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 <math-underflow.h> |
23 | #include <float.h> |
24 | |
25 | /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's |
26 | approximation to gamma function. */ |
27 | |
28 | static const long double gamma_coeff[] = |
29 | { |
30 | 0x1.5555555555555556p-4L, |
31 | -0xb.60b60b60b60b60bp-12L, |
32 | 0x3.4034034034034034p-12L, |
33 | -0x2.7027027027027028p-12L, |
34 | 0x3.72a3c5631fe46aep-12L, |
35 | -0x7.daac36664f1f208p-12L, |
36 | 0x1.a41a41a41a41a41ap-8L, |
37 | -0x7.90a1b2c3d4e5f708p-8L, |
38 | }; |
39 | |
40 | #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0])) |
41 | |
42 | /* Return gamma (X), for positive X less than 1766, in the form R * |
43 | 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to |
44 | avoid overflow or underflow in intermediate calculations. */ |
45 | |
46 | static long double |
47 | gammal_positive (long double x, int *exp2_adj) |
48 | { |
49 | int local_signgam; |
50 | if (x < 0.5L) |
51 | { |
52 | *exp2_adj = 0; |
53 | return __ieee754_expl (__ieee754_lgammal_r (x + 1, &local_signgam)) / x; |
54 | } |
55 | else if (x <= 1.5L) |
56 | { |
57 | *exp2_adj = 0; |
58 | return __ieee754_expl (__ieee754_lgammal_r (x, &local_signgam)); |
59 | } |
60 | else if (x < 7.5L) |
61 | { |
62 | /* Adjust into the range for using exp (lgamma). */ |
63 | *exp2_adj = 0; |
64 | long double n = __ceill (x - 1.5L); |
65 | long double x_adj = x - n; |
66 | long double eps; |
67 | long double prod = __gamma_productl (x_adj, 0, n, &eps); |
68 | return (__ieee754_expl (__ieee754_lgammal_r (x_adj, &local_signgam)) |
69 | * prod * (1.0L + eps)); |
70 | } |
71 | else |
72 | { |
73 | long double eps = 0; |
74 | long double x_eps = 0; |
75 | long double x_adj = x; |
76 | long double prod = 1; |
77 | if (x < 13.0L) |
78 | { |
79 | /* Adjust into the range for applying Stirling's |
80 | approximation. */ |
81 | long double n = __ceill (13.0L - x); |
82 | x_adj = x + n; |
83 | x_eps = (x - (x_adj - n)); |
84 | prod = __gamma_productl (x_adj - n, x_eps, n, &eps); |
85 | } |
86 | /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)). |
87 | Compute gamma (X_ADJ + X_EPS) using Stirling's approximation, |
88 | starting by computing pow (X_ADJ, X_ADJ) with a power of 2 |
89 | factored out. */ |
90 | long double exp_adj = -eps; |
91 | long double x_adj_int = __roundl (x_adj); |
92 | long double x_adj_frac = x_adj - x_adj_int; |
93 | int x_adj_log2; |
94 | long double x_adj_mant = __frexpl (x_adj, &x_adj_log2); |
95 | if (x_adj_mant < M_SQRT1_2l) |
96 | { |
97 | x_adj_log2--; |
98 | x_adj_mant *= 2.0L; |
99 | } |
100 | *exp2_adj = x_adj_log2 * (int) x_adj_int; |
101 | long double ret = (__ieee754_powl (x_adj_mant, x_adj) |
102 | * __ieee754_exp2l (x_adj_log2 * x_adj_frac) |
103 | * __ieee754_expl (-x_adj) |
104 | * sqrtl (2 * M_PIl / x_adj) |
105 | / prod); |
106 | exp_adj += x_eps * __ieee754_logl (x_adj); |
107 | long double bsum = gamma_coeff[NCOEFF - 1]; |
