k(1-((E+2**-k)-r))
//
// Special cases:
// expm1(INF) is INF, expm1(NaN) is NaN;
// expm1(-INF) is -1, and
// for finite argument, only expm1(0)=0 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 expm1(x) overflow
//
// 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.
//
// Expm1 returns e**x - 1, the base-e exponential of x minus 1.
// It is more accurate than Exp(x) - 1 when x is near zero.
//
// Special cases are:
//
// Expm1(+Inf) = +Inf
// Expm1(-Inf) = -1
// Expm1(NaN) = NaN
//
// Very large values overflow to -1 or +Inf.
func Expm1(x float64) float64 {
if haveArchExpm1 {
return archExpm1(x)
}
return expm1(x)
}
func expm1(x float64) float64 {
const (
Othreshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF
Ln2X56 = 3.88162421113569373274e+01 // 0x4043687a9f1af2b1
Ln2HalfX3 = 1.03972077083991796413e+00 // 0x3ff0a2b23f3bab73
Ln2Half = 3.46573590279972654709e-01 // 0x3fd62e42fefa39ef
Ln2Hi = 6.93147180369123816490e-01 // 0x3fe62e42fee00000
Ln2Lo = 1.90821492927058770002e-10 // 0x3dea39ef35793c76
InvLn2 = 1.44269504088896338700e+00 // 0x3ff71547652b82fe
Tiny = 1.0 / (1 << 54) // 2**-54 = 0x3c90000000000000
// scaled coefficients related to expm1
Q1 = -3.33333333333331316428e-02 // 0xBFA11111111110F4
Q2 = 1.58730158725481460165e-03 // 0x3F5A01A019FE5585
Q3 = -7.93650757867487942473e-05 // 0xBF14CE199EAADBB7
Q4 = 4.00821782732936239552e-06 // 0x3ED0CFCA86E65239
Q5 = -2.01099218183624371326e-07 // 0xBE8AFDB76E09C32D
)
// special cases
switch {
case IsInf(x, 1) || IsNaN(x):
return x
case IsInf(x, -1):
return -1
}
absx := x
sign := false
if x < 0 {
absx = -absx
sign = true
}
// filter out huge argument
if absx >= Ln2X56 { // if |x| >= 56 * ln2
if sign {
return -1 // x < -56*ln2, return -1
}
if absx >= Othreshold { // if |x| >= 709.78...
return Inf(1)
}
}
// argument reduction
var c float64
var k int
if absx > Ln2Half { // if |x| > 0.5 * ln2
var hi, lo float64
if absx < Ln2HalfX3 { // and |x| < 1.5 * ln2
if !sign {
hi = x - Ln2Hi
lo = Ln2Lo
k = 1
} else {
hi = x + Ln2Hi
lo = -Ln2Lo
k = -1
}
} else {
if !sign {
k = int(InvLn2*x + 0.5)
} else {
k = int(InvLn2*x - 0.5)
}
t := float64(k)
hi = x - t*Ln2Hi // t * Ln2Hi is exact here
lo = t * Ln2Lo
}
x = hi - lo
c = (hi - x) - lo
} else if absx < Tiny { // when |x| < 2**-54, return x
return x
} else {
k = 0
}
// x is now in primary range
hfx := 0.5 * x
hxs := x * hfx
r1 := 1 + hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))))
t := 3 - r1*hfx
e := hxs * ((r1 - t) / (6.0 - x*t))
if k == 0 {
return x - (x*e - hxs) // c is 0
}
e = (x*(e-c) - c)
e -= hxs
switch {
case k == -1:
return 0.5*(x-e) - 0.5
case k == 1:
if x < -0.25 {
return -2 * (e - (x + 0.5))
}
return 1 + 2*(x-e)
case k <= -2 || k > 56: // suffice to return exp(x)-1
y := 1 - (e - x)
y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
return y - 1
}
if k < 20 {
t := Float64frombits(0x3ff0000000000000 - (0x20000000000000 >> uint(k))) // t=1-2**-k
y := t - (e - x)
y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
return y
}
t = Float64frombits(uint64(0x3ff-k) << 52) // 2**-k
y := x - (e + t)
y++
y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
return y
}
|
