1 | /* Implementation of gamma function according to ISO C. |
2 | Copyright (C) 1997-2018 Free Software Foundation, Inc. |
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 |
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. |
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 |
14 | Lesser General Public License for more details. |
15 | |
16 | You should have received a copy of the GNU Lesser General Public |
17 | License along with the GNU C Library; if not, see |
18 | <http://www.gnu.org/licenses/>. */ |
19 | |
20 | #include <math.h> |
21 | #include <math_private.h> |
22 | #include <float.h> |
23 | |
24 | /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's |
25 | approximation to gamma function. */ |
26 | |
27 | static const double gamma_coeff[] = |
28 | { |
29 | 0x1.5555555555555p-4, |
30 | -0xb.60b60b60b60b8p-12, |
31 | 0x3.4034034034034p-12, |
32 | -0x2.7027027027028p-12, |
33 | 0x3.72a3c5631fe46p-12, |
34 | -0x7.daac36664f1f4p-12, |
35 | }; |
36 | |
37 | #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0])) |
38 | |
39 | /* Return gamma (X), for positive X less than 184, in the form R * |
40 | 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to |
41 | avoid overflow or underflow in intermediate calculations. */ |
42 | |
43 | static double |
44 | gamma_positive (double x, int *exp2_adj) |
45 | { |
46 | int local_signgam; |
47 | if (x < 0.5) |
48 | { |
49 | *exp2_adj = 0; |
50 | return __ieee754_exp (__ieee754_lgamma_r (x + 1, &local_signgam)) / x; |
51 | } |
52 | else if (x <= 1.5) |
53 | { |
54 | *exp2_adj = 0; |
55 | return __ieee754_exp (__ieee754_lgamma_r (x, &local_signgam)); |
56 | } |
57 | else if (x < 6.5) |
58 | { |
59 | /* Adjust into the range for using exp (lgamma). */ |
60 | *exp2_adj = 0; |
61 | double n = __ceil (x - 1.5); |
62 | double x_adj = x - n; |
63 | double eps; |
64 | double prod = __gamma_product (x_adj, 0, n, &eps); |
65 | return (__ieee754_exp (__ieee754_lgamma_r (x_adj, &local_signgam)) |
66 | * prod * (1.0 + eps)); |
67 | } |
68 | else |
69 | { |
70 | double eps = 0; |
71 | double x_eps = 0; |
72 | double x_adj = x; |
73 | double prod = 1; |
74 | if (x < 12.0) |
75 | { |
76 | /* Adjust into the range for applying Stirling's |
77 | approximation. */ |
78 | double n = __ceil (12.0 - x); |
79 | x_adj = math_narrow_eval (x + n); |
80 | x_eps = (x - (x_adj - n)); |
81 | prod = __gamma_product (x_adj - n, x_eps, n, &eps); |
82 | } |
83 | /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)). |
84 | Compute gamma (X_ADJ + X_EPS) using Stirling's approximation, |
85 | starting by computing pow (X_ADJ, X_ADJ) with a power of 2 |
86 | factored out. */ |
87 | double exp_adj = -eps; |
88 | double x_adj_int = __round (x_adj); |
89 | double x_adj_frac = x_adj - x_adj_int; |
90 | int x_adj_log2; |
91 | double x_adj_mant = __frexp (x_adj, &x_adj_log2); |
92 | if (x_adj_mant < M_SQRT1_2) |
93 | { |
94 | x_adj_log2--; |
95 | x_adj_mant *= 2.0; |
96 | } |
97 | *exp2_adj = x_adj_log2 * (int) x_adj_int; |
98 | double ret = (__ieee754_pow (x_adj_mant, x_adj) |
99 | * __ieee754_exp2 (x_adj_log2 * x_adj_frac) |
100 | * __ieee754_exp (-x_adj) |
101 | * __ieee754_sqrt (2 * M_PI / x_adj) |
102 | / prod); |
103 | exp_adj += x_eps * __ieee754_log (x_adj); |
