1/* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2017 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 <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 _Float128 gamma_coeff[] =
29 {
30 L(0x1.5555555555555555555555555555p-4),
31 L(-0xb.60b60b60b60b60b60b60b60b60b8p-12),
32 L(0x3.4034034034034034034034034034p-12),
33 L(-0x2.7027027027027027027027027028p-12),
34 L(0x3.72a3c5631fe46ae1d4e700dca8f2p-12),
35 L(-0x7.daac36664f1f207daac36664f1f4p-12),
36 L(0x1.a41a41a41a41a41a41a41a41a41ap-8),
37 L(-0x7.90a1b2c3d4e5f708192a3b4c5d7p-8),
38 L(0x2.dfd2c703c0cfff430edfd2c703cp-4),
39 L(-0x1.6476701181f39edbdb9ce625987dp+0),
40 L(0xd.672219167002d3a7a9c886459cp+0),
41 L(-0x9.cd9292e6660d55b3f712eb9e07c8p+4),
42 L(0x8.911a740da740da740da740da741p+8),
43 L(-0x8.d0cc570e255bf59ff6eec24b49p+12),
44 };
45
46#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
47
48/* Return gamma (X), for positive X less than 1775, 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
52static _Float128
53gammal_positive (_Float128 x, int *exp2_adj)
54{
55 int local_signgam;
56 if (x < L(0.5))
57 {
58 *exp2_adj = 0;
59 return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x;
60 }
61 else if (x <= L(1.5))
62 {
63 *exp2_adj = 0;
64 return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam));
65 }
66 else if (x < L(12.5))
67 {
68 /* Adjust into the range for using exp (lgamma). */
69 *exp2_adj = 0;
70 _Float128 n = __ceill (x - L(1.5));
71 _Float128 x_adj = x - n;
72 _Float128 eps;
73 _Float128 prod = __gamma_productl (x_adj, 0, n, &eps);
74 return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam))
75 * prod * (1 + eps));
76 }
77 else
78 {
79 _Float128 eps = 0;
80 _Float128 x_eps = 0;
81 _Float128 x_adj = x;
82 _Float128 prod = 1;
83 if (x < 24)
84 {
85 /* Adjust into the range for applying Stirling's
86 approximation. */
87 _Float128 n = __ceill (24 - 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 _Float128 exp_adj = -eps;
97 _Float128 x_adj_int = __roundl (x_adj);
98 _Float128 x_adj_frac = x_adj - x_adj_int;
99 int x_adj_log2;
100 _Float128 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;
105 }
106 *exp2_adj = x_adj_log2 * (int) x_adj_int;
107 _Float128 ret = (__ieee754_powl (x_adj_mant, x_adj)
108 * __ieee754_exp2l (x_adj_log2 * x_adj_frac)
109 * __ieee754_expl (-x_adj)
110 * __ieee754_sqrtl (2 * M_PIl / x_adj)
111 / prod);
112 exp_adj += x_eps * __ieee754_logl (x_adj);
113 _Float128 bsum = gamma_coeff[NCOEFF - 1];
114 _Float128 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_Float128
123__ieee754_gammal_r (_Float128 x, int *signgamp)
124{
125 int64_t hx;
126 u_int64_t lx;
127 _Float128 ret;
128
129 GET_LDOUBLE_WORDS64 (hx, lx, x);
130
131 if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
132 {
133 /* Return value for x == 0 is Inf with divide by zero exception. */
134 *signgamp = 0;
135 return 1.0 / x;
136 }
137 if (hx < 0 && (u_int64_t) hx < 0xffff000000000000ULL && __rintl (x) == x)
138 {
139 /* Return value for integer x < 0 is NaN with invalid exception. */
140 *signgamp = 0;
141 return (x - x) / (x - x);
142 }
143 if (hx == 0xffff000000000000ULL && lx == 0)
144 {
145 /* x == -Inf. According to ISO this is NaN. */
146 *signgamp = 0;
147 return x - x;
148 }
149 if ((hx & 0x7fff000000000000ULL) == 0x7fff000000000000ULL)
150 {
151 /* Positive infinity (return positive infinity) or NaN (return
152 NaN). */
153 *signgamp = 0;
154 return x + x;
155 }
156
157 if (x >= 1756)
158 {
159 /* Overflow. */
160 *signgamp = 0;
161 return LDBL_MAX * LDBL_MAX;
162 }
163 else
164 {
165 SET_RESTORE_ROUNDL (FE_TONEAREST);
166 if (x > 0)
167 {
168 *signgamp = 0;
169 int exp2_adj;
170 ret = gammal_positive (x, &exp2_adj);
171 ret = __scalbnl (ret, exp2_adj);
172 }
173 else if (x >= -LDBL_EPSILON / 4)
174 {
175 *signgamp = 0;
176 ret = 1 / x;
177 }
178 else
179 {
180 _Float128 tx = __truncl (x);
181 *signgamp = (tx == 2 * __truncl (tx / 2)) ? -1 : 1;
182 if (x <= -1775)
183 /* Underflow. */
184 ret = LDBL_MIN * LDBL_MIN;
185 else
186 {
187 _Float128 frac = tx - x;
188 if (frac > L(0.5))
189 frac = 1 - frac;
190 _Float128 sinpix = (frac <= L(0.25)
191 ? __sinl (M_PIl * frac)
192 : __cosl (M_PIl * (L(0.5) - frac)));
193 int exp2_adj;
194 ret = M_PIl / (-x * sinpix
195 * gammal_positive (-x, &exp2_adj));
196 ret = __scalbnl (ret, -exp2_adj);
197 math_check_force_underflow_nonneg (ret);
198 }
199 }
200 }
201 if (isinf (ret) && x != 0)
202 {
203 if (*signgamp < 0)
204 return -(-__copysignl (LDBL_MAX, ret) * LDBL_MAX);
205 else
206 return __copysignl (LDBL_MAX, ret) * LDBL_MAX;
207 }
208 else if (ret == 0)
209 {
210 if (*signgamp < 0)
211 return -(-__copysignl (LDBL_MIN, ret) * LDBL_MIN);
212 else
213 return __copysignl (LDBL_MIN, ret) * LDBL_MIN;
214 }
215 else
216 return ret;
217}
218strong_alias (__ieee754_gammal_r, __gammal_r_finite)
219