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
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
|
/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* or http://www.opensolaris.org/os/licensing.
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
#pragma weak __expm1 = expm1
/* INDENT OFF */
/*
* expm1(x)
* Returns exp(x)-1, the exponential of x minus 1.
*
* Method
* 1. Arugment 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.
*/
/* INDENT ON */
#include "libm_macros.h"
#include <math.h>
static const double xxx[] = {
/* one */ 1.0,
/* huge */ 1.0e+300,
/* tiny */ 1.0e-300,
/* o_threshold */ 7.09782712893383973096e+02, /* 40862E42 FEFA39EF */
/* ln2_hi */ 6.93147180369123816490e-01, /* 3FE62E42 FEE00000 */
/* ln2_lo */ 1.90821492927058770002e-10, /* 3DEA39EF 35793C76 */
/* invln2 */ 1.44269504088896338700e+00, /* 3FF71547 652B82FE */
/* scaled coefficients related to expm1 */
/* Q1 */ -3.33333333333331316428e-02, /* BFA11111 111110F4 */
/* Q2 */ 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
/* Q3 */ -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
/* Q4 */ 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
/* Q5 */ -2.01099218183624371326e-07 /* BE8AFDB7 6E09C32D */
};
#define one xxx[0]
#define huge xxx[1]
#define tiny xxx[2]
#define o_threshold xxx[3]
#define ln2_hi xxx[4]
#define ln2_lo xxx[5]
#define invln2 xxx[6]
#define Q1 xxx[7]
#define Q2 xxx[8]
#define Q3 xxx[9]
#define Q4 xxx[10]
#define Q5 xxx[11]
double
expm1(double x) {
double y, hi, lo, c = 0.0L, t, e, hxs, hfx, r1;
int k, xsb;
unsigned hx;
hx = ((unsigned *) &x)[HIWORD]; /* high word of x */
xsb = hx & 0x80000000; /* sign bit of x */
if (xsb == 0)
y = x;
else
y = -x; /* y = |x| */
hx &= 0x7fffffff; /* high word of |x| */
/* filter out huge and non-finite argument */
/* for example exp(38)-1 is approximately 3.1855932e+16 */
if (hx >= 0x4043687A) {
/* if |x|>=56*ln2 (~38.8162...) */
if (hx >= 0x40862E42) { /* if |x|>=709.78... -> inf */
if (hx >= 0x7ff00000) {
if (((hx & 0xfffff) | ((int *) &x)[LOWORD])
!= 0)
return (x * x); /* + -> * for Cheetah */
else
/* exp(+-inf)={inf,-1} */
return (xsb == 0 ? x : -1.0);
}
if (x > o_threshold)
return (huge * huge); /* overflow */
}
if (xsb != 0) { /* x < -56*ln2, return -1.0 w/inexact */
if (x + tiny < 0.0) /* raise inexact */
return (tiny - one); /* return -1 */
}
}
/* argument reduction */
if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
if (xsb == 0) { /* positive number */
hi = x - ln2_hi;
lo = ln2_lo;
k = 1;
} else {
/* negative number */
hi = x + ln2_hi;
lo = -ln2_lo;
k = -1;
}
} else {
/* |x| > 1.5 ln2 */
k = (int) (invln2 * x + (xsb == 0 ? 0.5 : -0.5));
t = k;
hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
lo = t * ln2_lo;
}
x = hi - lo;
c = (hi - x) - lo; /* still at |x| > 0.5 ln2 */
} else if (hx < 0x3c900000) {
/* when |x|<2**-54, return x */
t = huge + x; /* return x w/inexact when x != 0 */
return (x - (t - (huge + x)));
} else
/* |x| <= 0.5 ln2 */
k = 0;
/* x is now in primary range */
hfx = 0.5 * x;
hxs = x * hfx;
r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
t = 3.0 - r1 * hfx;
e = hxs * ((r1 - t) / (6.0 - x * t));
if (k == 0) /* |x| <= 0.5 ln2 */
return (x - (x * e - hxs));
else { /* |x| > 0.5 ln2 */
e = (x * (e - c) - c);
e -= hxs;
if (k == -1)
return (0.5 * (x - e) - 0.5);
if (k == 1) {
if (x < -0.25)
return (-2.0 * (e - (x + 0.5)));
else
return (one + 2.0 * (x - e));
}
if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */
y = one - (e - x);
((int *) &y)[HIWORD] += k << 20;
return (y - one);
}
t = one;
if (k < 20) {
((int *) &t)[HIWORD] = 0x3ff00000 - (0x200000 >> k);
/* t = 1 - 2^-k */
y = t - (e - x);
((int *) &y)[HIWORD] += k << 20;
} else {
((int *) &t)[HIWORD] = (0x3ff - k) << 20; /* 2^-k */
y = x - (e + t);
y += one;
((int *) &y)[HIWORD] += k << 20;
}
}
return (y);
}
|
