1/* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2022 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
9
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19#include <math.h>
20#include <math_private.h>
21#include <fenv_private.h>
22#include <math-underflow.h>
23#include <float.h>
24#include <libm-alias-finite.h>
25
26/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
27 approximation to gamma function. */
28
29static const long double gamma_coeff[] =
30 {
31 0x1.5555555555555556p-4L,
32 -0xb.60b60b60b60b60bp-12L,
33 0x3.4034034034034034p-12L,
34 -0x2.7027027027027028p-12L,
35 0x3.72a3c5631fe46aep-12L,
36 -0x7.daac36664f1f208p-12L,
37 0x1.a41a41a41a41a41ap-8L,
38 -0x7.90a1b2c3d4e5f708p-8L,
39 };
40
41#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
42
43/* Return gamma (X), for positive X less than 1766, 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 long double
48gammal_positive (long double x, int *exp2_adj)
49{
50 int local_signgam;
51 if (x < 0.5L)
52 {
53 *exp2_adj = 0;
54 return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
55 }
56 else if (x <= 1.5L)
57 {
58 *exp2_adj = 0;
59 return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
60 }
61 else if (x < 7.5L)
62 {
63 /* Adjust into the range for using exp (lgamma). */
64 *exp2_adj = 0;
65 long double n = ceill (x - 1.5L);
66 long double x_adj = x - n;
67 long double eps;
68 long double prod = __gamma_productl (x_adj, 0, n, &eps);
69 return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
70 * prod * (1.0L + eps));
71 }
72 else
73 {
74 long double eps = 0;
75 long double x_eps = 0;
76 long double x_adj = x;
77 long double prod = 1;
78 if (x < 13.0L)
79 {
80 /* Adjust into the range for applying Stirling's
81 approximation. */
82 long double n = ceill (13.0L - x);
83 x_adj = x + n;
84 x_eps = (x - (x_adj - n));
85 prod = __gamma_productl (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 long double exp_adj = -eps;
92 long double x_adj_int = roundl (x_adj);
93 long double x_adj_frac = x_adj - x_adj_int;
94 int x_adj_log2;
95 long double x_adj_mant = __frexpl (x_adj, &x_adj_log2);
96 if (x_adj_mant < M_SQRT1_2l)
97 {
98 x_adj_log2--;
99 x_adj_mant *= 2.0L;
100 }
101 *exp2_adj = x_adj_log2 * (int) x_adj_int;
102 long double ret = (__ieee754_powl (x_adj_mant, x_adj)
103 * __ieee754_exp2l (x_adj_log2 * x_adj_frac)
104 * __ieee754_expl (-x_adj)
105 * sqrtl (2 * M_PIl / x_adj)
106 / prod);
107 exp_adj += x_eps * __ieee754_logl (x_adj);
108 long double bsum = gamma_coeff[NCOEFF - 1];
109 long 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 * __expm1l (exp_adj);
114 }
115}
116
117long double
118__ieee754_gammal_r (long double x, int *signgamp)
119{
120 uint32_t es, hx, lx;
121 long double ret;
122
123 GET_LDOUBLE_WORDS (es, hx, lx, x);
124
125 if (__glibc_unlikely (((es & 0x7fff) | hx | lx) == 0))
126 {
127 /* Return value for x == 0 is Inf with divide by zero exception. */
128 *signgamp = 0;
129 return 1.0 / x;
130 }
131 if (__glibc_unlikely (es == 0xffffffff && ((hx & 0x7fffffff) | lx) == 0))
132 {
133 /* x == -Inf. According to ISO this is NaN. */
134 *signgamp = 0;
135 return x - x;
136 }
137 if (__glibc_unlikely ((es & 0x7fff) == 0x7fff))
138 {
139 /* Positive infinity (return positive infinity) or NaN (return
140 NaN). */
141 *signgamp = 0;
142 return x + x;
143 }
144 if (__builtin_expect ((es & 0x8000) != 0, 0) && rintl (x) == x)
145 {
146 /* Return value for integer x < 0 is NaN with invalid exception. */
147 *signgamp = 0;
148 return (x - x) / (x - x);
149 }
150
151 if (x >= 1756.0L)
152 {
153 /* Overflow. */
154 *signgamp = 0;
155 return LDBL_MAX * LDBL_MAX;
156 }
157 else
158 {
159 SET_RESTORE_ROUNDL (FE_TONEAREST);
160 if (x > 0.0L)
161 {
162 *signgamp = 0;
163 int exp2_adj;
164 ret = gammal_positive (x, &exp2_adj);
165 ret = __scalbnl (ret, exp2_adj);
166 }
167 else if (x >= -LDBL_EPSILON / 4.0L)
168 {
169 *signgamp = 0;
170 ret = 1.0L / x;
171 }
172 else
173 {
174 long double tx = truncl (x);
175 *signgamp = (tx == 2.0L * truncl (tx / 2.0L)) ? -1 : 1;
176 if (x <= -1766.0L)
177 /* Underflow. */
178 ret = LDBL_MIN * LDBL_MIN;
179 else
180 {
181 long double frac = tx - x;
182 if (frac > 0.5L)
183 frac = 1.0L - frac;
184 long double sinpix = (frac <= 0.25L
185 ? __sinl (M_PIl * frac)
186 : __cosl (M_PIl * (0.5L - frac)));
187 int exp2_adj;
188 ret = M_PIl / (-x * sinpix
189 * gammal_positive (-x, &exp2_adj));
190 ret = __scalbnl (ret, -exp2_adj);
191 math_check_force_underflow_nonneg (ret);
192 }
193 }
194 }
195 if (isinf (ret) && x != 0)
196 {
197 if (*signgamp < 0)
198 return -(-copysignl (LDBL_MAX, ret) * LDBL_MAX);
199 else
200 return copysignl (LDBL_MAX, ret) * LDBL_MAX;
201 }
202 else if (ret == 0)
203 {
204 if (*signgamp < 0)
205 return -(-copysignl (LDBL_MIN, ret) * LDBL_MIN);
206 else
207 return copysignl (LDBL_MIN, ret) * LDBL_MIN;
208 }
209 else
210 return ret;
211}
212libm_alias_finite (__ieee754_gammal_r, __gammal_r)
213