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d705269e | 1 | /* Implementation of gamma function according to ISO C. |
688903eb | 2 | Copyright (C) 1997-2018 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 PE |
17 | License along with the GNU C Library; if not, see |
18 | <http://www.gnu.org/licenses/>. */ | |
c131718c | 19 | |
d705269e | 20 | #include <math.h> |
aaee3cd8 | 21 | #include <math-narrow-eval.h> |
d705269e | 22 | #include <math_private.h> |
8f5b00d3 | 23 | #include <math-underflow.h> |
d8cd06db | 24 | #include <float.h> |
d705269e | 25 | |
d8cd06db JM |
26 | /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's |
27 | approximation to gamma function. */ | |
28 | ||
29 | static const double gamma_coeff[] = | |
30 | { | |
31 | 0x1.5555555555555p-4, | |
32 | -0xb.60b60b60b60b8p-12, | |
33 | 0x3.4034034034034p-12, | |
34 | -0x2.7027027027028p-12, | |
35 | 0x3.72a3c5631fe46p-12, | |
36 | -0x7.daac36664f1f4p-12, | |
37 | }; | |
38 | ||
39 | #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0])) | |
40 | ||
41 | /* Return gamma (X), for positive X less than 184, in the form R * | |
42 | 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to | |
43 | avoid overflow or underflow in intermediate calculations. */ | |
44 | ||
45 | static double | |
46 | gamma_positive (double x, int *exp2_adj) | |
47 | { | |
48 | int local_signgam; | |
49 | if (x < 0.5) | |
50 | { | |
51 | *exp2_adj = 0; | |
52 | return __ieee754_exp (__ieee754_lgamma_r (x + 1, &local_signgam)) / x; | |
53 | } | |
54 | else if (x <= 1.5) | |
55 | { | |
56 | *exp2_adj = 0; | |
57 | return __ieee754_exp (__ieee754_lgamma_r (x, &local_signgam)); | |
58 | } | |
59 | else if (x < 6.5) | |
60 | { | |
61 | /* Adjust into the range for using exp (lgamma). */ | |
62 | *exp2_adj = 0; | |
63 | double n = __ceil (x - 1.5); | |
64 | double x_adj = x - n; | |
65 | double eps; | |
66 | double prod = __gamma_product (x_adj, 0, n, &eps); | |
67 | return (__ieee754_exp (__ieee754_lgamma_r (x_adj, &local_signgam)) | |
68 | * prod * (1.0 + eps)); | |
69 | } | |
70 | else | |
71 | { | |
72 | double eps = 0; | |
73 | double x_eps = 0; | |
74 | double x_adj = x; | |
75 | double prod = 1; | |
76 | if (x < 12.0) | |
77 | { | |
78 | /* Adjust into the range for applying Stirling's | |
79 | approximation. */ | |
80 | double n = __ceil (12.0 - x); | |
54142c44 | 81 | x_adj = math_narrow_eval (x + n); |
d8cd06db JM |
82 | x_eps = (x - (x_adj - n)); |
83 | prod = __gamma_product (x_adj - n, x_eps, n, &eps); | |
84 | } | |
85 | /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)). | |
86 | Compute gamma (X_ADJ + X_EPS) using Stirling's approximation, | |
87 | starting by computing pow (X_ADJ, X_ADJ) with a power of 2 | |
88 | factored out. */ | |
89 | double exp_adj = -eps; | |
90 | double x_adj_int = __round (x_adj); | |
91 | double x_adj_frac = x_adj - x_adj_int; | |
92 | int x_adj_log2; | |
93 | double x_adj_mant = __frexp (x_adj, &x_adj_log2); | |
94 | if (x_adj_mant < M_SQRT1_2) | |
95 | { | |
96 | x_adj_log2--; | |
97 | x_adj_mant *= 2.0; | |
98 | } | |
99 | *exp2_adj = x_adj_log2 * (int) x_adj_int; | |
100 | double ret = (__ieee754_pow (x_adj_mant, x_adj) | |
101 | * __ieee754_exp2 (x_adj_log2 * x_adj_frac) | |
102 | * __ieee754_exp (-x_adj) | |
f67a8147 | 103 | * sqrt (2 * M_PI / x_adj) |
d8cd06db | 104 | / prod); |
e02920bc | 105 | exp_adj += x_eps * __ieee754_log (x_adj); |
d8cd06db JM |
106 | double bsum = gamma_coeff[NCOEFF - 1]; |
107 | double x_adj2 = x_adj * x_adj; | |
108 | for (size_t i = 1; i <= NCOEFF - 1; i++) | |
109 | bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i]; | |
110 | exp_adj += bsum / x_adj; | |
111 | return ret + ret * __expm1 (exp_adj); | |
