1/* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2023 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 _Float128 gamma_coeff[] =
30 {
31 L(0x1.5555555555555555555555555555p-4),
32 L(-0xb.60b60b60b60b60b60b60b60b60b8p-12),
33 L(0x3.4034034034034034034034034034p-12),
34 L(-0x2.7027027027027027027027027028p-12),
35 L(0x3.72a3c5631fe46ae1d4e700dca8f2p-12),
36 L(-0x7.daac36664f1f207daac36664f1f4p-12),
37 L(0x1.a41a41a41a41a41a41a41a41a41ap-8),
38 L(-0x7.90a1b2c3d4e5f708192a3b4c5d7p-8),
39 L(0x2.dfd2c703c0cfff430edfd2c703cp-4),
40 L(-0x1.6476701181f39edbdb9ce625987dp+0),
41 L(0xd.672219167002d3a7a9c886459cp+0),
42 L(-0x9.cd9292e6660d55b3f712eb9e07c8p+4),
43 L(0x8.911a740da740da740da740da741p+8),
44 L(-0x8.d0cc570e255bf59ff6eec24b49p+12),
45 };
46
47#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
48
49/* Return gamma (X), for positive X less than 1775, in the form R *
50 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
51 avoid overflow or underflow in intermediate calculations. */
52
53static _Float128
54gammal_positive (_Float128 x, int *exp2_adj)
55{
56 int local_signgam;
57 if (x < L(0.5))
58 {
59 *exp2_adj = 0;
60 return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
61 }
62 else if (x <= L(1.5))
63 {
64 *exp2_adj = 0;
65 return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
66 }
67 else if (x < L(12.5))
68 {
69 /* Adjust into the range for using exp (lgamma). */
70 *exp2_adj = 0;
71 _Float128 n = ceill (x - L(1.5));
72 _Float128 x_adj = x - n;
73 _Float128 eps;
74 _Float128 prod = __gamma_productl (x_adj, 0, n, &eps);
75 return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
76 * prod * (1 + eps));
77 }
78 else
79 {
80 _Float128 eps = 0;
81 _Float128 x_eps = 0;
82 _Float128 x_adj = x;
83 _Float128 prod = 1;
84 if (x < 24)
85 {
86 /* Adjust into the range for applying Stirling's
87 approximation. */
88 _Float128 n = ceill (24 - x);
89 x_adj = x + n;
90 x_eps = (x - (x_adj - n));
91 prod = __gamma_productl (x_adj - n, x_eps, n, &eps);
92 }
93 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
94 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
95 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
96 factored out. */
97 _Float128 exp_adj = -eps;
98 _Float128 x_adj_int = roundl (x_adj);
99 _Float128 x_adj_frac = x_adj - x_adj_int;
100 int x_adj_log2;
101 _Float128 x_adj_mant = __frexpl (x_adj, &x_adj_log2);
102 if (x_adj_mant < M_SQRT1_2l)
103 {
104 x_adj_log2--;
105 x_adj_mant *= 2;
106 }
107 *exp2_adj = x_adj_log2 * (int) x_adj_int;
108 _Float128 ret = (__ieee754_powl (x_adj_mant, x_adj)
109 * __ieee754_exp2l (x_adj_log2 * x_adj_frac)
110 * __ieee754_expl (-x_adj)
111 * sqrtl (2 * M_PIl / x_adj)
112 / prod);
113 exp_adj += x_eps * __ieee754_logl (x_adj);
114 _Float128 bsum = gamma_coeff[NCOEFF - 1];
115 _Float128 x_adj2 = x_adj * x_adj;
116 for (size_t i = 1; i <= NCOEFF - 1; i++)
117 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
118 exp_adj += bsum / x_adj;
119 return ret + ret * __expm1l (exp_adj);
120 }
121}
122
123_Float128
124__ieee754_gammal_r (_Float128 x, int *signgamp)
125{
126 int64_t hx;
127 uint64_t lx;
128 _Float128 ret;
129
130 GET_LDOUBLE_WORDS64 (hx, lx, x);
131
132 if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
133 {
134 /* Return value for x == 0 is Inf with divide by zero exception. */
135 *signgamp = 0;
136 return 1.0 / x;
137 }
138 if (hx < 0 && (uint64_t) hx < 0xffff000000000000ULL && rintl (x) == x)
139 {
140 /* Return value for integer x < 0 is NaN with invalid exception. */
141 *signgamp = 0;
142 return (x - x) / (x - x);
143 }
144 if (hx == 0xffff000000000000ULL && lx == 0)
145 {
146 /* x == -Inf. According to ISO this is NaN. */
147 *signgamp = 0;
148 return x - x;
149 }
150 if ((hx & 0x7fff000000000000ULL) == 0x7fff000000000000ULL)
151 {
152 /* Positive infinity (return positive infinity) or NaN (return
153 NaN). */
154 *signgamp = 0;
155 return x + x;
156 }
157
158 if (x >= 1756)
159 {
160 /* Overflow. */
161 *signgamp = 0;
162 return LDBL_MAX * LDBL_MAX;
163 }
164 else
165 {
166 SET_RESTORE_ROUNDL (FE_TONEAREST);
167 if (x > 0)
168 {
169 *signgamp = 0;
170 int exp2_adj;
171 ret = gammal_positive (x, &exp2_adj);
172 ret = __scalbnl (ret, exp2_adj);
173 }
174 else if (x >= -LDBL_EPSILON / 4)
175 {
176 *signgamp = 0;
177 ret = 1 / x;
178 }
179 else
180 {
181 _Float128 tx = truncl (x);
182 *signgamp = (tx == 2 * truncl (tx / 2)) ? -1 : 1;
183 if (x <= -1775)
184 /* Underflow. */
185 ret = LDBL_MIN * LDBL_MIN;
186 else
187 {
188 _Float128 frac = tx - x;
189 if (frac > L(0.5))
190 frac = 1 - frac;
191 _Float128 sinpix = (frac <= L(0.25)
192 ? __sinl (M_PIl * frac)
193 : __cosl (M_PIl * (L(0.5) - frac)));
194 int exp2_adj;
195 ret = M_PIl / (-x * sinpix
196 * gammal_positive (-x, &exp2_adj));
197 ret = __scalbnl (ret, -exp2_adj);
198 math_check_force_underflow_nonneg (ret);
199 }
200 }
201 }
202 if (isinf (ret) && x != 0)
203 {
204 if (*signgamp < 0)
205 return -(-copysignl (LDBL_MAX, ret) * LDBL_MAX);
206 else
207 return copysignl (LDBL_MAX, ret) * LDBL_MAX;
208 }
209 else if (ret == 0)
210 {
211 if (*signgamp < 0)
212 return -(-copysignl (LDBL_MIN, ret) * LDBL_MIN);
213 else
214 return copysignl (LDBL_MIN, ret) * LDBL_MIN;
215 }
216 else
217 return ret;
218}
219libm_alias_finite (__ieee754_gammal_r, __gammal_r)
220