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d705269e | 1 | /* Implementation of gamma function according to ISO C. |
2b778ceb | 2 | Copyright (C) 1997-2021 Free Software Foundation, Inc. |
c131718c UD |
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 | |
41bdb6e2 AJ |
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. | |
c131718c UD |
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 | |
41bdb6e2 | 14 | Lesser General Public License for more details. |
c131718c | 15 | |
41bdb6e2 | 16 | You should have received a copy of the GNU Lesser General Public |
59ba27a6 | 17 | License along with the GNU C Library; if not, see |
5a82c748 | 18 | <https://www.gnu.org/licenses/>. */ |
c131718c | 19 | |
d705269e | 20 | #include <math.h> |
aaee3cd8 | 21 | #include <math-narrow-eval.h> |
d705269e | 22 | #include <math_private.h> |
70e2ba33 | 23 | #include <fenv_private.h> |
8f5b00d3 | 24 | #include <math-underflow.h> |
d8cd06db | 25 | #include <float.h> |
220622dd | 26 | #include <libm-alias-finite.h> |
d705269e | 27 | |
d8cd06db JM |
28 | /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's |
29 | approximation to gamma function. */ | |
30 | ||
31 | static const double gamma_coeff[] = | |
32 | { | |
33 | 0x1.5555555555555p-4, | |
34 | -0xb.60b60b60b60b8p-12, | |
35 | 0x3.4034034034034p-12, | |
36 | -0x2.7027027027028p-12, | |
37 | 0x3.72a3c5631fe46p-12, | |
38 | -0x7.daac36664f1f4p-12, | |
39 | }; | |
40 | ||
41 | #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0])) | |
42 | ||
43 | /* Return gamma (X), for positive X less than 184, in the form R * | |
44 | 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to | |
45 | avoid overflow or underflow in intermediate calculations. */ | |
46 | ||
47 | static double | |
48 | gamma_positive (double x, int *exp2_adj) | |
49 | { | |
50 | int local_signgam; | |
51 | if (x < 0.5) | |
52 | { | |
53 | *exp2_adj = 0; | |
54 | return __ieee754_exp (__ieee754_lgamma_r (x + 1, &local_signgam)) / x; | |
55 | } | |
56 | else if (x <= 1.5) | |
57 | { | |
58 | *exp2_adj = 0; | |
59 | return __ieee754_exp (__ieee754_lgamma_r (x, &local_signgam)); | |
60 | } | |
61 | else if (x < 6.5) | |
62 | { | |
63 | /* Adjust into the range for using exp (lgamma). */ | |
64 | *exp2_adj = 0; | |
71223ef9 | 65 | double n = ceil (x - 1.5); |
d8cd06db JM |
66 | double x_adj = x - n; |
67 | double eps; | |
68 | double prod = __gamma_product (x_adj, 0, n, &eps); | |
69 | return (__ieee754_exp (__ieee754_lgamma_r (x_adj, &local_signgam)) | |
70 | * prod * (1.0 + eps)); | |
71 | } | |
72 | else | |
73 | { | |
74 | double eps = 0; | |
75 | double x_eps = 0; | |
76 | double x_adj = x; | |
77 | double prod = 1; | |
78 | if (x < 12.0) | |
79 | { | |
80 | /* Adjust into the range for applying Stirling's | |
81 | approximation. */ | |
71223ef9 | 82 | double n = ceil (12.0 - x); |
54142c44 | 83 | x_adj = math_narrow_eval (x + n); |
d8cd06db JM |
84 | x_eps = (x - (x_adj - n)); |
85 | prod = __gamma_product (x_adj - n, x_eps, n, &eps); | |
86 | } | |
87 | /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)). | |
88 | Compute gamma (X_ADJ + X_EPS) using Stirling's approximation, | |
89 | starting by computing pow (X_ADJ, X_ADJ) with a power of 2 | |
90 | factored out. */ | |
91 | double exp_adj = -eps; | |
9755bc46 | 92 | double x_adj_int = round (x_adj); |
d8cd06db JM |
93 | double x_adj_frac = x_adj - x_adj_int; |
94 | int x_adj_log2; | |
95 | double x_adj_mant = __frexp (x_adj, &x_adj_log2); | |
96 | if (x_adj_mant < M_SQRT1_2) | |
97 | { | |
98 | x_adj_log2--; | |
99 | x_adj_mant *= 2.0; | |
100 | } | |
101 | *exp2_adj = x_adj_log2 * (int) x_adj_int; | |
102 | double ret = (__ieee754_pow (x_adj_mant, x_adj) | |
103 | * __ieee754_exp2 (x_adj_log2 * x_adj_frac) | |
104 | * __ieee754_exp (-x_adj) | |
f67a8147 | 105 | * sqrt (2 * M_PI / x_adj) |
d8cd06db | 106 | / prod); |
e02920bc | 107 | exp_adj += x_eps * __ieee754_log (x_adj); |
d8cd06db JM |
108 | double bsum = gamma_coeff[NCOEFF - 1]; |
