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.
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.
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.
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/>. */
22 #include <math_private.h>
23 #include <fenv_private.h>
24 #include <math-underflow.h>
27 /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
28 approximation to gamma function. */
30 static const _Float128 gamma_coeff
[] =
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),
48 #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
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. */
55 gammal_positive (_Float128 x
, int *exp2_adj
)
61 return __ieee754_expl (__ieee754_lgammal_r (x
+ 1, &local_signgam
)) / x
;
66 return __ieee754_expl (__ieee754_lgammal_r (x
, &local_signgam
));
70 /* Adjust into the range for using exp (lgamma). */
72 _Float128 n
= ceill (x
- L(1.5));
73 _Float128 x_adj
= x
- n
;
75 _Float128 prod
= __gamma_productl (x_adj
, 0, n
, &eps
);
76 return (__ieee754_expl (__ieee754_lgammal_r (x_adj
, &local_signgam
))
87 /* Adjust into the range for applying Stirling's
89 _Float128 n
= ceill (24 - x
);
91 x_eps
= (x
- (x_adj
- n
));
92 prod
= __gamma_productl (x_adj
- n
, x_eps
, n
, &eps
);
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
98 _Float128 exp_adj
= -eps
;
99 _Float128 x_adj_int
= roundl (x_adj
);
100 _Float128 x_adj_frac
= x_adj
- x_adj_int
;
102 _Float128 x_adj_mant
= __frexpl (x_adj
, &x_adj_log2
);
103 if (x_adj_mant
< M_SQRT1_2l
)
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
)
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
);
125 __ieee754_gammal_r (_Float128 x
, int *signgamp
)
131 GET_LDOUBLE_WORDS64 (hx
, lx
, x
);
133 if (((hx
& 0x7fffffffffffffffLL
) | lx
) == 0)
135 /* Return value for x == 0 is Inf with divide by zero exception. */
139 if (hx
< 0 && (uint64_t) hx
< 0xffff000000000000ULL
&& rintl (x
) == x
)
141 /* Return value for integer x < 0 is NaN with invalid exception. */
143 return (x
- x
) / (x
- x
);
145 if (hx
== 0xffff000000000000ULL
&& lx
== 0)
147 /* x == -Inf. According to ISO this is NaN. */
151 if ((hx
& 0x7fff000000000000ULL
) == 0x7fff000000000000ULL
)
153 /* Positive infinity (return positive infinity) or NaN (return
163 return LDBL_MAX
* LDBL_MAX
;
167 SET_RESTORE_ROUNDL (FE_TONEAREST
);
172 ret
= gammal_positive (x
, &exp2_adj
);
173 ret
= __scalbnl (ret
, exp2_adj
);
175 else if (x
>= -LDBL_EPSILON
/ 4)
182 _Float128 tx
= truncl (x
);
183 *signgamp
= (tx
== 2 * truncl (tx
/ 2)) ? -1 : 1;
186 ret
= LDBL_MIN
* LDBL_MIN
;
189 _Float128 frac
= tx
- x
;
192 _Float128 sinpix
= (frac
<= L(0.25)
193 ? __sinl (M_PIl
* frac
)
194 : __cosl (M_PIl
* (L(0.5) - frac
)));
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
);
203 if (isinf (ret
) && x
!= 0)
206 return -(-__copysignl (LDBL_MAX
, ret
) * LDBL_MAX
);
208 return __copysignl (LDBL_MAX
, ret
) * LDBL_MAX
;
213 return -(-__copysignl (LDBL_MIN
, ret
) * LDBL_MIN
);
215 return __copysignl (LDBL_MIN
, ret
) * LDBL_MIN
;
220 strong_alias (__ieee754_gammal_r
, __gammal_r_finite
)