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.
4 
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.
9 
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.
14 
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/>.  */
18 
19 #include <math.h>
20 #include <math-narrow-eval.h>
21 #include <math_private.h>
22 #include <fenv_private.h>
23 #include <math-underflow.h>
24 #include <float.h>
25 #include <libm-alias-finite.h>
26 #include <mul_split.h>
27 
28 /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
29    approximation to gamma function.  */
30 
31 static const double gamma_coeff[] =
32   {
33     0x1.5555555555555p-4,
34     -0xb.60b60b60b60b8p-12,
35     0x3.4034034034034p-12,
36     -0x2.7027027027028p-12,
37     0x3.72a3c5631fe46p-12,
38     -0x7.daac36664f1f4p-12,
39   };
40 
41 #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
42 
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.  */
46 
47 static double
gamma_positive(double x,int * exp2_adj)48 gamma_positive (double x, int *exp2_adj)
49 {
50   int local_signgam;
51   if (x < 0.5)
52     {
53       *exp2_adj = 0;
54       return __ieee754_exp (__ieee754_lgamma_r (x + 1, &local_signgam)) / x;
55     }
56   else if (x <= 1.5)
57     {
58       *exp2_adj = 0;
59       return __ieee754_exp (__ieee754_lgamma_r (x, &local_signgam));
60     }
61   else if (x < 6.5)
62     {
63       /* Adjust into the range for using exp (lgamma).  */
64       *exp2_adj = 0;
65       double n = ceil (x - 1.5);
66       double x_adj = x - n;
67       double eps;
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));
71     }
72   else
73     {
74       double eps = 0;
75       double x_eps = 0;
76       double x_adj = x;
77       double prod = 1;
78       if (x < 12.0)
79 	{
80 	  /* Adjust into the range for applying Stirling's
81 	     approximation.  */
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);
86 	}
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
90 	 factored out.  */
91       double x_adj_int = round (x_adj);
92       double x_adj_frac = x_adj - x_adj_int;
93       int x_adj_log2;
94       double x_adj_mant = __frexp (x_adj, &x_adj_log2);
95       if (x_adj_mant < M_SQRT1_2)
96 	{
97 	  x_adj_log2--;
98 	  x_adj_mant *= 2.0;
99 	}
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);
116       return h1 + l1;
117     }
118 }
119 
120 double
__ieee754_gamma_r(double x,int * signgamp)121 __ieee754_gamma_r (double x, int *signgamp)
122 {
123   int32_t hx;
124   uint32_t lx;
125   double ret;
126 
127   EXTRACT_WORDS (hx, lx, x);
128 
129   if (__glibc_unlikely (((hx & 0x7fffffff) | lx) == 0))
130     {
131       /* Return value for x == 0 is Inf with divide by zero exception.  */
132       *signgamp = 0;
133       return 1.0 / x;
134     }
135   if (__builtin_expect (hx < 0, 0)
136       && (uint32_t) hx < 0xfff00000 && rint (x) == x)
137     {
138       /* Return value for integer x < 0 is NaN with invalid exception.  */
139       *signgamp = 0;
140       return (x - x) / (x - x);
141     }
142   if (__glibc_unlikely ((unsigned int) hx == 0xfff00000 && lx == 0))
143     {
144       /* x == -Inf.  According to ISO this is NaN.  */
145       *signgamp = 0;
146       return x - x;
147     }
148   if (__glibc_unlikely ((hx & 0x7ff00000) == 0x7ff00000))
149     {
150       /* Positive infinity (return positive infinity) or NaN (return
151 	 NaN).  */
152       *signgamp = 0;
153       return x + x;
154     }
155 
156   if (x >= 172.0)
157     {
158       /* Overflow.  */
159       *signgamp = 0;
160       ret = math_narrow_eval (DBL_MAX * DBL_MAX);
161       return ret;
162     }
163   else
164     {
165       SET_RESTORE_ROUND (FE_TONEAREST);
166       if (x > 0.0)
167 	{
168 	  *signgamp = 0;
169 	  int exp2_adj;
170 	  double tret = gamma_positive (x, &exp2_adj);
171 	  ret = __scalbn (tret, exp2_adj);
172 	}
173       else if (x >= -DBL_EPSILON / 4.0)
174 	{
175 	  *signgamp = 0;
176 	  ret = 1.0 / x;
177 	}
178       else
179 	{
180 	  double tx = trunc (x);
181 	  *signgamp = (tx == 2.0 * trunc (tx / 2.0)) ? -1 : 1;
182 	  if (x <= -184.0)
183 	    /* Underflow.  */
184 	    ret = DBL_MIN * DBL_MIN;
185 	  else
186 	    {
187 	      double frac = tx - x;
188 	      if (frac > 0.5)
189 		frac = 1.0 - frac;
190 	      double sinpix = (frac <= 0.25
191 			       ? __sin (M_PI * frac)
192 			       : __cos (M_PI * (0.5 - frac)));
193 	      int exp2_adj;
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);
199 	      /* h1*x = h2 + l2 */
200 	      /* (h1 + l1) * x = h1*x + l1*x = h2 + l2 + l1*x */
201 	      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);
209 	    }
210 	}
211       ret = math_narrow_eval (ret);
212     }
213   if (isinf (ret) && x != 0)
214     {
215       if (*signgamp < 0)
216 	{
217 	  ret = math_narrow_eval (-copysign (DBL_MAX, ret) * DBL_MAX);
218 	  ret = -ret;
219 	}
220       else
221 	ret = math_narrow_eval (copysign (DBL_MAX, ret) * DBL_MAX);
222       return ret;
223     }
224   else if (ret == 0)
225     {
226       if (*signgamp < 0)
227 	{
228 	  ret = math_narrow_eval (-copysign (DBL_MIN, ret) * DBL_MIN);
229 	  ret = -ret;
230 	}
231       else
232 	ret = math_narrow_eval (copysign (DBL_MIN, ret) * DBL_MIN);
233       return ret;
234     }
235   else
236     return ret;
237 }
238 libm_alias_finite (__ieee754_gamma_r, __gamma_r)
239