1 | /* Implementation of gamma function according to ISO C. |
2 | Copyright (C) 1997-2022 Free Software Foundation, Inc. |
3 | This file is part of the GNU C Library. |
4 | |
5 | The GNU C Library is free software; you can redistribute it and/or |
6 | modify it under the terms of the GNU Lesser General Public |
7 | License as published by the Free Software Foundation; either |
8 | version 2.1 of the License, or (at your option) any later version. |
9 | |
10 | The GNU C Library is distributed in the hope that it will be useful, |
11 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
12 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
13 | Lesser General Public License for more details. |
14 | |
15 | You should have received a copy of the GNU Lesser General Public |
16 | License along with the GNU C Library; if not, see |
17 | <https://www.gnu.org/licenses/>. */ |
18 | |
19 | #include <math.h> |
20 | #include <math_private.h> |
21 | #include <fenv_private.h> |
22 | #include <math-underflow.h> |
23 | #include <float.h> |
24 | #include <libm-alias-finite.h> |
25 | |
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 long double gamma_coeff[] = |
30 | { |
31 | 0x1.555555555555555555555555558p-4L, |
32 | -0xb.60b60b60b60b60b60b60b60b6p-12L, |
33 | 0x3.4034034034034034034034034p-12L, |
34 | -0x2.7027027027027027027027027p-12L, |
35 | 0x3.72a3c5631fe46ae1d4e700dca9p-12L, |
36 | -0x7.daac36664f1f207daac36664f2p-12L, |
37 | 0x1.a41a41a41a41a41a41a41a41a4p-8L, |
38 | -0x7.90a1b2c3d4e5f708192a3b4c5ep-8L, |
39 | 0x2.dfd2c703c0cfff430edfd2c704p-4L, |
40 | -0x1.6476701181f39edbdb9ce625988p+0L, |
41 | 0xd.672219167002d3a7a9c886459cp+0L, |
42 | -0x9.cd9292e6660d55b3f712eb9e08p+4L, |
43 | 0x8.911a740da740da740da740da74p+8L, |
44 | }; |
45 | |
46 | #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0])) |
47 | |
48 | /* Return gamma (X), for positive X less than 191, 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 | |
52 | static long double |
53 | gammal_positive (long double x, int *exp2_adj) |
54 | { |
55 | int local_signgam; |
56 | if (x < 0.5L) |
57 | { |
58 | *exp2_adj = 0; |
59 | return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x; |
60 | } |
61 | else if (x <= 1.5L) |
62 | { |
63 | *exp2_adj = 0; |
64 | return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam)); |
65 | } |
66 | else if (x < 11.5L) |
67 | { |
68 | /* Adjust into the range for using exp (lgamma). */ |
69 | *exp2_adj = 0; |
70 | long double n = ceill (x - 1.5L); |
71 | long double x_adj = x - n; |
72 | long double eps; |
73 | long double prod = __gamma_productl (x: x_adj, x_eps: 0, n, eps: &eps); |
74 | return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam)) |
75 | * prod * (1.0L + eps)); |
76 | } |
77 | else |
78 | { |
79 | long double eps = 0; |
80 | long double x_eps = 0; |
81 | long double x_adj = x; |
82 | long double prod = 1; |
83 | if (x < 23.0L) |
84 | { |
85 | /* Adjust into the range for applying Stirling's |
86 | approximation. */ |
87 | long double n = ceill (23.0L - x); |
88 | x_adj = x + n; |
89 | x_eps = (x - (x_adj - n)); |
90 | prod = __gamma_productl (x: x_adj - n, x_eps, n, eps: &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 | long double exp_adj = -eps; |
97 | long double x_adj_int = roundl (x_adj); |
98 | long double x_adj_frac = x_adj - x_adj_int; |
99 | int x_adj_log2; |
