1 /* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2024 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
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.
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.
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/>. */
20 #include <math_private.h>
21 #include <fenv_private.h>
22 #include <math-underflow.h>
24 #include <libm-alias-finite.h>
26 /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
27 approximation to gamma function. */
29 static const _Float128 gamma_coeff
[] =
31 L(0x1.5555555555555555555555555555p
-4),
32 L(-0xb.60b60b60b60b60b60b60b60b60b8p
-12),
33 L(0x3.4034034034034034034034034034p
-12),
34 L(-0x2.7027027027027027027027027028p
-12),
35 L(0x3.72a3c5631fe46ae1d4e700dca8f2p
-12),
36 L(-0x7.daac36664f1f207daac36664f1f4p
-12),
37 L(0x1.a41a41a41a41a41a41a41a41a41ap
-8),
38 L(-0x7.90a1b2c3d4e5f708192a3b4c5d7p
-8),
39 L(0x2.dfd2c703c0cfff430edfd2c703cp
-4),
40 L(-0x1.6476701181f39edbdb9ce625987dp
+0),
41 L(0xd.672219167002d3a7a9c886459cp
+0),
42 L(-0x9.cd9292e6660d55b3f712eb9e07c8p
+4),
43 L(0x8.911a740da740da740da740da741p
+8),
44 L(-0x8.d0cc570e255bf59ff6eec24b49p
+12),
47 #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
49 /* Return gamma (X), for positive X less than 1775, in the form R *
50 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
51 avoid overflow or underflow in intermediate calculations. */
54 gammal_positive (_Float128 x
, int *exp2_adj
)
60 return __ieee754_expl (__ieee754_lgammal_r (x
+ 1, &local_signgam
)) / x
;
65 return __ieee754_expl (__ieee754_lgammal_r (x
, &local_signgam
));
69 /* Adjust into the range for using exp (lgamma). */
71 _Float128 n
= ceill (x
- L(1.5));
72 _Float128 x_adj
= x
- n
;
74 _Float128 prod
= __gamma_productl (x_adj
, 0, n
, &eps
);
75 return (__ieee754_expl (__ieee754_lgammal_r (x_adj
, &local_signgam
))
86 /* Adjust into the range for applying Stirling's
88 _Float128 n
= ceill (24 - x
);
90 x_eps
= (x
- (x_adj
- n
));
91 prod
= __gamma_productl (x_adj
- n
, x_eps
, n
, &eps
);
93 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
94 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
95 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
97 _Float128 exp_adj
= -eps
;
98 _Float128 x_adj_int
= roundl (x_adj
);
99 _Float128 x_adj_frac
= x_adj
- x_adj_int
;
101 _Float128 x_adj_mant
= __frexpl (x_adj
, &x_adj_log2
);
102 if (x_adj_mant
< M_SQRT1_2l
)
107 *exp2_adj
= x_adj_log2
* (int) x_adj_int
;
108 _Float128 ret
= (__ieee754_powl (x_adj_mant
, x_adj
)
109 * __ieee754_exp2l (x_adj_log2
* x_adj_frac
)
110 * __ieee754_expl (-x_adj
)
111 * sqrtl (2 * M_PIl
/ x_adj
)
113 exp_adj
+= x_eps
* __ieee754_logl (x_adj
);
114 _Float128 bsum
= gamma_coeff
[NCOEFF
- 1];
115 _Float128 x_adj2
= x_adj
* x_adj
;
116 for (size_t i
= 1; i
<= NCOEFF
- 1; i
++)
117 bsum
= bsum
/ x_adj2
+ gamma_coeff
[NCOEFF
- 1 - i
];
118 exp_adj
+= bsum
/ x_adj
;
119 return ret
+ ret
* __expm1l (exp_adj
);
124 __ieee754_gammal_r (_Float128 x
, int *signgamp
)
130 GET_LDOUBLE_WORDS64 (hx
, lx
, x
);
132 if (((hx
& 0x7fffffffffffffffLL
) | lx
) == 0)
134 /* Return value for x == 0 is Inf with divide by zero exception. */
138 if (hx
< 0 && (uint64_t) hx
< 0xffff000000000000ULL
&& rintl (x
) == x
)
140 /* Return value for integer x < 0 is NaN with invalid exception. */
142 return (x
- x
) / (x
- x
);
144 if (hx
== 0xffff000000000000ULL
&& lx
== 0)
146 /* x == -Inf. According to ISO this is NaN. */
150 if ((hx
& 0x7fff000000000000ULL
) == 0x7fff000000000000ULL
)
152 /* Positive infinity (return positive infinity) or NaN (return
162 return LDBL_MAX
* LDBL_MAX
;
166 SET_RESTORE_ROUNDL (FE_TONEAREST
);
171 ret
= gammal_positive (x
, &exp2_adj
);
172 ret
= __scalbnl (ret
, exp2_adj
);
174 else if (x
>= -LDBL_EPSILON
/ 4)
181 _Float128 tx
= truncl (x
);
182 *signgamp
= (tx
== 2 * truncl (tx
/ 2)) ? -1 : 1;
185 ret
= LDBL_MIN
* LDBL_MIN
;
188 _Float128 frac
= tx
- x
;
191 _Float128 sinpix
= (frac
<= L(0.25)
192 ? __sinl (M_PIl
* frac
)
193 : __cosl (M_PIl
* (L(0.5) - frac
)));
195 ret
= M_PIl
/ (-x
* sinpix
196 * gammal_positive (-x
, &exp2_adj
));
197 ret
= __scalbnl (ret
, -exp2_adj
);
198 math_check_force_underflow_nonneg (ret
);
202 if (isinf (ret
) && x
!= 0)
205 return -(-copysignl (LDBL_MAX
, ret
) * LDBL_MAX
);
207 return copysignl (LDBL_MAX
, ret
) * LDBL_MAX
;
212 return -(-copysignl (LDBL_MIN
, ret
) * LDBL_MIN
);
214 return copysignl (LDBL_MIN
, ret
) * LDBL_MIN
;
219 libm_alias_finite (__ieee754_gammal_r
, __gammal_r
)