da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#include "FEATURE/uwin"
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#if !_UWIN || _lib_gamma
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinvoid _STUB_gamma(){}
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#else
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin/*-
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Copyright (c) 1992, 1993
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * The Regents of the University of California. All rights reserved.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin *
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Redistribution and use in source and binary forms, with or without
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * modification, are permitted provided that the following conditions
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * are met:
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * 1. Redistributions of source code must retain the above copyright
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * notice, this list of conditions and the following disclaimer.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * 2. Redistributions in binary form must reproduce the above copyright
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * notice, this list of conditions and the following disclaimer in the
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * documentation and/or other materials provided with the distribution.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * 3. Neither the name of the University nor the names of its contributors
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * may be used to endorse or promote products derived from this software
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * without specific prior written permission.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin *
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * SUCH DAMAGE.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#ifndef lint
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic char sccsid[] = "@(#)gamma.c 8.1 (Berkeley) 6/4/93";
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#endif /* not lint */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin/*
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * This code by P. McIlroy, Oct 1992;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin *
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * The financial support of UUNET Communications Services is greatfully
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * acknowledged.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define gamma ______gamma
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#include <math.h>
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#include <errno.h>
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#include "mathimpl.h"
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#undef gamma
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin/* METHOD:
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * At negative integers, return +Inf, and set errno.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin *
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x < 6.5:
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Use argument reduction G(x+1) = xG(x) to reach the
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * range [1.066124,2.066124]. Use a rational
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * approximation centered at the minimum (x0+1) to
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * ensure monotonicity.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin *
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * adjusted for equal-ripples:
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin *
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin *
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Keep extra precision in multiplying (x-.5)(log(x)-1), to
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * avoid premature round-off.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin *
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Special values:
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * non-positive integer: Set overflow trap; return +Inf;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x > 171.63: Set overflow trap; return +Inf;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * NaN: Set invalid trap; return NaN
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin *
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Accuracy: Gamma(x) is accurate to within
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x > 0: error provably < 0.9ulp.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Maximum observed in 1,000,000 trials was .87ulp.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x < 0:
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Maximum observed error < 4ulp in 1,000,000 trials.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic double neg_gam __P((double));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic double small_gam __P((double));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic double smaller_gam __P((double));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic struct Double large_gam __P((double));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic struct Double ratfun_gam __P((double, double));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin/*
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * [1.066.., 2.066..] accurate to 4.25e-19.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define LEFT -.3955078125 /* left boundary for rat. approx */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define x0 .461632144968362356785 /* xmin - 1 */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define a0_hi 0.88560319441088874992
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define a0_lo -.00000000000000004996427036469019695
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define P0 6.21389571821820863029017800727e-01
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define P1 2.65757198651533466104979197553e-01
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define P2 5.53859446429917461063308081748e-03
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define P3 1.38456698304096573887145282811e-03
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define P4 2.40659950032711365819348969808e-03
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Q0 1.45019531250000000000000000000e+00
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Q1 1.06258521948016171343454061571e+00
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Q2 -2.07474561943859936441469926649e-01
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Q3 -1.46734131782005422506287573015e-01
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Q4 3.07878176156175520361557573779e-02
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Q5 5.12449347980666221336054633184e-03
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Q6 -1.76012741431666995019222898833e-03
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Q7 9.35021023573788935372153030556e-05
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Q8 6.13275507472443958924745652239e-06
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin/*
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Constants for large x approximation (x in [6, Inf])
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * (Accurate to 2.8*10^-19 absolute)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define lns2pi_hi 0.418945312500000
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define lns2pi_lo -.000006779295327258219670263595
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Pa0 8.33333333333333148296162562474e-02
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Pa1 -2.77777777774548123579378966497e-03
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Pa2 7.93650778754435631476282786423e-04
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Pa3 -5.95235082566672847950717262222e-04
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Pa4 8.41428560346653702135821806252e-04
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Pa5 -1.89773526463879200348872089421e-03
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Pa6 5.69394463439411649408050664078e-03
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define Pa7 -1.44705562421428915453880392761e-02
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic const double zero = 0., one = 1.0, tiny = 1e-300;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic int endian;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin/*
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * TRUNC sets trailing bits in a floating-point number to zero.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * is a temporary variable.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#if defined(vax) || defined(tahoe)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define _IEEE 0
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define TRUNC(x) x = (double) (float) (x)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#else
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define _IEEE 1
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define infnan(x) 0.0
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#endif
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinextern double gamma(x)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin{
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin struct Double u;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin endian = (*(int *) &one) ? 1 : 0;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (x >= 6) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if(x > 171.63)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return(one/zero);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin u = large_gam(x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return(__exp__D(u.a, u.b));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin } else if (x >= 1.0 + LEFT + x0)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (small_gam(x));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin else if (x > 1.e-17)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (smaller_gam(x));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin else if (x > -1.e-17) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (x == 0.0)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (!_IEEE) return (infnan(ERANGE));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin else return (one/x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin one+1e-20; /* Raise inexact flag. */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (one/x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin } else if (!finite(x)) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (_IEEE) /* x = NaN, -Inf */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (x*x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin else
