1/* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2018 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 <http://www.gnu.org/licenses/>. */
19
20#include <math.h>
21#include <math_private.h>
22#include <math-underflow.h>
23#include <float.h>
24
25/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
26 approximation to gamma function. */
27
28static const long double gamma_coeff[] =
29 {
30 0x1.5555555555555556p-4L,
31 -0xb.60b60b60b60b60bp-12L,
32 0x3.4034034034034034p-12L,
33 -0x2.7027027027027028p-12L,
34 0x3.72a3c5631fe46aep-12L,
35 -0x7.daac36664f1f208p-12L,
36 0x1.a41a41a41a41a41ap-8L,
37 -0x7.90a1b2c3d4e5f708p-8L,
38 };
39
40#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
41
42/* Return gamma (X), for positive X less than 1766, in the form R *
43 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
44 avoid overflow or underflow in intermediate calculations. */
45
46static long double
47gammal_positive (long double x, int *exp2_adj)
48{
49 int local_signgam;
50 if (x < 0.5L)
51 {
52 *exp2_adj = 0;
53 return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
54 }
55 else if (x <= 1.5L)
56 {
57 *exp2_adj = 0;
58 return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
59 }
60 else if (x < 7.5L)
61 {
62 /* Adjust into the range for using exp (lgamma). */
63 *exp2_adj = 0;
64 long double n = __ceill (x - 1.5L);
65 long double x_adj = x - n;
66 long double eps;
67 long double prod = __gamma_productl (x_adj, 0, n, &eps);
68 return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
69 * prod * (1.0L + eps));
70 }
71 else
72 {
73 long double eps = 0;
74 long double x_eps = 0;
75 long double x_adj = x;
76 long double prod = 1;
77 if (x < 13.0L)
78 {
79 /* Adjust into the range for applying Stirling's
80 approximation. */
81 long double n = __ceill (13.0L - x);
82 x_adj = x + n;
83 x_eps = (x - (x_adj - n));
84 prod = __gamma_productl (x_adj - n, x_eps, n, &eps);
85 }
86 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
87 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
88 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
89 factored out. */
90 long double exp_adj = -eps;
91 long double x_adj_int = __roundl (x_adj);
92 long double x_adj_frac = x_adj - x_adj_int;
93 int x_adj_log2;
94 long double x_adj_mant = __frexpl (x_adj, &x_adj_log2);
95 if (x_adj_mant < M_SQRT1_2l)
96 {
97 x_adj_log2--;
98 x_adj_mant *= 2.0L;
99 }
100 *exp2_adj = x_adj_log2 * (int) x_adj_int;
101 long double ret = (__ieee754_powl (x_adj_mant, x_adj)
102 * __ieee754_exp2l (x_adj_log2 * x_adj_frac)
103 * __ieee754_expl (-x_adj)
104 * sqrtl (2 * M_PIl / x_adj)
105 / prod);
106 exp_adj += x_eps * __ieee754_logl (x_adj);
107 long double bsum = gamma_coeff[NCOEFF - 1];
108 long double x_adj2 = x_adj * x_adj;
109 for (size_t i = 1; i <= NCOEFF - 1; i++)
110 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
111 exp_adj += bsum / x_adj;
112 return ret + ret * __expm1l (exp_adj);
113 }
114}
115
116long double
117__ieee754_gammal_r (long double x, int *signgamp)
118{
119 uint32_t es, hx, lx;
120 long double ret;
121
122 GET_LDOUBLE_WORDS (es, hx, lx, x);
123
124 if (__glibc_unlikely (((es & 0x7fff) | hx | lx) == 0))
125 {
126 /* Return value for x == 0 is Inf with divide by zero exception. */
127 *signgamp = 0;
128 return 1.0 / x;
129 }
130 if (__glibc_unlikely (es == 0xffffffff && ((hx & 0x7fffffff) | lx) == 0))
131 {
132 /* x == -Inf. According to ISO this is NaN. */
133 *signgamp = 0;
134 return x - x;
135 }
136 if (__glibc_unlikely ((es & 0x7fff) == 0x7fff))
137 {
138 /* Positive infinity (return positive infinity) or NaN (return
139 NaN). */
140 *signgamp = 0;
141 return x + x;
142 }
143 if (__builtin_expect ((es & 0x8000) != 0, 0) && __rintl (x) == x)
144 {
145 /* Return value for integer x < 0 is NaN with invalid exception. */
146 *signgamp = 0;
147 return (x - x) / (x - x);
148 }
149
150 if (x >= 1756.0L)
151 {
152 /* Overflow. */
153 *signgamp = 0;
154 return LDBL_MAX * LDBL_MAX;
155 }
156 else
157 {
158 SET_RESTORE_ROUNDL (FE_TONEAREST);
159 if (x > 0.0L)
160 {
161 *signgamp = 0;
162 int exp2_adj;
163 ret = gammal_positive (x, &exp2_adj);
164 ret = __scalbnl (ret, exp2_adj);
165 }
166 else if (x >= -LDBL_EPSILON / 4.0L)
167 {
168 *signgamp = 0;
169 ret = 1.0L / x;
170 }
171 else
172 {
173 long double tx = __truncl (x);
174 *signgamp = (tx == 2.0L * __truncl (tx / 2.0L)) ? -1 : 1;
175 if (x <= -1766.0L)
176 /* Underflow. */
177 ret = LDBL_MIN * LDBL_MIN;
178 else
179 {
180 long double frac = tx - x;
181 if (frac > 0.5L)
182 frac = 1.0L - frac;
183 long double sinpix = (frac <= 0.25L
184 ? __sinl (M_PIl * frac)
185 : __cosl (M_PIl * (0.5L - frac)));
186 int exp2_adj;
187 ret = M_PIl / (-x * sinpix
188 * gammal_positive (-x, &exp2_adj));
189 ret = __scalbnl (ret, -exp2_adj);
190 math_check_force_underflow_nonneg (ret);
191 }
192 }
193 }
194 if (isinf (ret) && x != 0)
195 {
196 if (*signgamp < 0)
197 return -(-__copysignl (LDBL_MAX, ret) * LDBL_MAX);
198 else
199 return __copysignl (LDBL_MAX, ret) * LDBL_MAX;
200 }
201 else if (ret == 0)
202 {
203 if (*signgamp < 0)
204 return -(-__copysignl (LDBL_MIN, ret) * LDBL_MIN);
205 else
206 return __copysignl (LDBL_MIN, ret) * LDBL_MIN;
207 }
208 else
209 return ret;
210}
211strong_alias (__ieee754_gammal_r, __gammal_r_finite)
212