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n_lgamma.c revision 1.3 
      1 /*      $NetBSD: n_lgamma.c,v 1.3 1998/10/20 02:26:12 matt Exp $ */
      2 /*-
      3  * Copyright (c) 1992, 1993
      4  *	The Regents of the University of California.  All rights reserved.
      5  *
      6  * Redistribution and use in source and binary forms, with or without
      7  * modification, are permitted provided that the following conditions
      8  * are met:
      9  * 1. Redistributions of source code must retain the above copyright
     10  *    notice, this list of conditions and the following disclaimer.
     11  * 2. Redistributions in binary form must reproduce the above copyright
     12  *    notice, this list of conditions and the following disclaimer in the
     13  *    documentation and/or other materials provided with the distribution.
     14  * 3. All advertising materials mentioning features or use of this software
     15  *    must display the following acknowledgement:
     16  *	This product includes software developed by the University of
     17  *	California, Berkeley and its contributors.
     18  * 4. Neither the name of the University nor the names of its contributors
     19  *    may be used to endorse or promote products derived from this software
     20  *    without specific prior written permission.
     21  *
     22  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
     23  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
     24  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
     25  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
     26  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
     27  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
     28  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
     29  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
     30  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
     31  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
     32  * SUCH DAMAGE.
     33  */
     34 
     35 #ifndef lint
     36 #if 0
     37 static char sccsid[] = "@(#)lgamma.c	8.2 (Berkeley) 11/30/93";
     38 #endif
     39 #endif /* not lint */
     40 
     41 /*
     42  * Coded by Peter McIlroy, Nov 1992;
     43  *
     44  * The financial support of UUNET Communications Services is greatfully
     45  * acknowledged.
     46  */
     47 
     48 #include <math.h>
     49 #include <errno.h>
     50 
     51 #include "mathimpl.h"
     52 
     53 /* Log gamma function.
     54  * Error:  x > 0 error < 1.3ulp.
     55  *	   x > 4, error < 1ulp.
     56  *	   x > 9, error < .6ulp.
     57  * 	   x < 0, all bets are off. (When G(x) ~ 1, log(G(x)) ~ 0)
     58  * Method:
     59  *	x > 6:
     60  *		Use the asymptotic expansion (Stirling's Formula)
     61  *	0 < x < 6:
     62  *		Use gamma(x+1) = x*gamma(x) for argument reduction.
     63  *		Use rational approximation in
     64  *		the range 1.2, 2.5
     65  *		Two approximations are used, one centered at the
     66  *		minimum to ensure monotonicity; one centered at 2
     67  *		to maintain small relative error.
     68  *	x < 0:
     69  *		Use the reflection formula,
     70  *		G(1-x)G(x) = PI/sin(PI*x)
     71  * Special values:
     72  *	non-positive integer	returns +Inf.
     73  *	NaN			returns NaN
     74 */
     75 static int endian;
     76 #if defined(__vax__) || defined(tahoe)
     77 #define _IEEE		0
     78 /* double and float have same size exponent field */
     79 #define TRUNC(x)	x = (double) (float) (x)
     80 #else
     81 #define _IEEE		1
     82 #define TRUNC(x)	*(((int *) &x) + endian) &= 0xf8000000