109 | double x_adj2 = x_adj * x_adj; | |
110 | for (size_t i = 1; i <= NCOEFF - 1; i++) | |
111 | bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i]; | |
112 | exp_adj += bsum / x_adj; | |
113 | return ret + ret * __expm1 (exp_adj); | |
114 | } | |
115 | } | |
d705269e UD |
116 | |
117 | double | |
118 | __ieee754_gamma_r (double x, int *signgamp) | |
119 | { | |
d705269e | 120 | int32_t hx; |
24ab7723 | 121 | uint32_t lx; |
e02920bc | 122 | double ret; |
d705269e UD |
123 | |
124 | EXTRACT_WORDS (hx, lx, x); | |
125 | ||
a1ffb40e | 126 | if (__glibc_unlikely (((hx & 0x7fffffff) | lx) == 0)) |
b3fc5f84 | 127 | { |
52495f29 | 128 | /* Return value for x == 0 is Inf with divide by zero exception. */ |
b3fc5f84 | 129 | *signgamp = 0; |
52495f29 | 130 | return 1.0 / x; |
b3fc5f84 | 131 | } |
0ac5ae23 | 132 | if (__builtin_expect (hx < 0, 0) |
f29b6f17 | 133 | && (uint32_t) hx < 0xfff00000 && rint (x) == x) |
d705269e UD |
134 | { |
135 | /* Return value for integer x < 0 is NaN with invalid exception. */ | |
b3fc5f84 | 136 | *signgamp = 0; |
d705269e UD |
137 | return (x - x) / (x - x); |
138 | } | |
a1ffb40e | 139 | if (__glibc_unlikely ((unsigned int) hx == 0xfff00000 && lx == 0)) |
3bde1a69 UD |
140 | { |
141 | /* x == -Inf. According to ISO this is NaN. */ | |
142 | *signgamp = 0; | |
143 | return x - x; | |
144 | } | |
a1ffb40e | 145 | if (__glibc_unlikely ((hx & 0x7ff00000) == 0x7ff00000)) |
d8cd06db JM |
146 | { |
147 | /* Positive infinity (return positive infinity) or NaN (return | |
148 | NaN). */ | |
149 | *signgamp = 0; | |
150 | return x + x; | |
151 | } | |
d705269e | 152 | |
d8cd06db JM |
153 | if (x >= 172.0) |
154 | { | |
155 | /* Overflow. */ | |
156 | *signgamp = 0; | |
54142c44 | 157 | ret = math_narrow_eval (DBL_MAX * DBL_MAX); |
e02920bc | 158 | return ret; |
d8cd06db | 159 | } |
e02920bc | 160 | else |
d8cd06db | 161 | { |
e02920bc JM |
162 | SET_RESTORE_ROUND (FE_TONEAREST); |
163 | if (x > 0.0) | |
164 | { | |
165 | *signgamp = 0; | |
166 | int exp2_adj; | |
167 | double tret = gamma_positive (x, &exp2_adj); | |
168 | ret = __scalbn (tret, exp2_adj); | |
169 | } | |
170 | else if (x >= -DBL_EPSILON / 4.0) | |
171 | { | |
172 | *signgamp = 0; | |
173 | ret = 1.0 / x; | |
174 | } | |
175 | else | |
176 | { | |
7abf97be JM |
177 | double tx = trunc (x); |
178 | *signgamp = (tx == 2.0 * trunc (tx / 2.0)) ? -1 : 1; | |
e02920bc JM |
179 | if (x <= -184.0) |
180 | /* Underflow. */ | |
181 | ret = DBL_MIN * DBL_MIN; | |
182 | else | |
183 | { | |
184 | double frac = tx - x; | |
185 | if (frac > 0.5) | |
186 | frac = 1.0 - frac; | |
187 | double sinpix = (frac <= 0.25 | |
188 | ? __sin (M_PI * frac) | |
189 | : __cos (M_PI * (0.5 - frac))); | |
190 | int exp2_adj; | |
191 | double tret = M_PI / (-x * sinpix | |
192 | * gamma_positive (-x, &exp2_adj)); | |
193 | ret = __scalbn (tret, -exp2_adj); | |
d96164c3 | 194 | math_check_force_underflow_nonneg (ret); |
e02920bc JM |
195 | } |
196 | } | |
54142c44 | 197 | ret = math_narrow_eval (ret); |
d8cd06db | 198 | } |
e02920bc | 199 | if (isinf (ret) && x != 0) |
d8cd06db | 200 | { |
e02920bc JM |
201 | if (*signgamp < 0) |
202 | { | |
81dca813 | 203 | ret = math_narrow_eval (-copysign (DBL_MAX, ret) * DBL_MAX); |
e02920bc JM |
204 | ret = -ret; |
205 | } | |
206 | else | |
81dca813 | 207 | ret = math_narrow_eval (copysign (DBL_MAX, ret) * DBL_MAX); |
e02920bc | 208 | return ret; |
d8cd06db | 209 | } |
e02920bc | 210 | else if (ret == 0) |
d8cd06db | 211 | { |
e02920bc JM |
212 | if (*signgamp < 0) |
213 | { | |
81dca813 | 214 | ret = math_narrow_eval (-copysign (DBL_MIN, ret) * DBL_MIN); |
e02920bc JM |
215 | ret = -ret; |
216 | } | |
217 | else | |
81dca813 | 218 | ret = math_narrow_eval (copysign (DBL_MIN, ret) * DBL_MIN); |
e02920bc | 219 | return ret; |
d8cd06db | 220 | } |
e02920bc JM |
221 | else |
222 | return ret; | |
d705269e | 223 | } |
220622dd | 224 | libm_alias_finite (__ieee754_gamma_r, __gamma_r) |