1/* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2021 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4 Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997.
5
6 The GNU C Library is free software; you can redistribute it and/or
7 modify it under the terms of the GNU Lesser General Public
8 License as published by the Free Software Foundation; either
9 version 2.1 of the License, or (at your option) any later version.
10
11 The GNU C Library is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 Lesser General Public License for more details.
15
16 You should have received a copy of the GNU Lesser General Public
17 License along with the GNU C Library; if not, see
18 <https://www.gnu.org/licenses/>. */
19
20#include <math.h>
21#include <math-narrow-eval.h>
22#include <math_private.h>
23#include <fenv_private.h>
24#include <math-underflow.h>
25#include <float.h>
26#include <libm-alias-finite.h>
27
28/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
29 approximation to gamma function. */
30
31static const double gamma_coeff[] =
32 {
33 0x1.5555555555555p-4,
34 -0xb.60b60b60b60b8p-12,
35 0x3.4034034034034p-12,
36 -0x2.7027027027028p-12,
37 0x3.72a3c5631fe46p-12,
38 -0x7.daac36664f1f4p-12,
39 };
40
41#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
42
43/* Return gamma (X), for positive X less than 184, in the form R *
44 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
45 avoid overflow or underflow in intermediate calculations. */
46
47static double
48gamma_positive (double x, int *exp2_adj)
49{
50 int local_signgam;
51 if (x < 0.5)
52 {
53 *exp2_adj = 0;
54 return __ieee754_exp (__ieee754_lgamma_r (x + 1, &local_signgam)) / x;
55 }
56 else if (x <= 1.5)
57 {
58 *exp2_adj = 0;
59 return __ieee754_exp (__ieee754_lgamma_r (x, &local_signgam));
60 }
61 else if (x < 6.5)
62 {
63 /* Adjust into the range for using exp (lgamma). */
64 *exp2_adj = 0;
65 double n = ceil (x - 1.5);
66 double x_adj = x - n;
67 double eps;
68 double prod = __gamma_product (x_adj, 0, n, &eps);
69 return (__ieee754_exp (__ieee754_lgamma_r (x_adj, &local_signgam))
70 * prod * (1.0 + eps));
71 }
72 else
73 {
74 double eps = 0;
75 double x_eps = 0;
76 double x_adj = x;
77 double prod = 1;
78 if (x < 12.0)
79 {
80 /* Adjust into the range for applying Stirling's
81 approximation. */
82 double n = ceil (12.0 - x);
83 x_adj = math_narrow_eval (x + n);
84 x_eps = (x - (x_adj - n));
85 prod = __gamma_product (x_adj - n, x_eps, n, &eps);
86 }
87 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
88 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
89 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
90 factored out. */
91 double exp_adj = -eps;
92 double x_adj_int = round (x_adj);
93 double x_adj_frac = x_adj - x_adj_int;
94 int x_adj_log2;
95 double x_adj_mant = __frexp (x_adj, &x_adj_log2);
96 if (x_adj_mant < M_SQRT1_2)
97 {
98 x_adj_log2--;
99 x_adj_mant *= 2.0;
100 }
101 *exp2_adj = x_adj_log2 * (int) x_adj_int;
102 double ret = (__ieee754_pow (x_adj_mant, x_adj)
103 * __ieee754_exp2 (x_adj_log2 * x_adj_frac)
104 * __ieee754_exp (-x_adj)
105 * sqrt (2 * M_PI / x_adj)
106 / prod);
107 exp_adj += x_eps * __ieee754_log (x_adj);
108 double bsum = gamma_coeff[NCOEFF - 1];
109 double x_adj2 = x_adj * x_adj;
110 for (size_t i = 1; i <= NCOEFF - 1; i++)
111 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
112 exp_adj += bsum / x_adj;
113 return ret + ret * __expm1 (exp_adj);
114 }
115}
116
117double
118__ieee754_gamma_r (double x, int *signgamp)
119{
120 int32_t hx;
121 uint32_t lx;
122 double ret;
123
124 EXTRACT_WORDS (hx, lx, x);
125
126 if (__glibc_unlikely (((hx & 0x7fffffff) | lx) == 0))
127 {
128 /* Return value for x == 0 is Inf with divide by zero exception. */
129 *signgamp = 0;
130 return 1.0 / x;
131 }
132 if (__builtin_expect (hx < 0, 0)
133 && (uint32_t) hx < 0xfff00000 && rint (x) == x)
134 {
135 /* Return value for integer x < 0 is NaN with invalid exception. */
136 *signgamp = 0;
137 return (x - x) / (x - x);
138 }
139 if (__glibc_unlikely ((unsigned int) hx == 0xfff00000 && lx == 0))
140 {
141 /* x == -Inf. According to ISO this is NaN. */
142 *signgamp = 0;
143 return x - x;
144 }
145 if (__glibc_unlikely ((hx & 0x7ff00000) == 0x7ff00000))
146 {
147 /* Positive infinity (return positive infinity) or NaN (return
148 NaN). */
149 *signgamp = 0;
150 return x + x;
151 }
152
153 if (x >= 172.0)
154 {
155 /* Overflow. */
156 *signgamp = 0;
157 ret = math_narrow_eval (DBL_MAX * DBL_MAX);
158 return ret;
159 }
160 else
161 {
162 SET_RESTORE_ROUND (FE_TONEAREST);
163 if (x > 0.0)
164 {
165 *signgamp = 0;
166 int exp2_adj;
167 double tret = gamma_positive (x, &exp2_adj);
168 ret = __scalbn (tret, exp2_adj);
169 }
170 else if (x >= -DBL_EPSILON / 4.0)
171 {
172 *signgamp = 0;
173 ret = 1.0 / x;
174 }
175 else
176 {
177 double tx = trunc (x);
178 *signgamp = (tx == 2.0 * trunc (tx / 2.0)) ? -1 : 1;
179 if (x <= -184.0)
180 /* Underflow. */
181 ret = DBL_MIN * DBL_MIN;
182 else
183 {
184 double frac = tx - x;
185 if (frac > 0.5)
186 frac = 1.0 - frac;
187 double sinpix = (frac <= 0.25
188 ? __sin (M_PI * frac)
189 : __cos (M_PI * (0.5 - frac)));
190 int exp2_adj;
191 double tret = M_PI / (-x * sinpix
192 * gamma_positive (-x, &exp2_adj));
193 ret = __scalbn (tret, -exp2_adj);
194 math_check_force_underflow_nonneg (ret);
195 }
196 }
197 ret = math_narrow_eval (ret);
198 }
199 if (isinf (ret) && x != 0)
200 {
201 if (*signgamp < 0)
202 {
203 ret = math_narrow_eval (-copysign (DBL_MAX, ret) * DBL_MAX);
204 ret = -ret;
205 }
206 else
207 ret = math_narrow_eval (copysign (DBL_MAX, ret) * DBL_MAX);
208 return ret;
209 }
210 else if (ret == 0)
211 {
212 if (*signgamp < 0)
213 {
214 ret = math_narrow_eval (-copysign (DBL_MIN, ret) * DBL_MIN);
215 ret = -ret;
216 }
217 else
218 ret = math_narrow_eval (copysign (DBL_MIN, ret) * DBL_MIN);
219 return ret;
220 }
221 else
222 return ret;
223}
224libm_alias_finite (__ieee754_gamma_r, __gamma_r)
225