     83 #define infnan(x)	0.0
     84 #endif
     85 
     86 static double small_lgam(double);
     87 static double large_lgam(double);
     88 static double neg_lgam(double);
     89 static double one = 1.0;
     90 int signgam;
     91 
     92 #define UNDERFL (1e-1020 * 1e-1020)
     93 
     94 #define LEFT	(1.0 - (x0 + .25))
     95 #define RIGHT	(x0 - .218)
     96 /*
     97  * Constants for approximation in [1.244,1.712]
     98 */
     99 #define x0	0.461632144968362356785
    100 #define x0_lo	-.000000000000000015522348162858676890521
    101 #define a0_hi	-0.12148629128932952880859
    102 #define a0_lo	.0000000007534799204229502
    103 #define r0	-2.771227512955130520e-002
    104 #define r1	-2.980729795228150847e-001
    105 #define r2	-3.257411333183093394e-001
    106 #define r3	-1.126814387531706041e-001
    107 #define r4	-1.129130057170225562e-002
    108 #define r5	-2.259650588213369095e-005
    109 #define s0	 1.714457160001714442e+000
    110 #define s1	 2.786469504618194648e+000
    111 #define s2	 1.564546365519179805e+000
    112 #define s3	 3.485846389981109850e-001
    113 #define s4	 2.467759345363656348e-002
    114 /*
    115  * Constants for approximation in [1.71, 2.5]
    116 */
    117 #define a1_hi	4.227843350984671344505727574870e-01
    118 #define a1_lo	4.670126436531227189e-18
    119 #define p0	3.224670334241133695662995251041e-01
    120 #define p1	3.569659696950364669021382724168e-01
    121 #define p2	1.342918716072560025853732668111e-01
    122 #define p3	1.950702176409779831089963408886e-02
    123 #define p4	8.546740251667538090796227834289e-04
    124 #define q0	1.000000000000000444089209850062e+00
    125 #define q1	1.315850076960161985084596381057e+00
    126 #define q2	6.274644311862156431658377186977e-01
    127 #define q3	1.304706631926259297049597307705e-01
    128 #define q4	1.102815279606722369265536798366e-02
    129 #define q5	2.512690594856678929537585620579e-04
    130 #define q6	-1.003597548112371003358107325598e-06
    131 /*
    132  * Stirling's Formula, adjusted for equal-ripple. x in [6,Inf].
    133 */
    134 #define lns2pi	.418938533204672741780329736405
    135 #define pb0	 8.33333333333333148296162562474e-02
    136 #define pb1	-2.77777777774548123579378966497e-03
    137 #define pb2	 7.93650778754435631476282786423e-04
    138 #define pb3	-5.95235082566672847950717262222e-04
    139 #define pb4	 8.41428560346653702135821806252e-04
    140 #define pb5	-1.89773526463879200348872089421e-03
    141 #define pb6	 5.69394463439411649408050664078e-03
    142 #define pb7	-1.44705562421428915453880392761e-02
    143 
    144 __pure double
    145 lgamma(double x)
    146 {
    147 	double r;
    148 
    149 	signgam = 1;
    150 	endian = ((*(int *) &one)) ? 1 : 0;
    151 
    152 	if (!finite(x)) {
    153 		if (_IEEE)
    154 			return (x+x);
    155 		else return (infnan(EDOM));
    156 	}
    157 
    158 	if (x > 6 + RIGHT) {
    159 		r = large_lgam(x);
    160 		return (r);
    161 	} else if (x > 1e-16)
    162 		return (small_lgam(x));
    163 	else if (x > -1e-16) {
    164 		if (x < 0)
    165 			signgam = -1, x = -x;
    166 		return (-log(x));
    167 	} else
    168 		return (neg_lgam(x));
    169 }
    170 
    171 static double
    172 large_lgam(double x)
    173 {
    174 	double z, p, x1;
    175 	struct Double t, u, v;
    176 	u = __log__D(x);
    177 	u.a -= 1.0;
    178 	if (x > 1e15) {
    179 		v.a = x - 0.5;
    180 		TRUNC(v.a);
    181 		v.b = (x - v.a) - 0.5;
    182 		t.a = u.a*v.a;
    183 		t.b = x*u.b + v.b*u.a;
    184 		if (_IEEE == 0 && !finite(t.a))
    185 			return(infnan(ERANGE));