104 | double bsum = gamma_coeff[NCOEFF - 1]; |
105 | double x_adj2 = x_adj * x_adj; |
106 | for (size_t i = 1; i <= NCOEFF - 1; i++) |
107 | bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i]; |
108 | exp_adj += bsum / x_adj; |
109 | return ret + ret * __expm1 (exp_adj); |
110 | } |
111 | } |
112 | |
113 | double |
114 | __ieee754_gamma_r (double x, int *signgamp) |
115 | { |
116 | int32_t hx; |
117 | uint32_t lx; |
118 | double ret; |
119 | |
120 | EXTRACT_WORDS (hx, lx, x); |
121 | |
122 | if (__glibc_unlikely (((hx & 0x7fffffff) | lx) == 0)) |
123 | { |
124 | /* Return value for x == 0 is Inf with divide by zero exception. */ |
125 | *signgamp = 0; |
126 | return 1.0 / x; |
127 | } |
128 | if (__builtin_expect (hx < 0, 0) |
129 | && (uint32_t) hx < 0xfff00000 && __rint (x) == x) |
130 | { |
131 | /* Return value for integer x < 0 is NaN with invalid exception. */ |
132 | *signgamp = 0; |
133 | return (x - x) / (x - x); |
134 | } |
135 | if (__glibc_unlikely ((unsigned int) hx == 0xfff00000 && lx == 0)) |
136 | { |
137 | /* x == -Inf. According to ISO this is NaN. */ |
138 | *signgamp = 0; |
139 | return x - x; |
140 | } |
141 | if (__glibc_unlikely ((hx & 0x7ff00000) == 0x7ff00000)) |
142 | { |
143 | /* Positive infinity (return positive infinity) or NaN (return |
144 | NaN). */ |
145 | *signgamp = 0; |
146 | return x + x; |
147 | } |
148 | |
149 | if (x >= 172.0) |
150 | { |
151 | /* Overflow. */ |
152 | *signgamp = 0; |
153 | ret = math_narrow_eval (DBL_MAX * DBL_MAX); |
154 | return ret; |
155 | } |
156 | else |
157 | { |
158 | SET_RESTORE_ROUND (FE_TONEAREST); |
159 | if (x > 0.0) |
160 | { |
161 | *signgamp = 0; |
162 | int exp2_adj; |
163 | double tret = gamma_positive (x, &exp2_adj); |
164 | ret = __scalbn (tret, exp2_adj); |
165 | } |
166 | else if (x >= -DBL_EPSILON / 4.0) |
167 | { |
168 | *signgamp = 0; |
169 | ret = 1.0 / x; |
170 | } |
171 | else |
172 | { |
173 | double tx = __trunc (x); |
174 | *signgamp = (tx == 2.0 * __trunc (tx / 2.0)) ? -1 : 1; |
175 | if (x <= -184.0) |
176 | /* Underflow. */ |
177 | ret = DBL_MIN * DBL_MIN; |
178 | else |
179 | { |
180 | double frac = tx - x; |
181 | if (frac > 0.5) |
182 | frac = 1.0 - frac; |
183 | double sinpix = (frac <= 0.25 |
184 | ? __sin (M_PI * frac) |
185 | : __cos (M_PI * (0.5 - frac))); |
186 | int exp2_adj; |
187 | double tret = M_PI / (-x * sinpix |
188 | * gamma_positive (-x, &exp2_adj)); |
189 | ret = __scalbn (tret, -exp2_adj); |
190 | math_check_force_underflow_nonneg (ret); |
191 | } |
192 | } |
193 | ret = math_narrow_eval (ret); |
194 | } |
195 | if (isinf (ret) && x != 0) |
196 | { |
197 | if (*signgamp < 0) |
198 | { |
199 | ret = math_narrow_eval (-__copysign (DBL_MAX, ret) * DBL_MAX); |
200 | ret = -ret; |
201 | } |
202 | else |
203 | ret = math_narrow_eval (__copysign (DBL_MAX, ret) * DBL_MAX); |
204 | return ret; |
205 | } |
206 | else if (ret == 0) |
207 | { |
208 | if (*signgamp < 0) |
209 | { |
210 | ret = math_narrow_eval (-__copysign (DBL_MIN, ret) * DBL_MIN); |
211 | ret = -ret; |
212 | } |
213 | else |
214 | ret = math_narrow_eval (__copysign (DBL_MIN, ret) * DBL_MIN); |
215 | return ret; |
216 | } |
217 | else |
218 | return ret; |
219 | } |
220 | strong_alias (__ieee754_gamma_r, __gamma_r_finite) |
221 | |