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
29 static const long double gamma_coeff[] =
30 {
31 0x1.555555555555555555555555558p-4L,
32 -0xb.60b60b60b60b60b60b60b60b6p-12L,
33 0x3.4034034034034034034034034p-12L,
34 -0x2.7027027027027027027027027p-12L,
35 0x3.72a3c5631fe46ae1d4e700dca9p-12L,
36 -0x7.daac36664f1f207daac36664f2p-12L,
37 0x1.a41a41a41a41a41a41a41a41a4p-8L,
38 -0x7.90a1b2c3d4e5f708192a3b4c5ep-8L,
39 0x2.dfd2c703c0cfff430edfd2c704p-4L,
40 -0x1.6476701181f39edbdb9ce625988p+0L,
41 0xd.672219167002d3a7a9c886459cp+0L,
42 -0x9.cd9292e6660d55b3f712eb9e08p+4L,
43 0x8.911a740da740da740da740da74p+8L,
44 };
45
46 #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
47
48 /* Return gamma (X), for positive X less than 191, in the form R *
49 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
50 avoid overflow or underflow in intermediate calculations. */
51
52 static long double
53 gammal_positive (long double x, int *exp2_adj)
54 {
55 int local_signgam;
56 if (x < 0.5L)
57 {
58 *exp2_adj = 0;
59 return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
60 }
61 else if (x <= 1.5L)
62 {
63 *exp2_adj = 0;
64 return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
65 }
66 else if (x < 11.5L)
67 {
68 /* Adjust into the range for using exp (lgamma). */
69 *exp2_adj = 0;
70 long double n = ceill (x - 1.5L);
71 long double x_adj = x - n;
72 long double eps;
73 long double prod = __gamma_productl (x_adj, 0, n, &eps);
74 return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
75 * prod * (1.0L + eps));
76 }
77 else
78 {
79 long double eps = 0;
80 long double x_eps = 0;
81 long double x_adj = x;
82 long double prod = 1;
83 if (x < 23.0L)
84 {
85 /* Adjust into the range for applying Stirling's
86 approximation. */
87 long double n = ceill (23.0L - x);
88 x_adj = x + n;
89 x_eps = (x - (x_adj - n));
90 prod = __gamma_productl (x_adj - n, x_eps, n, &eps);
91 }
92 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
93 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
94 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
95 factored out. */
96 long double exp_adj = -eps;
97 long double x_adj_int = roundl (x_adj);
98 long double x_adj_frac = x_adj - x_adj_int;
99 int x_adj_log2;
100 long double x_adj_mant = __frexpl (x_adj, &x_adj_log2);
101 if (x_adj_mant < M_SQRT1_2l)
102 {
103 x_adj_log2--;
104 x_adj_mant *= 2.0L;
105 }
106 *exp2_adj = x_adj_log2 * (int) x_adj_int;
107 long double ret = (__ieee754_powl (x_adj_mant, x_adj)
108 * __ieee754_exp2l (x_adj_log2 * x_adj_frac)
109 * __ieee754_expl (-x_adj)
110 * sqrtl (2 * M_PIl / x_adj)
111 / prod);
112 exp_adj += x_eps * __ieee754_logl (x_adj);
113 long double bsum = gamma_coeff[NCOEFF - 1];
114 long double x_adj2 = x_adj * x_adj;
115 for (size_t i = 1; i <= NCOEFF - 1; i++)
116 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
117 exp_adj += bsum / x_adj;
118 return ret + ret * __expm1l (exp_adj);
119 }
120 }
121
122 long double
123 __ieee754_gammal_r (long double x, int *signgamp)
124 {
125 int64_t hx;
126 double xhi;
127 long double ret;
128
129 xhi = ldbl_high (x);
130 EXTRACT_WORDS64 (hx, xhi);
131
132 if ((hx & 0x7fffffffffffffffLL) == 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 < 0xfff0000000000000ULL && 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 == 0xfff0000000000000ULL)
145 {
146 /* x == -Inf. According to ISO this is NaN. */
147 *signgamp = 0;
148 return x - x;
149 }
150 if ((hx & 0x7ff0000000000000ULL) == 0x7ff0000000000000ULL)
151 {
152 /* Positive infinity (return positive infinity) or NaN (return
153 NaN). */
154 *signgamp = 0;
155 return x + x;
156 }
157
158 if (x >= 172.0L)
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 > 0.0L)
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 >= -0x1p-110L)
175 {
176 *signgamp = 0;
177 ret = 1.0L / x;
178 }
179 else
180 {
181 long double tx = truncl (x);
182 *signgamp = (tx == 2.0L * truncl (tx / 2.0L)) ? -1 : 1;
183 if (x <= -191.0L)
184 /* Underflow. */
185 ret = LDBL_MIN * LDBL_MIN;
186 else
187 {
188 long double frac = tx - x;
189 if (frac > 0.5L)
190 frac = 1.0L - frac;
191 long double sinpix = (frac <= 0.25L
192 ? __sinl (M_PIl * frac)
193 : __cosl (M_PIl * (0.5L - 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 }
219 libm_alias_finite (__ieee754_gammal_r, __gammal_r)