text: /*	$NetBSD: b_tgamma.c,v 1.1 2012/05/05 17:54:14 christos Exp $	*/
  
  /*-
   * Copyright (c) 1992, 1993
   *	The Regents of the University of California.  All rights reserved.
   *
   * Redistribution and use in source and binary forms, with or without
   * modification, are permitted provided that the following conditions
   * are met:
   * 1. Redistributions of source code must retain the above copyright
   *    notice, this list of conditions and the following disclaimer.
   * 2. Redistributions in binary form must reproduce the above copyright
   *    notice, this list of conditions and the following disclaimer in the
   *    documentation and/or other materials provided with the distribution.
   * 3. All advertising materials mentioning features or use of this software
   *    must display the following acknowledgement:
   *	This product includes software developed by the University of
   *	California, Berkeley and its contributors.
   * 4. Neither the name of the University nor the names of its contributors
   *    may be used to endorse or promote products derived from this software
   *    without specific prior written permission.
   *
   * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
   * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
   * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
   * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
   * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
   * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
   * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
   * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
   * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
   * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
   * SUCH DAMAGE.
   */
  
  /* @(#)gamma.c	8.1 (Berkeley) 6/4/93 */
  #include <sys/cdefs.h>
  #if 0
  __FBSDID("$FreeBSD: release/9.0.0/lib/msun/bsdsrc/b_tgamma.c 176449 2008-02-22 02:26:51Z das $");
  #else
  __RCSID("$NetBSD: b_tgamma.c,v 1.1 2012/05/05 17:54:14 christos Exp $");
  #endif
  
  /*
   * This code by P. McIlroy, Oct 1992;
   *
   * The financial support of UUNET Communications Services is greatfully
   * acknowledged.
   */
  
  #include "math.h"
  #include "math_private.h"
  
  /* METHOD:
   * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
   * 	At negative integers, return NaN and raise invalid.
   *
   * x < 6.5:
   *	Use argument reduction G(x+1) = xG(x) to reach the
   *	range [1.066124,2.066124].  Use a rational
   *	approximation centered at the minimum (x0+1) to
   *	ensure monotonicity.
   *
   * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
   *	adjusted for equal-ripples:
   *
   *	log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
   *
   *	Keep extra precision in multiplying (x-.5)(log(x)-1), to
   *	avoid premature round-off.
   *
   * Special values:
   *	-Inf:			return NaN and raise invalid;
   *	negative integer:	return NaN and raise invalid;
   *	other x ~< 177.79:	return +-0 and raise underflow;
   *	+-0:			return +-Inf and raise divide-by-zero;
   *	finite x ~> 171.63:	return +Inf and raise overflow;
   *	+Inf:			return +Inf;
   *	NaN: 			return NaN.
   *
   * Accuracy: tgamma(x) is accurate to within
   *	x > 0:  error provably < 0.9ulp.
   *	Maximum observed in 1,000,000 trials was .87ulp.
   *	x < 0:
   *	Maximum observed error < 4ulp in 1,000,000 trials.
   */
  
  static double neg_gam(double);
  static double small_gam(double);
  static double smaller_gam(double);
  static struct Double large_gam(double);
  static struct Double ratfun_gam(double, double);
  
  /*
   * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
   * [1.066.., 2.066..] accurate to 4.25e-19.
   */
  #define LEFT -.3955078125	/* left boundary for rat. approx */
  #define x0 .461632144968362356785	/* xmin - 1 */
  
  #define a0_hi 0.88560319441088874992
  #define a0_lo -.00000000000000004996427036469019695
  #define P0	 6.21389571821820863029017800727e-01
  #define P1	 2.65757198651533466104979197553e-01
  #define P2	 5.53859446429917461063308081748e-03
  #define P3	 1.38456698304096573887145282811e-03
  #define P4	 2.40659950032711365819348969808e-03
  #define Q0	 1.45019531250000000000000000000e+00
  #define Q1	 1.06258521948016171343454061571e+00
  #define Q2	-2.07474561943859936441469926649e-01
  #define Q3	-1.46734131782005422506287573015e-01
  #define Q4	 3.07878176156175520361557573779e-02
  #define Q5	 5.12449347980666221336054633184e-03
  #define Q6	-1.76012741431666995019222898833e-03
  #define Q7	 9.35021023573788935372153030556e-05
  #define Q8	 6.13275507472443958924745652239e-06
  /*
   * Constants for large x approximation (x in [6, Inf])
   * (Accurate to 2.8*10^-19 absolute)
   */
  #define lns2pi_hi 0.418945312500000
  #define lns2pi_lo -.000006779295327258219670263595
  #define Pa0	 8.33333333333333148296162562474e-02
  #define Pa1	-2.77777777774548123579378966497e-03
  #define Pa2	 7.93650778754435631476282786423e-04
  #define Pa3	-5.95235082566672847950717262222e-04
  #define Pa4	 8.41428560346653702135821806252e-04
  #define Pa5	-1.89773526463879200348872089421e-03
  #define Pa6	 5.69394463439411649408050664078e-03
  #define Pa7	-1.44705562421428915453880392761e-02
  
  static const double zero = 0., one = 1.0, tiny = 1e-300;
  
  double
  tgamma(double x)
  {
  	struct Double u;
  