112 | } | |
113 | } | |
d705269e UD |
114 | |
115 | double | |
116 | __ieee754_gamma_r (double x, int *signgamp) | |
117 | { | |
d705269e | 118 | int32_t hx; |
24ab7723 | 119 | uint32_t lx; |
e02920bc | 120 | double ret; |
d705269e UD |
121 | |
122 | EXTRACT_WORDS (hx, lx, x); | |
123 | ||
a1ffb40e | 124 | if (__glibc_unlikely (((hx & 0x7fffffff) | lx) == 0)) |
b3fc5f84 | 125 | { |
52495f29 | 126 | /* Return value for x == 0 is Inf with divide by zero exception. */ |
b3fc5f84 | 127 | *signgamp = 0; |
52495f29 | 128 | return 1.0 / x; |
b3fc5f84 | 129 | } |
0ac5ae23 | 130 | if (__builtin_expect (hx < 0, 0) |
24ab7723 | 131 | && (uint32_t) hx < 0xfff00000 && __rint (x) == x) |
d705269e UD |
132 | { |
133 | /* Return value for integer x < 0 is NaN with invalid exception. */ | |
b3fc5f84 | 134 | *signgamp = 0; |
d705269e UD |
135 | return (x - x) / (x - x); |
136 | } | |
a1ffb40e | 137 | if (__glibc_unlikely ((unsigned int) hx == 0xfff00000 && lx == 0)) |
3bde1a69 UD |
138 | { |
139 | /* x == -Inf. According to ISO this is NaN. */ | |
140 | *signgamp = 0; | |
141 | return x - x; | |
142 | } | |
a1ffb40e | 143 | if (__glibc_unlikely ((hx & 0x7ff00000) == 0x7ff00000)) |
d8cd06db JM |
144 | { |
145 | /* Positive infinity (return positive infinity) or NaN (return | |
146 | NaN). */ | |
147 | *signgamp = 0; | |
148 | return x + x; | |
149 | } | |
d705269e | 150 | |
d8cd06db JM |
151 | if (x >= 172.0) |
152 | { | |
153 | /* Overflow. */ | |
154 | *signgamp = 0; | |
54142c44 | 155 | ret = math_narrow_eval (DBL_MAX * DBL_MAX); |
e02920bc | 156 | return ret; |
d8cd06db | 157 | } |
e02920bc | 158 | else |
d8cd06db | 159 | { |
e02920bc JM |
160 | SET_RESTORE_ROUND (FE_TONEAREST); |
161 | if (x > 0.0) | |
162 | { | |
163 | *signgamp = 0; | |
164 | int exp2_adj; | |
165 | double tret = gamma_positive (x, &exp2_adj); | |
166 | ret = __scalbn (tret, exp2_adj); | |
167 | } | |
168 | else if (x >= -DBL_EPSILON / 4.0) | |
169 | { | |
170 | *signgamp = 0; | |
171 | ret = 1.0 / x; | |
172 | } | |
173 | else | |
174 | { | |
175 | double tx = __trunc (x); | |
176 | *signgamp = (tx == 2.0 * __trunc (tx / 2.0)) ? -1 : 1; | |
177 | if (x <= -184.0) | |
178 | /* Underflow. */ | |
179 | ret = DBL_MIN * DBL_MIN; | |
180 | else | |
181 | { | |
182 | double frac = tx - x; | |
183 | if (frac > 0.5) | |
184 | frac = 1.0 - frac; | |
185 | double sinpix = (frac <= 0.25 | |
186 | ? __sin (M_PI * frac) | |
187 | : __cos (M_PI * (0.5 - frac))); | |
188 | int exp2_adj; | |
189 | double tret = M_PI / (-x * sinpix | |
190 | * gamma_positive (-x, &exp2_adj)); | |
191 | ret = __scalbn (tret, -exp2_adj); | |
d96164c3 | 192 | math_check_force_underflow_nonneg (ret); |
e02920bc JM |
193 | } |
194 | } | |
54142c44 | 195 | ret = math_narrow_eval (ret); |
d8cd06db | 196 | } |
e02920bc | 197 | if (isinf (ret) && x != 0) |
d8cd06db | 198 | { |
e02920bc JM |
199 | if (*signgamp < 0) |
200 | { | |
54142c44 | 201 | ret = math_narrow_eval (-__copysign (DBL_MAX, ret) * DBL_MAX); |
e02920bc JM |
202 | ret = -ret; |
203 | } | |
204 | else | |
54142c44 | 205 | ret = math_narrow_eval (__copysign (DBL_MAX, ret) * DBL_MAX); |
e02920bc | 206 | return ret; |
d8cd06db | 207 | } |
e02920bc | 208 | else if (ret == 0) |
d8cd06db | 209 | { |
e02920bc JM |
210 | if (*signgamp < 0) |
211 | { | |
54142c44 | 212 | ret = math_narrow_eval (-__copysign (DBL_MIN, ret) * DBL_MIN); |
e02920bc JM |
213 | ret = -ret; |
214 | } | |
215 | else | |
54142c44 | 216 | ret = math_narrow_eval (__copysign (DBL_MIN, ret) * DBL_MIN); |
e02920bc | 217 | return ret; |
d8cd06db | 218 | } |
e02920bc JM |
219 | else |
220 | return ret; | |
d705269e | 221 | } |
0ac5ae23 | 222 | strong_alias (__ieee754_gamma_r, __gamma_r_finite) |