1/* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2021 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 <https://www.gnu.org/licenses/>. */
19
20#include <math.h>
21#include <math-narrow-eval.h>
22#include <math_private.h>
23#include <fenv_private.h>
24#include <math-underflow.h>
25#include <float.h>
26#include <libm-alias-finite.h>
27
28/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
29 approximation to gamma function. */
30
31static const float gamma_coeff[] =
32 {
33 0x1.555556p-4f,
34 -0xb.60b61p-12f,
35 0x3.403404p-12f,
36 };
37
38#define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
39
40/* Return gamma (X), for positive X less than 42, in the form R *
41 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
42 avoid overflow or underflow in intermediate calculations. */
43
44static float
45gammaf_positive (float x, int *exp2_adj)
46{
47 int local_signgam;
48 if (x < 0.5f)
49 {
50 *exp2_adj = 0;
51 return __ieee754_expf (__ieee754_lgammaf_r (x + 1, &local_signgam)) / x;
52 }
53 else if (x <= 1.5f)
54 {
55 *exp2_adj = 0;
56 return __ieee754_expf (__ieee754_lgammaf_r (x, &local_signgam));
57 }
58 else if (x < 2.5f)
59 {
60 *exp2_adj = 0;
61 float x_adj = x - 1;
62 return (__ieee754_expf (__ieee754_lgammaf_r (x_adj, &local_signgam))
63 * x_adj);
64 }
65 else
66 {
67 float eps = 0;
68 float x_eps = 0;
69 float x_adj = x;
70 float prod = 1;
71 if (x < 4.0f)
72 {
73 /* Adjust into the range for applying Stirling's
74 approximation. */
75 float n = ceilf (4.0f - x);
76 x_adj = math_narrow_eval (x + n);
77 x_eps = (x - (x_adj - n));
78 prod = __gamma_productf (x_adj - n, x_eps, n, &eps);
79 }
80 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
81 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
82 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
83 factored out. */
84 float exp_adj = -eps;
85 float x_adj_int = roundf (x_adj);
86 float x_adj_frac = x_adj - x_adj_int;
87 int x_adj_log2;
88 float x_adj_mant = __frexpf (x_adj, &x_adj_log2);
89 if (x_adj_mant < (float) M_SQRT1_2)
90 {
91 x_adj_log2--;
92 x_adj_mant *= 2.0f;
93 }
94 *exp2_adj = x_adj_log2 * (int) x_adj_int;
95 float ret = (__ieee754_powf (x_adj_mant, x_adj)
96 * __ieee754_exp2f (x_adj_log2 * x_adj_frac)
97 * __ieee754_expf (-x_adj)
98 * sqrtf (2 * (float) M_PI / x_adj)
99 / prod);
100 exp_adj += x_eps * __ieee754_logf (x_adj);
101 float bsum = gamma_coeff[NCOEFF - 1];
102 float x_adj2 = x_adj * x_adj;
103 for (size_t i = 1; i <= NCOEFF - 1; i++)
104 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
105 exp_adj += bsum / x_adj;
106 return ret + ret * __expm1f (exp_adj);
107 }
108}
109
110float
111__ieee754_gammaf_r (float x, int *signgamp)
112{
113 int32_t hx;
114 float ret;
115
116 GET_FLOAT_WORD (hx, x);
117
118 if (__glibc_unlikely ((hx & 0x7fffffff) == 0))
119 {
120 /* Return value for x == 0 is Inf with divide by zero exception. */
121 *signgamp = 0;
122 return 1.0 / x;
123 }
124 if (__builtin_expect (hx < 0, 0)
125 && (uint32_t) hx < 0xff800000 && rintf (x) == x)
126 {
127 /* Return value for integer x < 0 is NaN with invalid exception. */
128 *signgamp = 0;
129 return (x - x) / (x - x);
130 }
131 if (__glibc_unlikely (hx == 0xff800000))
132 {
133 /* x == -Inf. According to ISO this is NaN. */
134 *signgamp = 0;
135 return x - x;
136 }
137 if (__glibc_unlikely ((hx & 0x7f800000) == 0x7f800000))
138 {
139 /* Positive infinity (return positive infinity) or NaN (return
140 NaN). */
141 *signgamp = 0;
142 return x + x;
143 }
144
145 if (x >= 36.0f)
146 {
147 /* Overflow. */
148 *signgamp = 0;
149 ret = math_narrow_eval (FLT_MAX * FLT_MAX);
150 return ret;
151 }
152 else
153 {
154 SET_RESTORE_ROUNDF (FE_TONEAREST);
155 if (x > 0.0f)
156 {
157 *signgamp = 0;
158 int exp2_adj;
159 float tret = gammaf_positive (x, &exp2_adj);
160 ret = __scalbnf (tret, exp2_adj);
161 }
162 else if (x >= -FLT_EPSILON / 4.0f)
163 {
164 *signgamp = 0;
165 ret = 1.0f / x;
166 }
167 else
168 {
169 float tx = truncf (x);
170 *signgamp = (tx == 2.0f * truncf (tx / 2.0f)) ? -1 : 1;
171 if (x <= -42.0f)
172 /* Underflow. */
173 ret = FLT_MIN * FLT_MIN;
174 else
175 {
176 float frac = tx - x;
177 if (frac > 0.5f)
178 frac = 1.0f - frac;
179 float sinpix = (frac <= 0.25f
180 ? __sinf ((float) M_PI * frac)
181 : __cosf ((float) M_PI * (0.5f - frac)));
182 int exp2_adj;
183 float tret = (float) M_PI / (-x * sinpix
184 * gammaf_positive (-x, &exp2_adj));
185 ret = __scalbnf (tret, -exp2_adj);
186 math_check_force_underflow_nonneg (ret);
187 }
188 }
189 ret = math_narrow_eval (ret);
190 }
191 if (isinf (ret) && x != 0)
192 {
193 if (*signgamp < 0)
194 {
195 ret = math_narrow_eval (-copysignf (FLT_MAX, ret) * FLT_MAX);
196 ret = -ret;
197 }
198 else
199 ret = math_narrow_eval (copysignf (FLT_MAX, ret) * FLT_MAX);
200 return ret;
201 }
202 else if (ret == 0)
203 {
204 if (*signgamp < 0)
205 {
206 ret = math_narrow_eval (-copysignf (FLT_MIN, ret) * FLT_MIN);
207 ret = -ret;
208 }
209 else
210 ret = math_narrow_eval (copysignf (FLT_MIN, ret) * FLT_MIN);
211 return ret;
212 }
213 else
214 return ret;
215}
216libm_alias_finite (__ieee754_gammaf_r, __gammaf_r)
217