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f964490f | 1 | /* Implementation of gamma function according to ISO C. |
d4697bc9 | 2 | Copyright (C) 1997-2014 Free Software Foundation, Inc. |
f964490f RM |
3 | This file is part of the GNU C Library. |
4 | Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and | |
0ac5ae23 | 5 | Jakub Jelinek <jj@ultra.linux.cz, 1999. |
f964490f RM |
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 | |
59ba27a6 PE |
18 | License along with the GNU C Library; if not, see |
19 | <http://www.gnu.org/licenses/>. */ | |
f964490f RM |
20 | |
21 | #include <math.h> | |
22 | #include <math_private.h> | |
d8cd06db | 23 | #include <float.h> |
f964490f | 24 | |
d8cd06db JM |
25 | /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's |
26 | approximation to gamma function. */ | |
27 | ||
28 | static const long double gamma_coeff[] = | |
29 | { | |
30 | 0x1.555555555555555555555555558p-4L, | |
31 | -0xb.60b60b60b60b60b60b60b60b6p-12L, | |
32 | 0x3.4034034034034034034034034p-12L, | |
33 | -0x2.7027027027027027027027027p-12L, | |
34 | 0x3.72a3c5631fe46ae1d4e700dca9p-12L, | |
35 | -0x7.daac36664f1f207daac36664f2p-12L, | |
36 | 0x1.a41a41a41a41a41a41a41a41a4p-8L, | |
37 | -0x7.90a1b2c3d4e5f708192a3b4c5ep-8L, | |
38 | 0x2.dfd2c703c0cfff430edfd2c704p-4L, | |
39 | -0x1.6476701181f39edbdb9ce625988p+0L, | |
40 | 0xd.672219167002d3a7a9c886459cp+0L, | |
41 | -0x9.cd9292e6660d55b3f712eb9e08p+4L, | |
42 | 0x8.911a740da740da740da740da74p+8L, | |
43 | }; | |
44 | ||
45 | #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0])) | |
46 | ||
47 | /* Return gamma (X), for positive X less than 191, in the form R * | |
48 | 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to | |
49 | avoid overflow or underflow in intermediate calculations. */ | |
50 | ||
51 | static long double | |
52 | gammal_positive (long double x, int *exp2_adj) | |
53 | { | |
54 | int local_signgam; | |
55 | if (x < 0.5L) | |
56 | { | |
57 | *exp2_adj = 0; | |
58 | return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x; | |
59 | } | |
60 | else if (x <= 1.5L) | |
61 | { | |
62 | *exp2_adj = 0; | |
63 | return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam)); | |
64 | } | |
65 | else if (x < 11.5L) | |
66 | { | |
67 | /* Adjust into the range for using exp (lgamma). */ | |
68 | *exp2_adj = 0; | |
69 | long double n = __ceill (x - 1.5L); | |
70 | long double x_adj = x - n; | |
71 | long double eps; | |
72 | long double prod = __gamma_productl (x_adj, 0, n, &eps); | |
73 | return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam)) | |
74 | * prod * (1.0L + eps)); | |
75 | } | |
76 | else | |
77 | { | |
78 | long double eps = 0; | |
79 | long double x_eps = 0; | |
80 | long double x_adj = x; | |
81 | long double prod = 1; | |
82 | if (x < 23.0L) | |
83 | { | |
84 | /* Adjust into the range for applying Stirling's | |
85 | approximation. */ | |
86 | long double n = __ceill (23.0L - x); | |
87 | x_adj = x + n; | |
88 | x_eps = (x - (x_adj - n)); | |
89 | prod = __gamma_productl (x_adj - n, x_eps, n, &eps); | |
90 | } | |
91 | /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)). | |
92 | Compute gamma (X_ADJ + X_EPS) using Stirling's approximation, | |
93 | starting by computing pow (X_ADJ, X_ADJ) with a power of 2 | |
94 | factored out. */ | |
95 | long double exp_adj = -eps; | |
