1 /* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2023 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-narrow-eval.h>
21 #include <math_private.h>
22 #include <fenv_private.h>
23 #include <math-underflow.h>
24 #include <float.h>
25 #include <libm-alias-finite.h>
26 #include <mul_split.h>
27
28 /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
29 approximation to gamma function. */
30
31 static const double gamma_coeff[] =
32 {
33 0x1.5555555555555p-4,
34 -0xb.60b60b60b60b8p-12,
35 0x3.4034034034034p-12,
36 -0x2.7027027027028p-12,
37 0x3.72a3c5631fe46p-12,
38 -0x7.daac36664f1f4p-12,
39 };
40
41 #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
42
43 /* Return gamma (X), for positive X less than 184, in the form R *
44 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
45 avoid overflow or underflow in intermediate calculations. */
46
47 static double
48 gamma_positive (double x, int *exp2_adj)
49 {
50 int local_signgam;
51 if (x < 0.5)
52 {
53 *exp2_adj = 0;
54 return __ieee754_exp (__ieee754_lgamma_r (x + 1, &local_signgam)) / x;
55 }
56 else if (x <= 1.5)
57 {
58 *exp2_adj = 0;
59 return __ieee754_exp (__ieee754_lgamma_r (x, &local_signgam));
60 }
61 else if (x < 6.5)
62 {
63 /* Adjust into the range for using exp (lgamma). */
64 *exp2_adj = 0;
65 double n = ceil (x - 1.5);
66 double x_adj = x - n;
67 double eps;
68 double prod = __gamma_product (x_adj, 0, n, &eps);
69 return (__ieee754_exp (__ieee754_lgamma_r (x_adj, &local_signgam))
70 * prod * (1.0 + eps));
71 }
72 else
73 {
74 double eps = 0;
75 double x_eps = 0;
76 double x_adj = x;
77 double prod = 1;
78 if (x < 12.0)
79 {
80 /* Adjust into the range for applying Stirling's
81 approximation. */
82 double n = ceil (12.0 - x);
83 x_adj = math_narrow_eval (x + n);
84 x_eps = (x - (x_adj - n));
85 prod = __gamma_product (x_adj - n, x_eps, n, &eps);
86 }
87 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
88 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
89 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
90 factored out. */
91 double x_adj_int = round (x_adj);
92 double x_adj_frac = x_adj - x_adj_int;
93 int x_adj_log2;
94 double x_adj_mant = __frexp (x_adj, &x_adj_log2);
95 if (x_adj_mant < M_SQRT1_2)
96 {
97 x_adj_log2--;
98 x_adj_mant *= 2.0;
99 }
100 *exp2_adj = x_adj_log2 * (int) x_adj_int;
101 double h1, l1, h2, l2;
102 mul_split (&h1, &l1, __ieee754_pow (x_adj_mant, x_adj),
103 __ieee754_exp2 (x_adj_log2 * x_adj_frac));
104 mul_split (&h2, &l2, __ieee754_exp (-x_adj), sqrt (2 * M_PI / x_adj));
105 mul_expansion (&h1, &l1, h1, l1, h2, l2);
106 /* Divide by prod * (1 + eps). */
107 div_expansion (&h1, &l1, h1, l1, prod, prod * eps);
108 double exp_adj = x_eps * __ieee754_log (x_adj);
109 double bsum = gamma_coeff[NCOEFF - 1];
110 double x_adj2 = x_adj * x_adj;
111 for (size_t i = 1; i <= NCOEFF - 1; i++)
112 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
113 exp_adj += bsum / x_adj;
114 /* Now return (h1+l1) * exp(exp_adj), where exp_adj is small. */
