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      1 /*      $NetBSD: n_gamma.c,v 1.9 2013/11/09 21:41:03 christos 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. Neither the name of the University nor the names of its contributors
     15  *    may be used to endorse or promote products derived from this software
     16  *    without specific prior written permission.
     17  *
     18  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
     19  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
     20  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
     21  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
     22  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
     23  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
     24  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
     25  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
     26  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
     27  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
     28  * SUCH DAMAGE.
     29  */
     30 
     31 #ifndef lint
     32 #if 0
     33 static char sccsid[] = "@(#)gamma.c	8.1 (Berkeley) 6/4/93";
     34 #endif
     35 #endif /* not lint */
     36 
     37 /*
     38  * This code by P. McIlroy, Oct 1992;
     39  *
     40  * The financial support of UUNET Communications Services is gratefully
     41  * acknowledged.
     42  */
     43 
     44 #include <math.h>
     45 #include "mathimpl.h"
     46 #include <errno.h>
     47 
     48 /* METHOD:
     49  * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
     50  * 	At negative integers, return +Inf, and set errno.
     51  *
     52  * x < 6.5:
     53  *	Use argument reduction G(x+1) = xG(x) to reach the
     54  *	range [1.066124,2.066124].  Use a rational
     55  *	approximation centered at the minimum (x0+1) to
     56  *	ensure monotonicity.
     57  *
     58  * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
     59  *	adjusted for equal-ripples:
     60  *
     61  *	log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
     62  *
     63  *	Keep extra precision in multiplying (x-.5)(log(x)-1), to
     64  *	avoid premature round-off.
     65  *
     66  * Special values:
     67  *	non-positive integer:	Set overflow trap; return +Inf;
     68  *	x > 171.63:		Set overflow trap; return +Inf;
     69  *	NaN: 			Set invalid trap;  return NaN
     70  *
     71  * Accuracy: Gamma(x) is accurate to within
     72  *	x > 0:  error provably < 0.9ulp.
     73  *	Maximum observed in 1,000,000 trials was .87ulp.
     74  *	x < 0:
     75  *	Maximum observed error < 4ulp in 1,000,000 trials.
     76  */
     77 
     78 static double neg_gam (double);
     79 static double small_gam (double);
     80 static double smaller_gam (double);
     81 static struct Double large_gam (double);
     82 static struct Double ratfun_gam (double, double);
     83 
     84 /*
     85  * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
     86  * [1.066.., 2.066..] accurate to 4.25e-19.
     87  */
     88 #define LEFT -.3955078125	/* left boundary for rat. approx */
     89 #define x0 .461632144968362356785	/* xmin - 1 */
     90 
     91 #define a0_hi 0.88560319441088874992
     92 #define a0_lo -.00000000000000004996427036469019695
     93 #define P0	 6.21389571821820863029017800727e-01
     94 #define P1	 2.65757198651533466104979197553e-01
     95 #define P2	 5.53859446429917461063308081748e-03
     96 #define P3	 1.38456698304096573887145282811e-03
     97 #define P4	 2.40659950032711365819348969808e-03
     98 #define Q0	 1.45019531250000000000000000000e+00
     99 #define Q1	 1.06258521948016171343454061571e+00
    100 #define Q2	-2.07474561943859936441469926649e-01
    101 #define Q3	-1.46734131782005422506287573015e-01
    102 #define Q4	 3.07878176156175520361557573779e-02
    103 #define Q5	 5.12449347980666221336054633184e-03
    104 #define Q6	-1.76012741431666995019222898833e-03
    105 #define Q7	 9.35021023573788935372153030556e-05
    106 #define Q8	 6.13275507472443958924745652239e-06
    107 /*
    108  * Constants for large x approximation (x in [6, Inf])
    109  * (Accurate to 2.8*10^-19 absolute)
    110  */
    111 #define lns2pi_hi 0.418945312500000
    112 #define lns2pi_lo -.000006779295327258219670263595
    113 #define Pa0	 8.33333333333333148296162562474e-02
    114 #define Pa1	-2.77777777774548123579378966497e-03
    115 #define Pa2	 7.93650778754435631476282786423e-04
    116 #define Pa3	-5.95235082566672847950717262222e-04
    117 #define Pa4	 8.41428560346653702135821806252e-04
    118 #define Pa5	-1.89773526463879200348872089421e-03
    119 #define Pa6	 5.69394463439411649408050664078e-03
    120 #define Pa7	-1.44705562421428915453880392761e-02
    121 
    122 static const double zero = 0., one = 1.0, tiny = _TINY;
    123 /*
    124  * TRUNC sets trailing bits in a floating-point number to zero.
