1/* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2020 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_private.h>
22#include <fenv_private.h>
23#include <math-underflow.h>
24#include <float.h>
25#include <libm-alias-finite.h>
26
27/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
28 approximation to gamma function. */
29
30static const long double gamma_coeff[] =
31 {
32 0x1.5555555555555556p-4L,
33 -0xb.60b60b60b60b60bp-12L,
34 0x3.4034034034034034p-12L,
35 -0x2.7027027027027028p-12L,
36 0x3.72a3c5631fe46aep-12L,
37 -0x7.daac36664f1f208p-12L,
38 0x1.a41a41a41a41a41ap-8L,
39 -0x7.90a1b2c3d4e5f708p-8L,
40 };
41
42#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
43
44/* Return gamma (X), for positive X less than 1766, in the form R *
45 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
46 avoid overflow or underflow in intermediate calculations. */
47
48static long double
49gammal_positive (long double x, int *exp2_adj)
50{
51 int local_signgam;
52 if (x < 0.5L)
53 {
54 *exp2_adj = 0;
55 return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
56 }
57 else if (x <= 1.5L)
58 {
59 *exp2_adj = 0;
60 return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
61 }
62 else if (x < 7.5L)
63 {
64 /* Adjust into the range for using exp (lgamma). */
65 *exp2_adj = 0;
66 long double n = ceill (x - 1.5L);
67 long double x_adj = x - n;
68 long double eps;
69 long double prod = __gamma_productl (x_adj, 0, n, &eps);
70 return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
71 * prod * (1.0L + eps));
72 }
73 else
74 {
75 long double eps = 0;
76 long double x_eps = 0;
77 long double x_adj = x;
78 long double prod = 1;
79 if (x < 13.0L)
80 {
81 /* Adjust into the range for applying Stirling's
82 approximation. */
83 long double n = ceill (13.0L - x);
84 x_adj = x + n;
85 x_eps = (x - (x_adj - n));
86 prod = __gamma_productl (x_adj - n, x_eps, n, &eps);
87 }
88 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
89 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
90 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
91 factored out. */
92 long double exp_adj = -eps;
93 long double x_adj_int = roundl (x_adj);
94 long double x_adj_frac = x_adj - x_adj_int;
95 int x_adj_log2;
96 long double x_adj_mant = __frexpl (x_adj, &x_adj_log2);
97 if (x_adj_mant < M_SQRT1_2l)
98 {
99 x_adj_log2--;
100 x_adj_mant *= 2.0L;
101 }
102 *exp2_adj = x_adj_log2 * (int) x_adj_int;
103 long double ret = (__ieee754_powl (x_adj_mant, x_adj)
104 * __ieee754_exp2l (x_adj_log2 * x_adj_frac)
105 * __ieee754_expl (-x_adj)
106 * sqrtl (2 * M_PIl / x_adj)
107 / prod);
108 exp_adj += x_eps * __ieee754_logl (x_adj);
109 long double bsum = gamma_coeff[NCOEFF - 1];
110 long double x_adj2 = x_adj * x_adj;
111 for (size_t i = 1; i <= NCOEFF - 1; i++)
112 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
113 exp_adj += bsum / x_adj;
114 return ret + ret * __expm1l (exp_adj);
115 }
116}
117
118long double
119__ieee754_gammal_r (long double x, int *signgamp)
120{
121 uint32_t es, hx, lx;
122 long double ret;
123
124 GET_LDOUBLE_WORDS (es, hx, lx, x);
125
126 if (__glibc_unlikely (((es & 0x7fff) | hx | 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 (__glibc_unlikely (es == 0xffffffff && ((hx & 0x7fffffff) | lx) == 0))
133 {
134 /* x == -Inf. According to ISO this is NaN. */
135 *signgamp = 0;
136 return x - x;
137 }
138 if (__glibc_unlikely ((es & 0x7fff) == 0x7fff))
139 {
140 /* Positive infinity (return positive infinity) or NaN (return
141 NaN). */
142 *signgamp = 0;
143 return x + x;
144 }
145 if (__builtin_expect ((es & 0x8000) != 0, 0) && rintl (x) == x)
146 {
147 /* Return value for integer x < 0 is NaN with invalid exception. */
148 *signgamp = 0;
149 return (x - x) / (x - x);
150 }
151
152 if (x >= 1756.0L)
153 {
154 /* Overflow. */
155 *signgamp = 0;
156 return LDBL_MAX * LDBL_MAX;
157 }
158 else
159 {
160 SET_RESTORE_ROUNDL (FE_TONEAREST);
161 if (x > 0.0L)
162 {
163 *signgamp = 0;
164 int exp2_adj;
165 ret = gammal_positive (x, &exp2_adj);
166 ret = __scalbnl (ret, exp2_adj);
167 }
168 else if (x >= -LDBL_EPSILON / 4.0L)
169 {
170 *signgamp = 0;
171 ret = 1.0L / x;
172 }
173 else
174 {
175 long double tx = truncl (x);
176 *signgamp = (tx == 2.0L * truncl (tx / 2.0L)) ? -1 : 1;
177 if (x <= -1766.0L)
178 /* Underflow. */
179 ret = LDBL_MIN * LDBL_MIN;
180 else
181 {
182 long double frac = tx - x;
183 if (frac > 0.5L)
184 frac = 1.0L - frac;
185 long double sinpix = (frac <= 0.25L
186 ? __sinl (M_PIl * frac)
187 : __cosl (M_PIl * (0.5L - frac)));
188 int exp2_adj;
189 ret = M_PIl / (-x * sinpix
190 * gammal_positive (-x, &exp2_adj));
191 ret = __scalbnl (ret, -exp2_adj);
192 math_check_force_underflow_nonneg (ret);
193 }
194 }
195 }
196 if (isinf (ret) && x != 0)
197 {
198 if (*signgamp < 0)
199 return -(-copysignl (LDBL_MAX, ret) * LDBL_MAX);
200 else
201 return copysignl (LDBL_MAX, ret) * LDBL_MAX;
202 }
203 else if (ret == 0)
204 {
205 if (*signgamp < 0)
206 return -(-copysignl (LDBL_MIN, ret) * LDBL_MIN);
207 else
208 return copysignl (LDBL_MIN, ret) * LDBL_MIN;
209 }
210 else
211 return ret;
212}
213libm_alias_finite (__ieee754_gammal_r, __gammal_r)
214