  	if (x >= 6) {
  		if(x > 171.63)
  			return (x / zero);
  		u = large_gam(x);
  		return(__exp__D(u.a, u.b));
  	} else if (x >= 1.0 + LEFT + x0)
  		return (small_gam(x));
  	else if (x > 1.e-17)
  		return (smaller_gam(x));
  	else if (x > -1.e-17) {
  		if (x != 0.0)
  			u.a = one - tiny;	/* raise inexact */
  		return (one/x);
  	} else if (!finite(x))
  		return (x - x);		/* x is NaN or -Inf */
  	else
  		return (neg_gam(x));
  }
  /*
   * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
   */
  static struct Double
  large_gam(double x)
  {
  	double z, p;
  	struct Double t, u, v;
  
  	z = one/(x*x);
  	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
  	p = p/x;
  
  	u = __log__D(x);
  	u.a -= one;
  	v.a = (x -= .5);
  	TRUNC(v.a);
  	v.b = x - v.a;
  	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
  	t.b = v.b*u.a + x*u.b;
  	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
  	t.b += lns2pi_lo; t.b += p;
  	u.a = lns2pi_hi + t.b; u.a += t.a;
  	u.b = t.a - u.a;
  	u.b += lns2pi_hi; u.b += t.b;
  	return (u);
  }
  /*
   * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
   * It also has correct monotonicity.
   */
  static double
  small_gam(double x)
  {
  	double y, ym1, t;
  	struct Double yy, r;
  	y = x - one;
  	ym1 = y - one;
  	if (y <= 1.0 + (LEFT + x0)) {
  		yy = ratfun_gam(y - x0, 0);
  		return (yy.a + yy.b);
  	}
  	r.a = y;
  	TRUNC(r.a);
  	yy.a = r.a - one;
  	y = ym1;
  	yy.b = r.b = y - yy.a;
  	/* Argument reduction: G(x+1) = x*G(x) */
  	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
  		t = r.a*yy.a;
  		r.b = r.a*yy.b + y*r.b;
  		r.a = t;
  		TRUNC(r.a);
  		r.b += (t - r.a);
  	}
  	/* Return r*tgamma(y). */
  	yy = ratfun_gam(y - x0, 0);
  	y = r.b*(yy.a + yy.b) + r.a*yy.b;
  	y += yy.a*r.a;
  	return (y);
  }
  /*
   * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
   */
  static double
  smaller_gam(double x)
  {
  	double t, d;
  	struct Double r, xx;
  	if (x < x0 + LEFT) {
  		t = x, TRUNC(t);
  		d = (t+x)*(x-t);
  		t *= t;
  		xx.a = (t + x), TRUNC(xx.a);
  		xx.b = x - xx.a; xx.b += t; xx.b += d;
  		t = (one-x0); t += x;
  		d = (one-x0); d -= t; d += x;
  		x = xx.a + xx.b;
  	} else {
  		xx.a =  x, TRUNC(xx.a);
  		xx.b = x - xx.a;
  		t = x - x0;
  		d = (-x0 -t); d += x;
  	}
  	r = ratfun_gam(t, d);
  	d = r.a/x, TRUNC(d);
  	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
  	return (d + r.a/x);
  }
  /*
   * returns (z+c)^2 * P(z)/Q(z) + a0
   */
  static struct Double
  ratfun_gam(double z, double c)
  {
  	double p, q;
  	struct Double r, t;
  
  	q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
  	p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
  
  	/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
  	p = p/q;
  	t.a = z, TRUNC(t.a);		/* t ~= z + c */
  	t.b = (z - t.a) + c;
  	t.b *= (t.a + z);
  	q = (t.a *= t.a);		/* t = (z+c)^2 */
  	TRUNC(t.a);
  	t.b += (q - t.a);
  	r.a = p, TRUNC(r.a);		/* r = P/Q */
  	r.b = p - r.a;
  	t.b = t.b*p + t.a*r.b + a0_lo;
  	t.a *= r.a;			/* t = (z+c)^2*(P/Q) */
  	r.a = t.a + a0_hi, TRUNC(r.a);
  	r.b = ((a0_hi-r.a) + t.a) + t.b;
  	return (r);			/* r = a0 + t */
  }
  
  static double
  neg_gam(double x)
  {
  	int sgn = 1;
  	struct Double lg, lsine;
  	double y, z;
  
  	y = ceil(x);
  	if (y == x)		/* Negative integer. */
  		return ((x - x) / zero);
  	z = y - x;
  	if (z > 0.5)
  		z = one - z;
  	y = 0.5 * y;
  	if (y == ceil(y))
  		sgn = -1;
  	if (z < .25)
  		z = sin(M_PI*z);
  	else
  		z = cos(M_PI*(0.5-z));
  	/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
  	if (x < -170) {
  		if (x < -190)
  			return ((double)sgn*tiny*tiny);
  		y = one - x;		/* exact: 128 < |x| < 255 */
  		lg = large_gam(y);
  		lsine = __log__D(M_PI/z);	/* = TRUNC(log(u)) + small */
  		lg.a -= lsine.a;		/* exact (opposite signs) */
  		lg.b -= lsine.b;
  		y = -(lg.a + lg.b);
  		z = (y + lg.a) + lg.b;
  		y = __exp__D(y, z);
  		if (sgn < 0) y = -y;
  		return (y);
  	}
  	y = one-x;
  	if (one-y == x)
  		y = tgamma(y);
  	else		/* 1-x is inexact */
  		y = -x*tgamma(-x);
  	if (sgn < 0) y = -y;
  	return (M_PI / (y*z));
  }
  