108 | long double x_adj2 = x_adj * x_adj; |
109 | for (size_t i = 1; i <= NCOEFF - 1; i++) |
110 | bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i]; |
111 | exp_adj += bsum / x_adj; |
112 | return ret + ret * __expm1l (exp_adj); |
113 | } |
114 | } |
115 | |
116 | long double |
117 | __ieee754_gammal_r (long double x, int *signgamp) |
118 | { |
119 | uint32_t es, hx, lx; |
120 | long double ret; |
121 | |
122 | GET_LDOUBLE_WORDS (es, hx, lx, x); |
123 | |
124 | if (__glibc_unlikely (((es & 0x7fff) | hx | lx) == 0)) |
125 | { |
126 | /* Return value for x == 0 is Inf with divide by zero exception. */ |
127 | *signgamp = 0; |
128 | return 1.0 / x; |
129 | } |
130 | if (__glibc_unlikely (es == 0xffffffff && ((hx & 0x7fffffff) | lx) == 0)) |
131 | { |
132 | /* x == -Inf. According to ISO this is NaN. */ |
133 | *signgamp = 0; |
134 | return x - x; |
135 | } |
136 | if (__glibc_unlikely ((es & 0x7fff) == 0x7fff)) |
137 | { |
138 | /* Positive infinity (return positive infinity) or NaN (return |
139 | NaN). */ |
140 | *signgamp = 0; |
141 | return x + x; |
142 | } |
143 | if (__builtin_expect ((es & 0x8000) != 0, 0) && __rintl (x) == x) |
144 | { |
145 | /* Return value for integer x < 0 is NaN with invalid exception. */ |
146 | *signgamp = 0; |
147 | return (x - x) / (x - x); |
148 | } |
149 | |
150 | if (x >= 1756.0L) |
151 | { |
152 | /* Overflow. */ |
153 | *signgamp = 0; |
154 | return LDBL_MAX * LDBL_MAX; |
155 | } |
156 | else |
157 | { |
158 | SET_RESTORE_ROUNDL (FE_TONEAREST); |
159 | if (x > 0.0L) |
160 | { |
161 | *signgamp = 0; |
162 | int exp2_adj; |
163 | ret = gammal_positive (x, &exp2_adj); |
164 | ret = __scalbnl (ret, exp2_adj); |
165 | } |
166 | else if (x >= -LDBL_EPSILON / 4.0L) |
167 | { |
168 | *signgamp = 0; |
169 | ret = 1.0L / x; |
170 | } |
171 | else |
172 | { |
173 | long double tx = __truncl (x); |
174 | *signgamp = (tx == 2.0L * __truncl (tx / 2.0L)) ? -1 : 1; |
175 | if (x <= -1766.0L) |
176 | /* Underflow. */ |
177 | ret = LDBL_MIN * LDBL_MIN; |
178 | else |
179 | { |
180 | long double frac = tx - x; |
181 | if (frac > 0.5L) |
182 | frac = 1.0L - frac; |
183 | long double sinpix = (frac <= 0.25L |
184 | ? __sinl (M_PIl * frac) |
185 | : __cosl (M_PIl * (0.5L - frac))); |
186 | int exp2_adj; |
187 | ret = M_PIl / (-x * sinpix |
188 | * gammal_positive (-x, &exp2_adj)); |
189 | ret = __scalbnl (ret, -exp2_adj); |
190 | math_check_force_underflow_nonneg (ret); |
191 | } |
192 | } |
193 | } |
194 | if (isinf (ret) && x != 0) |
195 | { |
196 | if (*signgamp < 0) |
197 | return -(-__copysignl (LDBL_MAX, ret) * LDBL_MAX); |
198 | else |
199 | return __copysignl (LDBL_MAX, ret) * LDBL_MAX; |
200 | } |
201 | else if (ret == 0) |
202 | { |
203 | if (*signgamp < 0) |
204 | return -(-__copysignl (LDBL_MIN, ret) * LDBL_MIN); |
205 | else |
206 | return __copysignl (LDBL_MIN, ret) * LDBL_MIN; |
207 | } |
208 | else |
209 | return ret; |
210 | } |
211 | strong_alias (__ieee754_gammal_r, __gammal_r_finite) |
212 | |