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
|
/* Implementation of gamma function according to ISO C.
Copyright (C) 1997-2023 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<https://www.gnu.org/licenses/>. */
#include <math.h>
#include <math-narrow-eval.h>
#include <math_private.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <float.h>
#include <libm-alias-finite.h>
#include <mul_split.h>
/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
approximation to gamma function. */
static const double gamma_coeff[] =
{
0x1.5555555555555p-4,
-0xb.60b60b60b60b8p-12,
0x3.4034034034034p-12,
-0x2.7027027027028p-12,
0x3.72a3c5631fe46p-12,
-0x7.daac36664f1f4p-12,
};
#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
/* Return gamma (X), for positive X less than 184, in the form R *
2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
avoid overflow or underflow in intermediate calculations. */
static double
gamma_positive (double x, int *exp2_adj)
{
int local_signgam;
if (x < 0.5)
{
*exp2_adj = 0;
return __ieee754_exp (__ieee754_lgamma_r (x + 1, &local_signgam)) / x;
}
else if (x <= 1.5)
{
*exp2_adj = 0;
return __ieee754_exp (__ieee754_lgamma_r (x, &local_signgam));
}
else if (x < 6.5)
{
/* Adjust into the range for using exp (lgamma). */
*exp2_adj = 0;
double n = ceil (x - 1.5);
double x_adj = x - n;
double eps;
double prod = __gamma_product (x_adj, 0, n, &eps);
return (__ieee754_exp (__ieee754_lgamma_r (x_adj, &local_signgam))
* prod * (1.0 + eps));
}
else
{
double eps = 0;
double x_eps = 0;
double x_adj = x;
double prod = 1;
if (x < 12.0)
{
/* Adjust into the range for applying Stirling's
approximation. */
double n = ceil (12.0 - x);
x_adj = math_narrow_eval (x + n);
x_eps = (x - (x_adj - n));
prod = __gamma_product (x_adj - n, x_eps, n, &eps);
}
/* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
starting by computing pow (X_ADJ, X_ADJ) with a power of 2
factored out. */
double x_adj_int = round (x_adj);
double x_adj_frac = x_adj - x_adj_int;
int x_adj_log2;
double x_adj_mant = __frexp (x_adj, &x_adj_log2);
if (x_adj_mant < M_SQRT1_2)
{
x_adj_log2--;
x_adj_mant *= 2.0;
}
*exp2_adj = x_adj_log2 * (int) x_adj_int;
double h1, l1, h2, l2;
mul_split (&h1, &l1, __ieee754_pow (x_adj_mant, x_adj),
__ieee754_exp2 (x_adj_log2 * x_adj_frac));
mul_split (&h2, &l2, __ieee754_exp (-x_adj), sqrt (2 * M_PI / x_adj));
mul_expansion (&h1, &l1, h1, l1, h2, l2);
/* Divide by prod * (1 + eps). */
div_expansion (&h1, &l1, h1, l1, prod, prod * eps);
double exp_adj = x_eps * __ieee754_log (x_adj);
double bsum = gamma_coeff[NCOEFF - 1];
double x_adj2 = x_adj * x_adj;
for (size_t i = 1; i <= NCOEFF - 1; i++)
bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
exp_adj += bsum / x_adj;
/* Now return (h1+l1) * exp(exp_adj), where exp_adj is small. */
l1 += h1 * __expm1 (exp_adj);
return h1 + l1;
}
}
double
__ieee754_gamma_r (double x, int *signgamp)
{
int32_t hx;
uint32_t lx;
double ret;
EXTRACT_WORDS (hx, lx, x);
if (__glibc_unlikely (((hx & 0x7fffffff) | lx) == 0))
{
/* Return value for x == 0 is Inf with divide by zero exception. */
*signgamp = 0;
return 1.0 / x;
}
if (__builtin_expect (hx < 0, 0)
&& (uint32_t) hx < 0xfff00000 && rint (x) == x)
{
/* Return value for integer x < 0 is NaN with invalid exception. */
*signgamp = 0;
return (x - x) / (x - x);
}
if (__glibc_unlikely ((unsigned int) hx == 0xfff00000 && lx == 0))
{
/* x == -Inf. According to ISO this is NaN. */
*signgamp = 0;
return x - x;
}
if (__glibc_unlikely ((hx & 0x7ff00000) == 0x7ff00000))
{
/* Positive infinity (return positive infinity) or NaN (return
NaN). */
*signgamp = 0;
return x + x;
}
if (x >= 172.0)
{
/* Overflow. */
*signgamp = 0;
ret = math_narrow_eval (DBL_MAX * DBL_MAX);
return ret;
}
else
{
SET_RESTORE_ROUND (FE_TONEAREST);
if (x > 0.0)
{
*signgamp = 0;
int exp2_adj;
double tret = gamma_positive (x, &exp2_adj);
ret = __scalbn (tret, exp2_adj);
}
else if (x >= -DBL_EPSILON / 4.0)
{
*signgamp = 0;
ret = 1.0 / x;
}
else
{
double tx = trunc (x);
*signgamp = (tx == 2.0 * trunc (tx / 2.0)) ? -1 : 1;
if (x <= -184.0)
/* Underflow. */
ret = DBL_MIN * DBL_MIN;
else
{
double frac = tx - x;
if (frac > 0.5)
frac = 1.0 - frac;
double sinpix = (frac <= 0.25
? __sin (M_PI * frac)
: __cos (M_PI * (0.5 - frac)));
int exp2_adj;
double h1, l1, h2, l2;
h2 = gamma_positive (-x, &exp2_adj);
mul_split (&h1, &l1, sinpix, h2);
/* sinpix*gamma_positive(.) = h1 + l1 */
mul_split (&h2, &l2, h1, x);
/* h1*x = h2 + l2 */
/* (h1 + l1) * x = h1*x + l1*x = h2 + l2 + l1*x */
l2 += l1 * x;
/* x*sinpix*gamma_positive(.) ~ h2 + l2 */
h1 = 0x3.243f6a8885a3p+0; /* binary64 approximation of Pi */
l1 = 0x8.d313198a2e038p-56; /* |h1+l1-Pi| < 3e-33 */
/* Now we divide h1 + l1 by h2 + l2. */
div_expansion (&h1, &l1, h1, l1, h2, l2);
ret = __scalbn (-h1, -exp2_adj);
math_check_force_underflow_nonneg (ret);
}
}
ret = math_narrow_eval (ret);
}
if (isinf (ret) && x != 0)
{
if (*signgamp < 0)
{
ret = math_narrow_eval (-copysign (DBL_MAX, ret) * DBL_MAX);
ret = -ret;
}
else
ret = math_narrow_eval (copysign (DBL_MAX, ret) * DBL_MAX);
return ret;
}
else if (ret == 0)
{
if (*signgamp < 0)
{
ret = math_narrow_eval (-copysign (DBL_MIN, ret) * DBL_MIN);
ret = -ret;
}
else
ret = math_narrow_eval (copysign (DBL_MIN, ret) * DBL_MIN);
return ret;
}
else
return ret;
}
libm_alias_finite (__ieee754_gamma_r, __gamma_r)
|
