1 | /* Implementation of gamma function according to ISO C. |
2 | Copyright (C) 1997-2019 Free Software Foundation, Inc. |
3 | This file is part of the GNU C Library. |
4 | Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and |
5 | Jakub Jelinek <jj@ultra.linux.cz, 1999. |
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 |
18 | License along with the GNU C Library; if not, see |
19 | <http://www.gnu.org/licenses/>. */ |
20 | |
21 | #include <math.h> |
22 | #include <math_private.h> |
23 | #include <fenv_private.h> |
24 | #include <math-underflow.h> |
25 | #include <float.h> |
26 | |
27 | /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's |
28 | approximation to gamma function. */ |
29 | |
30 | static const _Float128 gamma_coeff[] = |
31 | { |
32 | L(0x1.5555555555555555555555555555p-4), |
33 | L(-0xb.60b60b60b60b60b60b60b60b60b8p-12), |
34 | L(0x3.4034034034034034034034034034p-12), |
35 | L(-0x2.7027027027027027027027027028p-12), |
36 | L(0x3.72a3c5631fe46ae1d4e700dca8f2p-12), |
37 | L(-0x7.daac36664f1f207daac36664f1f4p-12), |
38 | L(0x1.a41a41a41a41a41a41a41a41a41ap-8), |
39 | L(-0x7.90a1b2c3d4e5f708192a3b4c5d7p-8), |
40 | L(0x2.dfd2c703c0cfff430edfd2c703cp-4), |
41 | L(-0x1.6476701181f39edbdb9ce625987dp+0), |
42 | L(0xd.672219167002d3a7a9c886459cp+0), |
43 | L(-0x9.cd9292e6660d55b3f712eb9e07c8p+4), |
44 | L(0x8.911a740da740da740da740da741p+8), |
45 | L(-0x8.d0cc570e255bf59ff6eec24b49p+12), |
46 | }; |
47 | |
48 | #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0])) |
49 | |
50 | /* Return gamma (X), for positive X less than 1775, in the form R * |
51 | 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to |
52 | avoid overflow or underflow in intermediate calculations. */ |
53 | |
54 | static _Float128 |
55 | gammal_positive (_Float128 x, int *exp2_adj) |
56 | { |
57 | int local_signgam; |
58 | if (x < L(0.5)) |
59 | { |
60 | *exp2_adj = 0; |
61 | return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x; |
62 | } |
63 | else if (x <= L(1.5)) |
64 | { |
65 | *exp2_adj = 0; |
66 | return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam)); |
67 | } |
68 | else if (x < L(12.5)) |
69 | { |
70 | /* Adjust into the range for using exp (lgamma). */ |
71 | *exp2_adj = 0; |
72 | _Float128 n = ceill (x - L(1.5)); |
73 | _Float128 x_adj = x - n; |
74 | _Float128 eps; |
75 | _Float128 prod = __gamma_productl (x_adj, 0, n, &eps); |
76 | return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam)) |
77 | * prod * (1 + eps)); |
78 | } |
79 | else |
80 | { |
81 | _Float128 eps = 0; |
82 | _Float128 x_eps = 0; |
83 | _Float128 x_adj = x; |
84 | _Float128 prod = 1; |
85 | if (x < 24) |
86 | { |
87 | /* Adjust into the range for applying Stirling's |
88 | approximation. */ |
89 | _Float128 n = ceill (24 - x); |
90 | x_adj = x + n; |
91 | x_eps = (x - (x_adj - n)); |
92 | prod = __gamma_productl (x_adj - n, x_eps, n, &eps); |
93 | } |
94 | /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)). |
95 | Compute gamma (X_ADJ + X_EPS) using Stirling's approximation, |
96 | starting by computing pow (X_ADJ, X_ADJ) with a power of 2 |
97 | factored out. */ |
98 | _Float128 exp_adj = -eps; |
99 | _Float128 x_adj_int = roundl (x_adj); |
100 | _Float128 x_adj_frac = x_adj - x_adj_int; |
101 | int x_adj_log2; |
102 | _Float128 x_adj_mant = __frexpl (x_adj, &x_adj_log2); |
103 | if (x_adj_mant < M_SQRT1_2l) |
