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 and
5 Jakub Jelinek <jj@ultra.linux.cz, 1999.
6
7 The GNU C Library is free software; you can redistribute it and/or
8 modify it under the terms of the GNU Lesser General Public
9 License as published by the Free Software Foundation; either
10 version 2.1 of the License, or (at your option) any later version.
11
12 The GNU C Library is distributed in the hope that it will be useful,
13 but WITHOUT ANY WARRANTY; without even the implied warranty of
14 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
15 Lesser General Public License for more details.
16
17 You should have received a copy of the GNU Lesser General Public
18 License along with the GNU C Library; if not, see
19 <https://www.gnu.org/licenses/>. */
20
21#include <math.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 _Float128 gamma_coeff[] =
32 {
33 L(0x1.5555555555555555555555555555p-4),
34 L(-0xb.60b60b60b60b60b60b60b60b60b8p-12),
35 L(0x3.4034034034034034034034034034p-12),
36 L(-0x2.7027027027027027027027027028p-12),
37 L(0x3.72a3c5631fe46ae1d4e700dca8f2p-12),
38 L(-0x7.daac36664f1f207daac36664f1f4p-12),
39 L(0x1.a41a41a41a41a41a41a41a41a41ap-8),
40 L(-0x7.90a1b2c3d4e5f708192a3b4c5d7p-8),
41 L(0x2.dfd2c703c0cfff430edfd2c703cp-4),
42 L(-0x1.6476701181f39edbdb9ce625987dp+0),
43 L(0xd.672219167002d3a7a9c886459cp+0),
44 L(-0x9.cd9292e6660d55b3f712eb9e07c8p+4),
45 L(0x8.911a740da740da740da740da741p+8),
46 L(-0x8.d0cc570e255bf59ff6eec24b49p+12),
47 };
48
49#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
50
51/* Return gamma (X), for positive X less than 1775, in the form R *
52 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
53 avoid overflow or underflow in intermediate calculations. */
54
55static _Float128
56gammal_positive (_Float128 x, int *exp2_adj)
57{
58 int local_signgam;
59 if (x < L(0.5))
60 {
61 *exp2_adj = 0;
62 return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
63 }
64 else if (x <= L(1.5))
65 {
66 *exp2_adj = 0;
67 return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
68 }
69 else if (x < L(12.5))
70 {
71 /* Adjust into the range for using exp (lgamma). */
72 *exp2_adj = 0;
73 _Float128 n = ceill (x - L(1.5));
74 _Float128 x_adj = x - n;
75 _Float128 eps;
76 _Float128 prod = __gamma_productl (x_adj, 0, n, &eps);
77 return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
78 * prod * (1 + eps));
79 }
80 else
81 {
82 _Float128 eps = 0;
83 _Float128 x_eps = 0;
84 _Float128 x_adj = x;
85 _Float128 prod = 1;
86 if (x < 24)
87 {
88 /* Adjust into the range for applying Stirling's
89 approximation. */
90 _Float128 n = ceill (24 - x);
91 x_adj = x + n;
92 x_eps = (x - (x_adj - n));
93 prod = __gamma_productl (x_adj - n, x_eps, n, &eps);
94 }
95 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
96 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
97 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
98 factored out. */
99 _Float128 exp_adj = -eps;
100 _Float128 x_adj_int = roundl (x_adj);
101 _Float128 x_adj_frac = x_adj - x_adj_int;
102 int x_adj_log2;
103 _Float128 x_adj_mant = __frexpl (x_adj, &x_adj_log2);
104 if (x_adj_mant < M_SQRT1_2l)
105 {
106 x_adj_log2--;
107 x_adj_mant *= 2;
108 }
109 *exp2_adj = x_adj_log2 * (int) x_adj_int;
110 _Float128 ret = (__ieee754_powl (x_adj_mant, x_adj)
111 * __ieee754_exp2l (x_adj_log2 * x_adj_frac)
112 * __ieee754_expl (-x_adj)
113 * sqrtl (2 * M_PIl / x_adj)
114 / prod);
115 exp_adj += x_eps * __ieee754_logl (x_adj);
116 _Float128 bsum = gamma_coeff[NCOEFF - 1];
117 _Float128 x_adj2 = x_adj * x_adj;
118 for (size_t i = 1; i <= NCOEFF - 1; i++)
119 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
120 exp_adj += bsum / x_adj;
121 return ret + ret * __expm1l (exp_adj);
122 }
123}
124
125_Float128
126__ieee754_gammal_r (_Float128 x, int *signgamp)
127{
128 int64_t hx;
129 uint64_t lx;
130 _Float128 ret;
131
132 GET_LDOUBLE_WORDS64 (hx, lx, x);
133
134 if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
135 {
136 /* Return value for x == 0 is Inf with divide by zero exception. */
137 *signgamp = 0;
138 return 1.0 / x;
139 }
140 if (hx < 0 && (uint64_t) hx < 0xffff000000000000ULL && rintl (x) == x)
141 {
142 /* Return value for integer x < 0 is NaN with invalid exception. */
143 *signgamp = 0;
144 return (x - x) / (x - x);
145 }
146 if (hx == 0xffff000000000000ULL && lx == 0)
147 {
148 /* x == -Inf. According to ISO this is NaN. */
149 *signgamp = 0;
150 return x - x;
151 }
152 if ((hx & 0x7fff000000000000ULL) == 0x7fff000000000000ULL)
153 {
154 /* Positive infinity (return positive infinity) or NaN (return
155 NaN). */
156 *signgamp = 0;
157 return x + x;
158 }
159
160 if (x >= 1756)
161 {
162 /* Overflow. */
163 *signgamp = 0;
164 return LDBL_MAX * LDBL_MAX;
165 }
166 else
167 {
168 SET_RESTORE_ROUNDL (FE_TONEAREST);
169 if (x > 0)
170 {
171 *signgamp = 0;
172 int exp2_adj;
173 ret = gammal_positive (x, &exp2_adj);
174 ret = __scalbnl (ret, exp2_adj);
175 }
176 else if (x >= -LDBL_EPSILON / 4)
177 {
178 *signgamp = 0;
179 ret = 1 / x;
180 }
181 else
182 {
183 _Float128 tx = truncl (x);
184 *signgamp = (tx == 2 * truncl (tx / 2)) ? -1 : 1;
185 if (x <= -1775)
186 /* Underflow. */
187 ret = LDBL_MIN * LDBL_MIN;
188 else
189 {
190 _Float128 frac = tx - x;
191 if (frac > L(0.5))
192 frac = 1 - frac;
193 _Float128 sinpix = (frac <= L(0.25)
194 ? __sinl (M_PIl * frac)
195 : __cosl (M_PIl * (L(0.5) - frac)));
196 int exp2_adj;
197 ret = M_PIl / (-x * sinpix
198 * gammal_positive (-x, &exp2_adj));
199 ret = __scalbnl (ret, -exp2_adj);
200 math_check_force_underflow_nonneg (ret);
201 }
202 }
203 }
204 if (isinf (ret) && x != 0)
205 {
206 if (*signgamp < 0)
207 return -(-copysignl (LDBL_MAX, ret) * LDBL_MAX);
208 else
209 return copysignl (LDBL_MAX, ret) * LDBL_MAX;
210 }
211 else if (ret == 0)
212 {
213 if (*signgamp < 0)
214 return -(-copysignl (LDBL_MIN, ret) * LDBL_MIN);
215 else
216 return copysignl (LDBL_MIN, ret) * LDBL_MIN;
217 }
218 else
219 return ret;
220}
221libm_alias_finite (__ieee754_gammal_r, __gammal_r)
222