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1 /* Implementation of gamma function according to ISO C.
2 Copyright (C) 1997-2022 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-narrow-eval.h>
21 #include <math_private.h>
22 #include <fenv_private.h>
23 #include <math-underflow.h>
25 #include <libm-alias-finite.h>
26 #include <mul_split.h>
28 /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
29 approximation to gamma function. */
31 static const double gamma_coeff
[] =
34 -0xb.60b60b60b60b8p
-12,
35 0x3.4034034034034p
-12,
36 -0x2.7027027027028p
-12,
37 0x3.72a3c5631fe46p
-12,
38 -0x7.daac36664f1f4p
-12,
41 #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
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. */
48 gamma_positive (double x
, int *exp2_adj
)
54 return __ieee754_exp (__ieee754_lgamma_r (x
+ 1, &local_signgam
)) / x
;
59 return __ieee754_exp (__ieee754_lgamma_r (x
, &local_signgam
));
63 /* Adjust into the range for using exp (lgamma). */
65 double n
= ceil (x
- 1.5);
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
));
80 /* Adjust into the range for applying Stirling's
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
);
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
91 double x_adj_int
= round (x_adj
);
92 double x_adj_frac
= x_adj
- x_adj_int
;
94 double x_adj_mant
= __frexp (x_adj
, &x_adj_log2
);
95 if (x_adj_mant
< M_SQRT1_2
)
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
);
121 __ieee754_gamma_r (double x
, int *signgamp
)
127 EXTRACT_WORDS (hx
, lx
, x
);
129 if (__glibc_unlikely (((hx
& 0x7fffffff) | lx
) == 0))
131 /* Return value for x == 0 is Inf with divide by zero exception. */
135 if (__builtin_expect (hx
< 0, 0)
136 && (uint32_t) hx
< 0xfff00000 && rint (x
) == x
)
138 /* Return value for integer x < 0 is NaN with invalid exception. */
140 return (x
- x
) / (x
- x
);
142 if (__glibc_unlikely ((unsigned int) hx
== 0xfff00000 && lx
== 0))
144 /* x == -Inf. According to ISO this is NaN. */
148 if (__glibc_unlikely ((hx
& 0x7ff00000) == 0x7ff00000))
150 /* Positive infinity (return positive infinity) or NaN (return
160 ret
= math_narrow_eval (DBL_MAX
* DBL_MAX
);
165 SET_RESTORE_ROUND (FE_TONEAREST
);
170 double tret
= gamma_positive (x
, &exp2_adj
);
171 ret
= __scalbn (tret
, exp2_adj
);
173 else if (x
>= -DBL_EPSILON
/ 4.0)
180 double tx
= trunc (x
);
181 *signgamp
= (tx
== 2.0 * trunc (tx
/ 2.0)) ? -1 : 1;
184 ret
= DBL_MIN
* DBL_MIN
;
187 double frac
= tx
- x
;
190 double sinpix
= (frac
<= 0.25
191 ? __sin (M_PI
* frac
)
192 : __cos (M_PI
* (0.5 - frac
)));
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
);
200 /* (h1 + l1) * x = h1*x + l1*x = h2 + 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
);
211 ret
= math_narrow_eval (ret
);
213 if (isinf (ret
) && x
!= 0)
217 ret
= math_narrow_eval (-copysign (DBL_MAX
, ret
) * DBL_MAX
);
221 ret
= math_narrow_eval (copysign (DBL_MAX
, ret
) * DBL_MAX
);
228 ret
= math_narrow_eval (-copysign (DBL_MIN
, ret
) * DBL_MIN
);
232 ret
= math_narrow_eval (copysign (DBL_MIN
, ret
) * DBL_MIN
);
238 libm_alias_finite (__ieee754_gamma_r
, __gamma_r
)