104 | { |
105 | x_adj_log2--; |
106 | x_adj_mant *= 2; |
107 | } |
108 | *exp2_adj = x_adj_log2 * (int) x_adj_int; |
109 | _Float128 ret = (__ieee754_powl (x_adj_mant, x_adj) |
110 | * __ieee754_exp2l (x_adj_log2 * x_adj_frac) |
111 | * __ieee754_expl (-x_adj) |
112 | * sqrtl (2 * M_PIl / x_adj) |
113 | / prod); |
114 | exp_adj += x_eps * __ieee754_logl (x_adj); |
115 | _Float128 bsum = gamma_coeff[NCOEFF - 1]; |
116 | _Float128 x_adj2 = x_adj * x_adj; |
117 | for (size_t i = 1; i <= NCOEFF - 1; i++) |
118 | bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i]; |
119 | exp_adj += bsum / x_adj; |
120 | return ret + ret * __expm1l (exp_adj); |
121 | } |
122 | } |
123 | |
124 | _Float128 |
125 | __ieee754_gammal_r (_Float128 x, int *signgamp) |
126 | { |
127 | int64_t hx; |
128 | uint64_t lx; |
129 | _Float128 ret; |
130 | |
131 | GET_LDOUBLE_WORDS64 (hx, lx, x); |
132 | |
133 | if (((hx & 0x7fffffffffffffffLL) | lx) == 0) |
134 | { |
135 | /* Return value for x == 0 is Inf with divide by zero exception. */ |
136 | *signgamp = 0; |
137 | return 1.0 / x; |
138 | } |
139 | if (hx < 0 && (uint64_t) hx < 0xffff000000000000ULL && rintl (x) == x) |
140 | { |
141 | /* Return value for integer x < 0 is NaN with invalid exception. */ |
142 | *signgamp = 0; |
143 | return (x - x) / (x - x); |
144 | } |
145 | if (hx == 0xffff000000000000ULL && lx == 0) |
146 | { |
147 | /* x == -Inf. According to ISO this is NaN. */ |
148 | *signgamp = 0; |
149 | return x - x; |
150 | } |
151 | if ((hx & 0x7fff000000000000ULL) == 0x7fff000000000000ULL) |
152 | { |
153 | /* Positive infinity (return positive infinity) or NaN (return |
154 | NaN). */ |
155 | *signgamp = 0; |
156 | return x + x; |
157 | } |
158 | |
159 | if (x >= 1756) |
160 | { |
161 | /* Overflow. */ |
162 | *signgamp = 0; |
163 | return LDBL_MAX * LDBL_MAX; |
164 | } |
165 | else |
166 | { |
167 | SET_RESTORE_ROUNDL (FE_TONEAREST); |
168 | if (x > 0) |
169 | { |
170 | *signgamp = 0; |
171 | int exp2_adj; |
172 | ret = gammal_positive (x, &exp2_adj); |
173 | ret = __scalbnl (ret, exp2_adj); |
174 | } |
175 | else if (x >= -LDBL_EPSILON / 4) |
176 | { |
177 | *signgamp = 0; |
178 | ret = 1 / x; |
179 | } |
180 | else |
181 | { |
182 | _Float128 tx = truncl (x); |
183 | *signgamp = (tx == 2 * truncl (tx / 2)) ? -1 : 1; |
184 | if (x <= -1775) |
185 | /* Underflow. */ |
186 | ret = LDBL_MIN * LDBL_MIN; |
187 | else |
188 | { |
189 | _Float128 frac = tx - x; |
190 | if (frac > L(0.5)) |
191 | frac = 1 - frac; |
192 | _Float128 sinpix = (frac <= L(0.25) |
193 | ? __sinl (M_PIl * frac) |
194 | : __cosl (M_PIl * (L(0.5) - frac))); |
195 | int exp2_adj; |
196 | ret = M_PIl / (-x * sinpix |
197 | * gammal_positive (-x, &exp2_adj)); |
198 | ret = __scalbnl (ret, -exp2_adj); |
199 | math_check_force_underflow_nonneg (ret); |
200 | } |
201 | } |
202 | } |
203 | if (isinf (ret) && x != 0) |
204 | { |
205 | if (*signgamp < 0) |
206 | return -(-copysignl (LDBL_MAX, ret) * LDBL_MAX); |
207 | else |
208 | return copysignl (LDBL_MAX, ret) * LDBL_MAX; |
209 | } |
210 | else if (ret == 0) |
211 | { |
212 | if (*signgamp < 0) |
213 | return -(-copysignl (LDBL_MIN, ret) * LDBL_MIN); |
214 | else |
215 | return copysignl (LDBL_MIN, ret) * LDBL_MIN; |
216 | } |
217 | else |
218 | return ret; |
219 | } |
220 | strong_alias (__ieee754_gammal_r, __gammal_r_finite) |
221 | |