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lgamma.c

#include "FEATURE/uwin"

#if !_UWIN || _lib_lgamma

void _STUB_lgamma(){}

#else

/*-
 * 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. 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.
 */

#ifndef lint
static char sccsid[] = "@(#)lgamma.c      8.2 (Berkeley) 11/30/93";
#endif /* not lint */

/*
 * Coded by Peter McIlroy, Nov 1992;
 *
 * The financial support of UUNET Communications Services is greatfully
 * acknowledged.
 */

#define gamma     ______gamma
#define lgamma    ______lgamma

#include <math.h>
#include <errno.h>
#include "mathimpl.h"

#undef      gamma
#undef      lgamma

/* Log gamma function.
 * Error:  x > 0 error < 1.3ulp.
 *       x > 4, error < 1ulp.
 *       x > 9, error < .6ulp.
 *       x < 0, all bets are off. (When G(x) ~ 1, log(G(x)) ~ 0)
 * Method:
 *    x > 6:
 *          Use the asymptotic expansion (Stirling's Formula)
 *    0 < x < 6:
 *          Use gamma(x+1) = x*gamma(x) for argument reduction.
 *          Use rational approximation in
 *          the range 1.2, 2.5
 *          Two approximations are used, one centered at the
 *          minimum to ensure monotonicity; one centered at 2
 *          to maintain small relative error.
 *    x < 0:
 *          Use the reflection formula,
 *          G(1-x)G(x) = PI/sin(PI*x)
 * Special values:
 *    non-positive integer    returns +Inf.
 *    NaN               returns NaN
*/
static int endian;
#if defined(vax) || defined(tahoe)
#define _IEEE           0
/* double and float have same size exponent field */
#define TRUNC(x)  x = (double) (float) (x)
#else
#define _IEEE           1
#define TRUNC(x)  *(((int *) &x) + endian) &= 0xf8000000
#define infnan(x) 0.0
#endif

static double small_lgam(double);
static double large_lgam(double);
static double neg_lgam(double);
static double zero = 0.0, one = 1.0;
int signgam;

#define UNDERFL (1e-1020 * 1e-1020)

#define LEFT      (1.0 - (x0 + .25))
#define RIGHT     (x0 - .218)
/*
/* Constants for approximation in [1.244,1.712]
*/
#define x0  0.461632144968362356785
#define x0_lo     -.000000000000000015522348162858676890521
#define a0_hi     -0.12148629128932952880859
#define a0_lo     .0000000007534799204229502
#define r0  -2.771227512955130520e-002
#define r1  -2.980729795228150847e-001
#define r2  -3.257411333183093394e-001
#define r3  -1.126814387531706041e-001
#define r4  -1.129130057170225562e-002
#define r5  -2.259650588213369095e-005
#define s0   1.714457160001714442e+000
#define s1   2.786469504618194648e+000
#define s2   1.564546365519179805e+000
#define s3   3.485846389981109850e-001
#define s4   2.467759345363656348e-002
/*
 * Constants for approximation in [1.71, 2.5]
*/
#define a1_hi     4.227843350984671344505727574870e-01
#define a1_lo     4.670126436531227189e-18
#define p0  3.224670334241133695662995251041e-01
#define p1  3.569659696950364669021382724168e-01
#define p2  1.342918716072560025853732668111e-01
#define p3  1.950702176409779831089963408886e-02
#define p4  8.546740251667538090796227834289e-04
#define q0  1.000000000000000444089209850062e+00
#define q1  1.315850076960161985084596381057e+00
#define q2  6.274644311862156431658377186977e-01
#define q3  1.304706631926259297049597307705e-01
#define q4  1.102815279606722369265536798366e-02
#define q5  2.512690594856678929537585620579e-04
#define q6  -1.003597548112371003358107325598e-06
/*
 * Stirling's Formula, adjusted for equal-ripple. x in [6,Inf].
*/
#define lns2pi    .418938533204672741780329736405
#define pb0  8.33333333333333148296162562474e-02
#define pb1 -2.77777777774548123579378966497e-03
#define pb2  7.93650778754435631476282786423e-04
#define pb3 -5.95235082566672847950717262222e-04
#define pb4  8.41428560346653702135821806252e-04
#define pb5 -1.89773526463879200348872089421e-03
#define pb6  5.69394463439411649408050664078e-03
#define pb7 -1.44705562421428915453880392761e-02

