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 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 "quadmath-imp.h"
22 __float128
23 tgammaq (__float128 x)
24 {
25 int sign;
26 __float128 ret;
27 ret = __quadmath_gammaq_r (x, &sign);
28 return sign < 0 ? -ret : ret;
29 }
30
31 /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
32 approximation to gamma function. */
33
34 static const __float128 gamma_coeff[] =
35 {
36 0x1.5555555555555555555555555555p-4Q,
37 -0xb.60b60b60b60b60b60b60b60b60b8p-12Q,
38 0x3.4034034034034034034034034034p-12Q,
39 -0x2.7027027027027027027027027028p-12Q,
40 0x3.72a3c5631fe46ae1d4e700dca8f2p-12Q,
41 -0x7.daac36664f1f207daac36664f1f4p-12Q,
42 0x1.a41a41a41a41a41a41a41a41a41ap-8Q,
43 -0x7.90a1b2c3d4e5f708192a3b4c5d7p-8Q,
44 0x2.dfd2c703c0cfff430edfd2c703cp-4Q,
45 -0x1.6476701181f39edbdb9ce625987dp+0Q,
46 0xd.672219167002d3a7a9c886459cp+0Q,
47 -0x9.cd9292e6660d55b3f712eb9e07c8p+4Q,
48 0x8.911a740da740da740da740da741p+8Q,
49 -0x8.d0cc570e255bf59ff6eec24b49p+12Q,
50 };
51
52 #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
53
54 /* Return gamma (X), for positive X less than 1775, in the form R *
55 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
56 avoid overflow or underflow in intermediate calculations. */
57
58 static __float128
59 gammal_positive (__float128 x, int *exp2_adj)
60 {
61 int local_signgam;
62 if (x < 0.5Q)
63 {
64 *exp2_adj = 0;
65 return expq (__quadmath_lgammaq_r (x + 1, &local_signgam)) / x;
66 }
67 else if (x <= 1.5Q)
68 {
69 *exp2_adj = 0;
70 return expq (__quadmath_lgammaq_r (x, &local_signgam));
71 }
72 else if (x < 12.5Q)
73 {
74 /* Adjust into the range for using exp (lgamma). */
75 *exp2_adj = 0;
76 __float128 n = ceilq (x - 1.5Q);
77 __float128 x_adj = x - n;
78 __float128 eps;
79 __float128 prod = __quadmath_gamma_productq (x_adj, 0, n, &eps);
80 return (expq (__quadmath_lgammaq_r (x_adj, &local_signgam))
81 * prod * (1 + eps));
82 }
83 else
84 {
85 __float128 eps = 0;
86 __float128 x_eps = 0;
87 __float128 x_adj = x;
88 __float128 prod = 1;
89 if (x < 24)
90 {
91 /* Adjust into the range for applying Stirling's
92 approximation. */
93 __float128 n = ceilq (24 - x);
94 x_adj = x + n;
95 x_eps = (x - (x_adj - n));
96 prod = __quadmath_gamma_productq (x_adj - n, x_eps, n, &eps);
97 }
98 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
99 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
100 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
101 factored out. */
102 __float128 exp_adj = -eps;
103 __float128 x_adj_int = roundq (x_adj);
104 __float128 x_adj_frac = x_adj - x_adj_int;
105 int x_adj_log2;
106 __float128 x_adj_mant = frexpq (x_adj, &x_adj_log2);
107 if (x_adj_mant < M_SQRT1_2q)
108 {
109 x_adj_log2--;
110 x_adj_mant *= 2;
111 }
112 *exp2_adj = x_adj_log2 * (int) x_adj_int;
113 __float128 ret = (powq (x_adj_mant, x_adj)
114 * exp2q (x_adj_log2 * x_adj_frac)
115 * expq (-x_adj)
116 * sqrtq (2 * M_PIq / x_adj)
117 / prod);
118 exp_adj += x_eps * logq (x_adj);
119 __float128 bsum = gamma_coeff[NCOEFF - 1];
120 __float128 x_adj2 = x_adj * x_adj;
121 for (size_t i = 1; i <= NCOEFF - 1; i++)
122 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
123 exp_adj += bsum / x_adj;
124 return ret + ret * expm1q (exp_adj);
125 }
126 }
127
128 __float128
129 __quadmath_gammaq_r (__float128 x, int *signgamp)
130 {
131 int64_t hx;
132 uint64_t lx;
133 __float128 ret;
134
135 GET_FLT128_WORDS64 (hx, lx, x);
136
137 if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
138 {
139 /* Return value for x == 0 is Inf with divide by zero exception. */
140 *signgamp = 0;
141 return 1.0 / x;
142 }
143 if (hx < 0 && (uint64_t) hx < 0xffff000000000000ULL && rintq (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 if (hx == 0xffff000000000000ULL && lx == 0)
150 {
151 /* x == -Inf. According to ISO this is NaN. */
152 *signgamp = 0;
153 return x - x;
154 }
155 if ((hx & 0x7fff000000000000ULL) == 0x7fff000000000000ULL)
156 {
157 /* Positive infinity (return positive infinity) or NaN (return
158 NaN). */
159 *signgamp = 0;
160 return x + x;
161 }
162
163 if (x >= 1756)
164 {
165 /* Overflow. */
166 *signgamp = 0;
167 return FLT128_MAX * FLT128_MAX;
168 }
169 else
170 {
171 SET_RESTORE_ROUNDF128 (FE_TONEAREST);
172 if (x > 0)
173 {
174 *signgamp = 0;
175 int exp2_adj;
176 ret = gammal_positive (x, &exp2_adj);
177 ret = scalbnq (ret, exp2_adj);
178 }
179 else if (x >= -FLT128_EPSILON / 4)
180 {
181 *signgamp = 0;
182 ret = 1 / x;
183 }
184 else
185 {
186 __float128 tx = truncq (x);
187 *signgamp = (tx == 2 * truncq (tx / 2)) ? -1 : 1;
188 if (x <= -1775)
189 /* Underflow. */
190 ret = FLT128_MIN * FLT128_MIN;
191 else
192 {
193 __float128 frac = tx - x;
194 if (frac > 0.5Q)
195 frac = 1 - frac;
196 __float128 sinpix = (frac <= 0.25Q
197 ? sinq (M_PIq * frac)
198 : cosq (M_PIq * (0.5Q - frac)));
199 int exp2_adj;
200 ret = M_PIq / (-x * sinpix
201 * gammal_positive (-x, &exp2_adj));
202 ret = scalbnq (ret, -exp2_adj);
203 math_check_force_underflow_nonneg (ret);
204 }
205 }
206 }
207 if (isinfq (ret) && x != 0)
208 {
209 if (*signgamp < 0)
210 return -(-copysignq (FLT128_MAX, ret) * FLT128_MAX);
211 else
212 return copysignq (FLT128_MAX, ret) * FLT128_MAX;
213 }
214 else if (ret == 0)
215 {
216 if (*signgamp < 0)
217 return -(-copysignq (FLT128_MIN, ret) * FLT128_MIN);
218 else
219 return copysignq (FLT128_MIN, ret) * FLT128_MIN;
220 }
221 else
222 return ret;
223 }