    125  * is a temporary variable.
    126  */
    127 #if defined(__vax__) || defined(tahoe)
    128 #define _IEEE		0
    129 #define TRUNC(x)	x = (double) (float) (x)
    130 #else
    131 static int endian;
    132 #define _IEEE		1
    133 #define TRUNC(x)	*(((int *) &x) + endian) &= 0xf8000000
    134 #define infnan(x)	0.0
    135 #endif
    136 
    137 double
    138 gamma(double x)
    139 {
    140 	double b;
    141 	struct Double u;
    142 #if _IEEE
    143 	int endian = (*(int *) &one) ? 1 : 0;
    144 #endif
    145 
    146 	if (x >= 6) {
    147 		if(x > 171.63)
    148 			return(one/zero);
    149 		u = large_gam(x);
    150 		return(__exp__D(u.a, u.b));
    151 	} else if (x >= 1.0 + LEFT + x0) {
    152 		return (small_gam(x));
    153 	} else if (x > 1.e-17) {
    154 		return (smaller_gam(x));
    155 	} else if (x > -1.e-17) {
    156 		if (x == 0.0) {
    157 			if (!_IEEE) return (infnan(ERANGE));
    158 			else return (one/x);
    159 		}
    160 		b =one+1e-20;		/* Raise inexact flag. ??? -ragge */
    161 		__USE(b);
    162 		return (one/x);
    163 	} else if (!finite(x)) {
    164 		if (_IEEE)		/* x = NaN, -Inf */
    165 			return (x*x);
    166 		else
    167 			return (infnan(EDOM));
    168 	 } else
    169 		return (neg_gam(x));
    170 }
    171 /*
    172  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
    173  */
    174 static struct Double
    175 large_gam(double x)
    176 {
    177 	double z, p;
    178 	struct Double t, u, v;
    179 
    180 	z = one/(x*x);
    181 	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
    182 	p = p/x;
    183 
    184 	u = __log__D(x);
    185 	u.a -= one;
    186 	v.a = (x -= .5);
    187 	TRUNC(v.a);
    188 	v.b = x - v.a;
    189 	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
    190 	t.b = v.b*u.a + x*u.b;
    191 	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
    192 	t.b += lns2pi_lo; t.b += p;
    193 	u.a = lns2pi_hi + t.b; u.a += t.a;
    194 	u.b = t.a - u.a;
    195 	u.b += lns2pi_hi; u.b += t.b;
    196 	return (u);
    197 }
    198 /*
    199  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
    200  * It also has correct monotonicity.