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (infnan(EDOM));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin } else
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (neg_gam(x));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin}
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin/*
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic struct Double
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinlarge_gam(x)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin{
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double z, p;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin struct Double t, u, v;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin z = one/(x*x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin p = p/x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin u = __log__D(x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin u.a -= one;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin v.a = (x -= .5);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin TRUNC(v.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin v.b = x - v.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b = v.b*u.a + x*u.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b += lns2pi_lo; t.b += p;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin u.a = lns2pi_hi + t.b; u.a += t.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin u.b = t.a - u.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin u.b += lns2pi_hi; u.b += t.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (u);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin}
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin/*
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * It also has correct monotonicity.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic double
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinsmall_gam(x)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin{
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double y, ym1, t;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin struct Double yy, r;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = x - one;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin ym1 = y - one;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (y <= 1.0 + (LEFT + x0)) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin yy = ratfun_gam(y - x0, 0);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (yy.a + yy.b);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin }
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.a = y;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin TRUNC(r.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin yy.a = r.a - one;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = ym1;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin yy.b = r.b = y - yy.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin /* Argument reduction: G(x+1) = x*G(x) */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t = r.a*yy.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.b = r.a*yy.b + y*r.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.a = t;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin TRUNC(r.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.b += (t - r.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin }
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin /* Return r*gamma(y). */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin yy = ratfun_gam(y - x0, 0);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = r.b*(yy.a + yy.b) + r.a*yy.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y += yy.a*r.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (y);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin}
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin/*
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Good on (0, 1+x0+LEFT]. Accurate to 1ulp.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic double
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinsmaller_gam(x)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin{
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double t, d;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin struct Double r, xx;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (x < x0 + LEFT) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t = x, TRUNC(t);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin d = (t+x)*(x-t);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t *= t;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin xx.a = (t + x), TRUNC(xx.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin xx.b = x - xx.a; xx.b += t; xx.b += d;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t = (one-x0); t += x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin d = (one-x0); d -= t; d += x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin x = xx.a + xx.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin } else {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin xx.a = x, TRUNC(xx.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin xx.b = x - xx.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t = x - x0;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin d = (-x0 -t); d += x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin }
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r = ratfun_gam(t, d);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin d = r.a/x, TRUNC(d);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (d + r.a/x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin}
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin/*
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * returns (z+c)^2 * P(z)/Q(z) + a0
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic struct Double
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinratfun_gam(z, c)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double z, c;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin{
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double p, q;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin struct Double r, t;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin p = p/q;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.a = z, TRUNC(t.a); /* t ~= z + c */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b = (z - t.a) + c;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b *= (t.a + z);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin q = (t.a *= t.a); /* t = (z+c)^2 */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin TRUNC(t.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b += (q - t.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.a = p, TRUNC(r.a); /* r = P/Q */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.b = p - r.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b = t.b*p + t.a*r.b + a0_lo;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.a *= r.a; /* t = (z+c)^2*(P/Q) */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.a = t.a + a0_hi, TRUNC(r.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.b = ((a0_hi-r.a) + t.a) + t.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (r); /* r = a0 + t */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin}
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic double
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinneg_gam(x)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin{
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin int sgn = 1;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin struct Double lg, lsine;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double y, z;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = floor(x + .5);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (y == x) /* Negative integer. */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if(!_IEEE)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (infnan(ERANGE));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin else
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (one/zero);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin z = fabs(x - y);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = .5*ceil(x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (y == ceil(y))
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin sgn = -1;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (z < .25)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin z = sin(M_PI*z);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin else
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin z = cos(M_PI*(0.5-z));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (x < -170) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (x < -190)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return ((double)sgn*tiny*tiny);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = one - x; /* exact: 128 < |x| < 255 */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin lg = large_gam(y);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin lg.a -= lsine.a; /* exact (opposite signs) */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin lg.b -= lsine.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = -(lg.a + lg.b);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin z = (y + lg.a) + lg.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = __exp__D(y, z);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (sgn < 0) y = -y;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (y);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin }
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = one-x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (one-y == x)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = gamma(y);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin else /* 1-x is inexact */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = -x*gamma(-x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (sgn < 0) y = -y;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (M_PI / (y*z));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin}
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#endif