    186 		return(t.a + t.b);
    187 	}
    188 	x1 = 1./x;
    189 	z = x1*x1;
    190 	p = pb0+z*(pb1+z*(pb2+z*(pb3+z*(pb4+z*(pb5+z*(pb6+z*pb7))))));
    191 					/* error in approximation = 2.8e-19 */
    192 
    193 	p = p*x1;			/* error < 2.3e-18 absolute */
    194 					/* 0 < p < 1/64 (at x = 5.5) */
    195 	v.a = x = x - 0.5;
    196 	TRUNC(v.a);			/* truncate v.a to 26 bits. */
    197 	v.b = x - v.a;
    198 	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
    199 	t.b = v.b*u.a + x*u.b;
    200 	t.b += p; t.b += lns2pi;	/* return t + lns2pi + p */
    201 	return (t.a + t.b);
    202 }
    203 
    204 static double
    205 small_lgam(double x)
    206 {
    207 	int x_int;
    208 	double y, z, t, r = 0, p, q, hi, lo;
    209 	struct Double rr;
    210 	x_int = (x + .5);
    211 	y = x - x_int;
    212 	if (x_int <= 2 && y > RIGHT) {
    213 		t = y - x0;
    214 		y--; x_int++;
    215 		goto CONTINUE;
    216 	} else if (y < -LEFT) {
    217 		t = y +(1.0-x0);
    218 CONTINUE:
    219 		z = t - x0_lo;
    220 		p = r0+z*(r1+z*(r2+z*(r3+z*(r4+z*r5))));
    221 		q = s0+z*(s1+z*(s2+z*(s3+z*s4)));
    222 		r = t*(z*(p/q) - x0_lo);
    223 		t = .5*t*t;
    224 		z = 1.0;
    225 		switch (x_int) {
    226 		case 6:	z  = (y + 5);
    227 		case 5:	z *= (y + 4);
    228 		case 4:	z *= (y + 3);
    229 		case 3:	z *= (y + 2);
    230 			rr = __log__D(z);
    231 			rr.b += a0_lo; rr.a += a0_hi;
    232 			return(((r+rr.b)+t+rr.a));
    233 		case 2: return(((r+a0_lo)+t)+a0_hi);
    234 		case 0: r -= log1p(x);
    235 		default: rr = __log__D(x);
    236 			rr.a -= a0_hi; rr.b -= a0_lo;
    237 			return(((r - rr.b) + t) - rr.a);
    238 		}
    239 	} else {
    240 		p = p0+y*(p1+y*(p2+y*(p3+y*p4)));
    241 		q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*q6)))));
    242 		p = p*(y/q);
    243 		t = (double)(float) y;
    244 		z = y-t;
    245 		hi = (double)(float) (p+a1_hi);
    246 		lo = a1_hi - hi; lo += p; lo += a1_lo;
    247 		r = lo*y + z*hi;	/* q + r = y*(a0+p/q) */
    248 		q = hi*t;
    249 		z = 1.0;
    250 		switch (x_int) {
    251 		case 6:	z  = (y + 5);
    252 		case 5:	z *= (y + 4);
    253 		case 4:	z *= (y + 3);
    254 		case 3:	z *= (y + 2);
    255 			rr = __log__D(z);
    256 			r += rr.b; r += q;
    257 			return(rr.a + r);
    258 		case 2:	return (q+ r);
    259 		case 0: rr = __log__D(x);
    260 			r -= rr.b; r -= log1p(x);
    261 			r += q; r-= rr.a;
    262 			return(r);
    263 		default: rr = __log__D(x);
    264 			r -= rr.b;
    265 			q -= rr.a;
    266 			return (r+q);
    267 		}
    268 	}
    269 }
    270 
    271 static double
    272 neg_lgam(double x)
    273 {
    274 	int xi;
    275 	double y, z, one = 1.0, zero = 0.0;
    276 
    277 	/* avoid destructive cancellation as much as possible */
    278 	if (x > -170) {
    279 		xi = x;
    280 		if (xi == x) {
    281 			if (_IEEE)
    282 				return(one/zero);
    283 			else
    284 				return(infnan(ERANGE));
    285 		}
    286 		y = gamma(x);
    287 		if (y < 0)
    288 			y = -y, signgam = -1;
    289 		return (log(y));
    290 	}
    291 	z = floor(x + .5);
    292 	if (z == x) {		/* convention: G(-(integer)) -> +Inf */
    293 		if (_IEEE)
    294 			return (one/zero);
    295 		else
    296 			return (infnan(ERANGE));
    297 	}
    298 	y = .5*ceil(x);
    299 	if (y == ceil(y))
    300 		signgam = -1;
    301 	x = -x;
    302 	z = fabs(x + z);	/* 0 < z <= .5 */
    303 	if (z < .25)
    304 		z = sin(M_PI*z);
    305 	else
    306 		z = cos(M_PI*(0.5-z));
    307 	z = log(M_PI/(z*x));
    308 	y = large_lgam(x);
    309 	return (z - y);
    310 }
    311