115 l1 += h1 * __expm1 (exp_adj);
116 return h1 + l1;
117 }
118 }
119
120 double
121 __ieee754_gamma_r (double x, int *signgamp)
122 {
123 int32_t hx;
124 uint32_t lx;
125 double ret;
126
127 EXTRACT_WORDS (hx, lx, x);
128
129 if (__glibc_unlikely (((hx & 0x7fffffff) | lx) == 0))
130 {
131 /* Return value for x == 0 is Inf with divide by zero exception. */
132 *signgamp = 0;
133 return 1.0 / x;
134 }
135 if (__builtin_expect (hx < 0, 0)
136 && (uint32_t) hx < 0xfff00000 && rint (x) == x)
137 {
138 /* Return value for integer x < 0 is NaN with invalid exception. */
139 *signgamp = 0;
140 return (x - x) / (x - x);
141 }
142 if (__glibc_unlikely ((unsigned int) hx == 0xfff00000 && lx == 0))
143 {
144 /* x == -Inf. According to ISO this is NaN. */
145 *signgamp = 0;
146 return x - x;
147 }
148 if (__glibc_unlikely ((hx & 0x7ff00000) == 0x7ff00000))
149 {
150 /* Positive infinity (return positive infinity) or NaN (return
151 NaN). */
152 *signgamp = 0;
153 return x + x;
154 }
155
156 if (x >= 172.0)
157 {
158 /* Overflow. */
159 *signgamp = 0;
160 ret = math_narrow_eval (DBL_MAX * DBL_MAX);
161 return ret;
162 }
163 else
164 {
165 SET_RESTORE_ROUND (FE_TONEAREST);
166 if (x > 0.0)
167 {
168 *signgamp = 0;
169 int exp2_adj;
170 double tret = gamma_positive (x, &exp2_adj);
171 ret = __scalbn (tret, exp2_adj);
172 }
173 else if (x >= -DBL_EPSILON / 4.0)
174 {
175 *signgamp = 0;
176 ret = 1.0 / x;
177 }
178 else
179 {
180 double tx = trunc (x);
181 *signgamp = (tx == 2.0 * trunc (tx / 2.0)) ? -1 : 1;
182 if (x <= -184.0)
183 /* Underflow. */
184 ret = DBL_MIN * DBL_MIN;
185 else
186 {
187 double frac = tx - x;
188 if (frac > 0.5)
189 frac = 1.0 - frac;
190 double sinpix = (frac <= 0.25
191 ? __sin (M_PI * frac)
192 : __cos (M_PI * (0.5 - frac)));
193 int exp2_adj;
194 double h1, l1, h2, l2;
195 h2 = gamma_positive (-x, &exp2_adj);
196 mul_split (&h1, &l1, sinpix, h2);
197 /* sinpix*gamma_positive(.) = h1 + l1 */
198 mul_split (&h2, &l2, h1, x);
199 /* h1*x = h2 + l2 */
200 /* (h1 + l1) * x = h1*x + l1*x = h2 + l2 + l1*x */
201 l2 += l1 * x;
202 /* x*sinpix*gamma_positive(.) ~ h2 + l2 */
203 h1 = 0x3.243f6a8885a3p+0; /* binary64 approximation of Pi */
204 l1 = 0x8.d313198a2e038p-56; /* |h1+l1-Pi| < 3e-33 */
205 /* Now we divide h1 + l1 by h2 + l2. */
206 div_expansion (&h1, &l1, h1, l1, h2, l2);
207 ret = __scalbn (-h1, -exp2_adj);
208 math_check_force_underflow_nonneg (ret);
209 }
210 }
211 ret = math_narrow_eval (ret);
212 }
213 if (isinf (ret) && x != 0)
214 {
215 if (*signgamp < 0)
216 {
217 ret = math_narrow_eval (-copysign (DBL_MAX, ret) * DBL_MAX);
218 ret = -ret;
219 }
220 else
221 ret = math_narrow_eval (copysign (DBL_MAX, ret) * DBL_MAX);
222 return ret;
223 }
224 else if (ret == 0)
225 {
226 if (*signgamp < 0)
227 {
228 ret = math_narrow_eval (-copysign (DBL_MIN, ret) * DBL_MIN);
229 ret = -ret;
230 }
231 else
232 ret = math_narrow_eval (copysign (DBL_MIN, ret) * DBL_MIN);
233 return ret;
234 }
235 else
236 return ret;
237 }
238 libm_alias_finite (__ieee754_gamma_r, __gamma_r)