100 | long double x_adj_mant = __frexpl (x: x_adj, exponent: &x_adj_log2); |
101 | if (x_adj_mant < M_SQRT1_2l) |
102 | { |
103 | x_adj_log2--; |
104 | x_adj_mant *= 2.0L; |
105 | } |
106 | *exp2_adj = x_adj_log2 * (int) x_adj_int; |
107 | long double ret = (__ieee754_powl (x_adj_mant, x_adj) |
108 | * __ieee754_exp2l (x_adj_log2 * x_adj_frac) |
109 | * __ieee754_expl (-x_adj) |
110 | * sqrtl (2 * M_PIl / x_adj) |
111 | / prod); |
112 | exp_adj += x_eps * __ieee754_logl (x_adj); |
113 | long double bsum = gamma_coeff[NCOEFF - 1]; |
114 | long double 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 (x: exp_adj); |
119 | } |
120 | } |
121 | |
122 | long double |
123 | __ieee754_gammal_r (long double x, int *signgamp) |
124 | { |
125 | int64_t hx; |
126 | double xhi; |
127 | long double ret; |
128 | |
129 | xhi = ldbl_high (x); |
130 | EXTRACT_WORDS64 (hx, xhi); |
131 | |
132 | if ((hx & 0x7fffffffffffffffLL) == 0) |
133 | { |
134 | /* Return value for x == 0 is Inf with divide by zero exception. */ |
135 | *signgamp = 0; |
136 | return 1.0 / x; |
137 | } |
138 | if (hx < 0 && (uint64_t) hx < 0xfff0000000000000ULL && rintl (x) == x) |
139 | { |
140 | /* Return value for integer x < 0 is NaN with invalid exception. */ |
141 | *signgamp = 0; |
142 | return (x - x) / (x - x); |
143 | } |
144 | if (hx == 0xfff0000000000000ULL) |
145 | { |
146 | /* x == -Inf. According to ISO this is NaN. */ |
147 | *signgamp = 0; |
148 | return x - x; |
149 | } |
150 | if ((hx & 0x7ff0000000000000ULL) == 0x7ff0000000000000ULL) |
151 | { |
152 | /* Positive infinity (return positive infinity) or NaN (return |
153 | NaN). */ |
154 | *signgamp = 0; |
155 | return x + x; |
156 | } |
157 | |
158 | if (x >= 172.0L) |
159 | { |
160 | /* Overflow. */ |
161 | *signgamp = 0; |
162 | return LDBL_MAX * LDBL_MAX; |
163 | } |
164 | else |
165 | { |
166 | SET_RESTORE_ROUNDL (FE_TONEAREST); |
167 | if (x > 0.0L) |
168 | { |
169 | *signgamp = 0; |
170 | int exp2_adj; |
171 | ret = gammal_positive (x, exp2_adj: &exp2_adj); |
172 | ret = __scalbnl (x: ret, n: exp2_adj); |
173 | } |
174 | else if (x >= -0x1p-110L) |
175 | { |
176 | *signgamp = 0; |
177 | ret = 1.0L / x; |
178 | } |
179 | else |
180 | { |
181 | long double tx = truncl (x); |
182 | *signgamp = (tx == 2.0L * truncl (tx / 2.0L)) ? -1 : 1; |
183 | if (x <= -191.0L) |
184 | /* Underflow. */ |
185 | ret = LDBL_MIN * LDBL_MIN; |
186 | else |
187 | { |
188 | long double frac = tx - x; |
189 | if (frac > 0.5L) |
190 | frac = 1.0L - frac; |
191 | long double sinpix = (frac <= 0.25L |
192 | ? __sinl (M_PIl * frac) |
193 | : __cosl (M_PIl * (0.5L - frac))); |
194 | int exp2_adj; |
195 | ret = M_PIl / (-x * sinpix |
196 | * gammal_positive (x: -x, exp2_adj: &exp2_adj)); |
197 | ret = __scalbnl (x: ret, n: -exp2_adj); |
198 | math_check_force_underflow_nonneg (ret); |
199 | } |
200 | } |
201 | } |
202 | if (isinf (ret) && x != 0) |
203 | { |
204 | if (*signgamp < 0) |
205 | return -(-copysignl (LDBL_MAX, ret) * LDBL_MAX); |
206 | else |
207 | return copysignl (LDBL_MAX, ret) * LDBL_MAX; |
208 | } |
209 | else if (ret == 0) |
210 | { |
211 | if (*signgamp < 0) |
212 | return -(-copysignl (LDBL_MIN, ret) * LDBL_MIN); |
213 | else |
214 | return copysignl (LDBL_MIN, ret) * LDBL_MIN; |
215 | } |
216 | else |
217 | return ret; |
218 | } |
219 | libm_alias_finite (__ieee754_gammal_r, __gammal_r) |
220 | |