    201  */
    202 static double
    203 small_gam(double x)
    204 {
    205 	double y, ym1, t;
    206 	struct Double yy, r;
    207 	y = x - one;
    208 	ym1 = y - one;
    209 	if (y <= 1.0 + (LEFT + x0)) {
    210 		yy = ratfun_gam(y - x0, 0);
    211 		return (yy.a + yy.b);
    212 	}
    213 	r.a = y;
    214 	TRUNC(r.a);
    215 	yy.a = r.a - one;
    216 	y = ym1;
    217 	yy.b = r.b = y - yy.a;
    218 	/* Argument reduction: G(x+1) = x*G(x) */
    219 	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
    220 		t = r.a*yy.a;
    221 		r.b = r.a*yy.b + y*r.b;
    222 		r.a = t;
    223 		TRUNC(r.a);
    224 		r.b += (t - r.a);
    225 	}
    226 	/* Return r*gamma(y). */
    227 	yy = ratfun_gam(y - x0, 0);
    228 	y = r.b*(yy.a + yy.b) + r.a*yy.b;
    229 	y += yy.a*r.a;
    230 	return (y);
    231 }
    232 /*
    233  * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
    234  */
    235 static double
    236 smaller_gam(double x)
    237 {
    238 	double t, d;
    239 	struct Double r, xx;
    240 	if (x < x0 + LEFT) {
    241 		t = x, TRUNC(t);
    242 		d = (t+x)*(x-t);
    243 		t *= t;
    244 		xx.a = (t + x), TRUNC(xx.a);
    245 		xx.b = x - xx.a; xx.b += t; xx.b += d;
    246 		t = (one-x0); t += x;
    247 		d = (one-x0); d -= t; d += x;
    248 		x = xx.a + xx.b;
    249 	} else {
    250 		xx.a =  x, TRUNC(xx.a);
    251 		xx.b = x - xx.a;
    252 		t = x - x0;
    253 		d = (-x0 -t); d += x;
    254 	}
    255 	r = ratfun_gam(t, d);
    256 	d = r.a/x, TRUNC(d);
    257 	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
    258 	return (d + r.a/x);
    259 }
    260 /*
    261  * returns (z+c)^2 * P(z)/Q(z) + a0
    262  */
    263 static struct Double
    264 ratfun_gam(double z, double c)
    265 {
    266 	double p, q;
    267 	struct Double r, t;
    268 
    269 	q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
    270 	p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
    271 
    272 	/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
    273 	p = p/q;
    274 	t.a = z, TRUNC(t.a);		/* t ~= z + c */
    275 	t.b = (z - t.a) + c;
    276 	t.b *= (t.a + z);
    277 	q = (t.a *= t.a);		/* t = (z+c)^2 */
    278 	TRUNC(t.a);
    279 	t.b += (q - t.a);
    280 	r.a = p, TRUNC(r.a);		/* r = P/Q */
    281 	r.b = p - r.a;
    282 	t.b = t.b*p + t.a*r.b + a0_lo;
    283 	t.a *= r.a;			/* t = (z+c)^2*(P/Q) */
    284 	r.a = t.a + a0_hi, TRUNC(r.a);
    285 	r.b = ((a0_hi-r.a) + t.a) + t.b;
    286 	return (r);			/* r = a0 + t */
    287 }
    288 
    289 static double
    290 neg_gam(double x)
    291 {
    292 	int sgn = 1;
    293 	struct Double lg, lsine;
    294 	double y, z;
    295 
    296 	y = floor(x + .5);
    297 	if (y == x) {		/* Negative integer. */
    298 		if(!_IEEE)
    299 			return (infnan(ERANGE));
    300 		else
    301 			return (one/zero);
    302 	}
    303 	z = fabs(x - y);
    304 	y = .5*ceil(x);
    305 	if (y == ceil(y))
    306 		sgn = -1;
    307 	if (z < .25)
    308 		z = sin(M_PI*z);
    309 	else
    310 		z = cos(M_PI*(0.5-z));
    311 	/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
    312 	if (x < -170) {
    313 		if (x < -190)
    314 			return ((double)sgn*tiny*tiny);
    315 		y = one - x;		/* exact: 128 < |x| < 255 */
    316 		lg = large_gam(y);
    317 		lsine = __log__D(M_PI/z);	/* = TRUNC(log(u)) + small */
    318 		lg.a -= lsine.a;		/* exact (opposite signs) */
    319 		lg.b -= lsine.b;
    320 		y = -(lg.a + lg.b);
    321 		z = (y + lg.a) + lg.b;
    322 		y = __exp__D(y, z);
    323 		if (sgn < 0) y = -y;
    324 		return (y);
    325 	}
    326 	y = one-x;
    327 	if (one-y == x)
    328 		y = gamma(y);
    329 	else		/* 1-x is inexact */
    330 		y = -x*gamma(-x);
    331 	if (sgn < 0) y = -y;
    332 	return (M_PI / (y*z));
    333 }
    334