1/* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2019 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 <http://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
27/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
28 approximation to gamma function. */
29
30static const _Float128 gamma_coeff[] =
31 {
32 L(0x1.5555555555555555555555555555p-4),
33 L(-0xb.60b60b60b60b60b60b60b60b60b8p-12),
34 L(0x3.4034034034034034034034034034p-12),
35 L(-0x2.7027027027027027027027027028p-12),
36 L(0x3.72a3c5631fe46ae1d4e700dca8f2p-12),
37 L(-0x7.daac36664f1f207daac36664f1f4p-12),
38 L(0x1.a41a41a41a41a41a41a41a41a41ap-8),
39 L(-0x7.90a1b2c3d4e5f708192a3b4c5d7p-8),
40 L(0x2.dfd2c703c0cfff430edfd2c703cp-4),
41 L(-0x1.6476701181f39edbdb9ce625987dp+0),
42 L(0xd.672219167002d3a7a9c886459cp+0),
43 L(-0x9.cd9292e6660d55b3f712eb9e07c8p+4),
44 L(0x8.911a740da740da740da740da741p+8),
45 L(-0x8.d0cc570e255bf59ff6eec24b49p+12),
46 };
47
48#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
49
50/* Return gamma (X), for positive X less than 1775, in the form R *
51 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
52 avoid overflow or underflow in intermediate calculations. */
53
54static _Float128
55gammal_positive (_Float128 x, int *exp2_adj)
56{
57 int local_signgam;
58 if (x < L(0.5))
59 {
60 *exp2_adj = 0;
61 return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
62 }
63 else if (x <= L(1.5))
64 {
65 *exp2_adj = 0;
66 return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
67 }
68 else if (x < L(12.5))
69 {
70 /* Adjust into the range for using exp (lgamma). */
71 *exp2_adj = 0;
72 _Float128 n = ceill (x - L(1.5));
73 _Float128 x_adj = x - n;
74 _Float128 eps;
75 _Float128 prod = __gamma_productl (x_adj, 0, n, &eps);
76 return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
77 * prod * (1 + eps));
78 }
79 else
80 {
81 _Float128 eps = 0;
82 _Float128 x_eps = 0;
83 _Float128 x_adj = x;
84 _Float128 prod = 1;
85 if (x < 24)
86 {
87 /* Adjust into the range for applying Stirling's
88 approximation. */
89 _Float128 n = ceill (24 - x);
90 x_adj = x + n;
91 x_eps = (x - (x_adj - n));
92 prod = __gamma_productl (x_adj - n, x_eps, n, &eps);
93 }
94 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
95 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
96 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
97 factored out. */
98 _Float128 exp_adj = -eps;
99 _Float128 x_adj_int = roundl (x_adj);
100 _Float128 x_adj_frac = x_adj - x_adj_int;
101 int x_adj_log2;
102 _Float128 x_adj_mant = __frexpl (x_adj, &x_adj_log2);
103 if (x_adj_mant < M_SQRT1_2l)
104 {
105 x_adj_log2--;
106 x_adj_mant *= 2;
107 }
108 *exp2_adj = x_adj_log2 * (int) x_adj_int;
109 _Float128 ret = (__ieee754_powl (x_adj_mant, x_adj)
110 * __ieee754_exp2l (x_adj_log2 * x_adj_frac)
111 * __ieee754_expl (-x_adj)
112 * sqrtl (2 * M_PIl / x_adj)
113 / prod);
114 exp_adj += x_eps * __ieee754_logl (x_adj);
115 _Float128 bsum = gamma_coeff[NCOEFF - 1];
116 _Float128 x_adj2 = x_adj * x_adj;
117 for (size_t i = 1; i <= NCOEFF - 1; i++)
118 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
119 exp_adj += bsum / x_adj;
120 return ret + ret * __expm1l (exp_adj);
121 }
122}
123
124_Float128
125__ieee754_gammal_r (_Float128 x, int *signgamp)
126{
127 int64_t hx;
128 uint64_t lx;
129 _Float128 ret;
130
131 GET_LDOUBLE_WORDS64 (hx, lx, x);
132
133 if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
134 {
135 /* Return value for x == 0 is Inf with divide by zero exception. */
136 *signgamp = 0;
137 return 1.0 / x;
138 }
139 if (hx < 0 && (uint64_t) hx < 0xffff000000000000ULL && rintl (x) == x)
140 {
141 /* Return value for integer x < 0 is NaN with invalid exception. */
142 *signgamp = 0;
143 return (x - x) / (x - x);
144 }
145 if (hx == 0xffff000000000000ULL && lx == 0)
146 {
147 /* x == -Inf. According to ISO this is NaN. */
148 *signgamp = 0;
149 return x - x;
150 }
151 if ((hx & 0x7fff000000000000ULL) == 0x7fff000000000000ULL)
152 {
153 /* Positive infinity (return positive infinity) or NaN (return
154 NaN). */
155 *signgamp = 0;
156 return x + x;
157 }
158
159 if (x >= 1756)
160 {
161 /* Overflow. */
162 *signgamp = 0;
163 return LDBL_MAX * LDBL_MAX;
164 }
165 else
166 {
167 SET_RESTORE_ROUNDL (FE_TONEAREST);
168 if (x > 0)
169 {
170 *signgamp = 0;
171 int exp2_adj;
172 ret = gammal_positive (x, &exp2_adj);
173 ret = __scalbnl (ret, exp2_adj);
174 }
175 else if (x >= -LDBL_EPSILON / 4)
176 {
177 *signgamp = 0;
178 ret = 1 / x;
179 }
180 else
181 {
182 _Float128 tx = truncl (x);
183 *signgamp = (tx == 2 * truncl (tx / 2)) ? -1 : 1;
184 if (x <= -1775)
185 /* Underflow. */
186 ret = LDBL_MIN * LDBL_MIN;
187 else
188 {
189 _Float128 frac = tx - x;
190 if (frac > L(0.5))
191 frac = 1 - frac;
192 _Float128 sinpix = (frac <= L(0.25)
193 ? __sinl (M_PIl * frac)
194 : __cosl (M_PIl * (L(0.5) - frac)));
195 int exp2_adj;
196 ret = M_PIl / (-x * sinpix
197 * gammal_positive (-x, &exp2_adj));
198 ret = __scalbnl (ret, -exp2_adj);
199 math_check_force_underflow_nonneg (ret);
200 }
201 }
202 }
203 if (isinf (ret) && x != 0)
204 {
205 if (*signgamp < 0)
206 return -(-copysignl (LDBL_MAX, ret) * LDBL_MAX);
207 else
208 return copysignl (LDBL_MAX, ret) * LDBL_MAX;
209 }
210 else if (ret == 0)
211 {
212 if (*signgamp < 0)
213 return -(-copysignl (LDBL_MIN, ret) * LDBL_MIN);
214 else
215 return copysignl (LDBL_MIN, ret) * LDBL_MIN;
216 }
217 else
218 return ret;
219}
220strong_alias (__ieee754_gammal_r, __gammal_r_finite)
221