96 | long double x_adj_int = __roundl (x_adj); | |
97 | long double x_adj_frac = x_adj - x_adj_int; | |
98 | int x_adj_log2; | |
99 | long double x_adj_mant = __frexpl (x_adj, &x_adj_log2); | |
100 | if (x_adj_mant < M_SQRT1_2l) | |
101 | { | |
102 | x_adj_log2--; | |
103 | x_adj_mant *= 2.0L; | |
104 | } | |
105 | *exp2_adj = x_adj_log2 * (int) x_adj_int; | |
106 | long double ret = (__ieee754_powl (x_adj_mant, x_adj) | |
107 | * __ieee754_exp2l (x_adj_log2 * x_adj_frac) | |
108 | * __ieee754_expl (-x_adj) | |
109 | * __ieee754_sqrtl (2 * M_PIl / x_adj) | |
110 | / prod); | |
111 | exp_adj += x_eps * __ieee754_logl (x); | |
112 | long double bsum = gamma_coeff[NCOEFF - 1]; | |
113 | long double x_adj2 = x_adj * x_adj; | |
114 | for (size_t i = 1; i <= NCOEFF - 1; i++) | |
115 | bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i]; | |
116 | exp_adj += bsum / x_adj; | |
117 | return ret + ret * __expm1l (exp_adj); | |
118 | } | |
119 | } | |
f964490f RM |
120 | |
121 | long double | |
122 | __ieee754_gammal_r (long double x, int *signgamp) | |
123 | { | |
f964490f | 124 | int64_t hx; |
765714ca | 125 | double xhi; |
f964490f | 126 | |
765714ca AM |
127 | xhi = ldbl_high (x); |
128 | EXTRACT_WORDS64 (hx, xhi); | |
f964490f | 129 | |
765714ca | 130 | if ((hx & 0x7fffffffffffffffLL) == 0) |
f964490f RM |
131 | { |
132 | /* Return value for x == 0 is Inf with divide by zero exception. */ | |
133 | *signgamp = 0; | |
134 | return 1.0 / x; | |
135 | } | |
136 | if (hx < 0 && (u_int64_t) hx < 0xfff0000000000000ULL && __rintl (x) == x) | |
137 | { | |
138 | /* Return value for integer x < 0 is NaN with invalid exception. */ | |
139 | *signgamp = 0; | |
140 | return (x - x) / (x - x); | |
141 | } | |
142 | if (hx == 0xfff0000000000000ULL) | |
143 | { | |
144 | /* x == -Inf. According to ISO this is NaN. */ | |
145 | *signgamp = 0; | |
146 | return x - x; | |
147 | } | |
d8cd06db JM |
148 | if ((hx & 0x7ff0000000000000ULL) == 0x7ff0000000000000ULL) |
149 | { | |
150 | /* Positive infinity (return positive infinity) or NaN (return | |
151 | NaN). */ | |
152 | *signgamp = 0; | |
153 | return x + x; | |
154 | } | |
f964490f | 155 | |
d8cd06db JM |
156 | if (x >= 172.0L) |
157 | { | |
158 | /* Overflow. */ | |
159 | *signgamp = 0; | |
160 | return LDBL_MAX * LDBL_MAX; | |
161 | } | |
162 | else if (x > 0.0L) | |
163 | { | |
164 | *signgamp = 0; | |
165 | int exp2_adj; | |
166 | long double ret = gammal_positive (x, &exp2_adj); | |
167 | return __scalbnl (ret, exp2_adj); | |
168 | } | |
169 | else if (x >= -0x1p-110L) | |
170 | { | |
171 | *signgamp = 0; | |
172 | return 1.0f / x; | |
173 | } | |
174 | else | |
175 | { | |
176 | long double tx = __truncl (x); | |
177 | *signgamp = (tx == 2.0L * __truncl (tx / 2.0L)) ? -1 : 1; | |
178 | if (x <= -191.0L) | |
179 | /* Underflow. */ | |
180 | return LDBL_MIN * LDBL_MIN; | |
181 | long double frac = tx - x; | |
182 | if (frac > 0.5L) | |
183 | frac = 1.0L - frac; | |
184 | long double sinpix = (frac <= 0.25L | |
185 | ? __sinl (M_PIl * frac) | |
186 | : __cosl (M_PIl * (0.5L - frac))); | |
187 | int exp2_adj; | |
188 | long double ret = M_PIl / (-x * sinpix | |
189 | * gammal_positive (-x, &exp2_adj)); | |
190 | return __scalbnl (ret, -exp2_adj); | |
191 | } | |
f964490f | 192 | } |
0ac5ae23 | 193 | strong_alias (__ieee754_gammal_r, __gammal_r_finite) |