extern __pure double lgamma(double x)
{
      double r;

      signgam = 1;
      endian = ((*(int *) &one)) ? 1 : 0;

      if (!finite(x))
            if (_IEEE)
                  return (x+x);
            else return (infnan(EDOM));

      if (x > 6 + RIGHT) {
            r = large_lgam(x);
            return (r);
      } else if (x > 1e-16)
            return (small_lgam(x));
      else if (x > -1e-16) {
            if (x < 0)
                  signgam = -1, x = -x;
            return (-log(x));
      } else
            return (neg_lgam(x));
}

static double
large_lgam(double x)
{
      double z, p, x1;
      struct Double t, u, v;
      u = __log__D(x);
      u.a -= 1.0;
      if (x > 1e15) {
            v.a = x - 0.5;
            TRUNC(v.a);
            v.b = (x - v.a) - 0.5;
            t.a = u.a*v.a;
            t.b = x*u.b + v.b*u.a;
            if (_IEEE == 0 && !finite(t.a))
                  return(infnan(ERANGE));
            return(t.a + t.b);
      }
      x1 = 1./x;
      z = x1*x1;
      p = pb0+z*(pb1+z*(pb2+z*(pb3+z*(pb4+z*(pb5+z*(pb6+z*pb7))))));
                              /* error in approximation = 2.8e-19 */

      p = p*x1;               /* error < 2.3e-18 absolute */
                              /* 0 < p < 1/64 (at x = 5.5) */
      v.a = x = x - 0.5;
      TRUNC(v.a);             /* truncate v.a to 26 bits. */
      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;
      t.b += p; t.b += lns2pi;      /* return t + lns2pi + p */
      return (t.a + t.b);
}

static double
small_lgam(double x)
{
      int x_int;
      double y, z, t, r = 0, p, q, hi, lo;
      struct Double rr;
      x_int = (int)(x + .5);
      y = x - x_int;
      if (x_int <= 2 && y > RIGHT) {
            t = y - x0;
            y--; x_int++;
            goto CONTINUE;
      } else if (y < -LEFT) {
            t = y +(1.0-x0);
CONTINUE:
            z = t - x0_lo;
            p = r0+z*(r1+z*(r2+z*(r3+z*(r4+z*r5))));
            q = s0+z*(s1+z*(s2+z*(s3+z*s4)));
            r = t*(z*(p/q) - x0_lo);
            t = .5*t*t;
            z = 1.0;
            switch (x_int) {
            case 6:     z  = (y + 5);
            case 5:     z *= (y + 4);
            case 4:     z *= (y + 3);
            case 3:     z *= (y + 2);
                  rr = __log__D(z);
                  rr.b += a0_lo; rr.a += a0_hi;
                  return(((r+rr.b)+t+rr.a));
            case 2: return(((r+a0_lo)+t)+a0_hi);
            case 0: r -= log1p(x);
            default: rr = __log__D(x);
                  rr.a -= a0_hi; rr.b -= a0_lo;
                  return(((r - rr.b) + t) - rr.a);
            }
      } else {
            p = p0+y*(p1+y*(p2+y*(p3+y*p4)));
            q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*q6)))));
            p = p*(y/q);
            t = (double)(float) y;
            z = y-t;
            hi = (double)(float) (p+a1_hi);
            lo = a1_hi - hi; lo += p; lo += a1_lo;
            r = lo*y + z*hi;  /* q + r = y*(a0+p/q) */
            q = hi*t;
            z = 1.0;
            switch (x_int) {
            case 6:     z  = (y + 5);
            case 5:     z *= (y + 4);
            case 4:     z *= (y + 3);
            case 3:     z *= (y + 2);
                  rr = __log__D(z);
                  r += rr.b; r += q;
                  return(rr.a + r);
            case 2:     return (q+ r);
            case 0: rr = __log__D(x);
                  r -= rr.b; r -= log1p(x);
                  r += q; r-= rr.a;
                  return(r);
            default: rr = __log__D(x);
                  r -= rr.b;
                  q -= rr.a;
                  return (r+q);
            }
      }
}

static double
neg_lgam(double x)
{
      int xi;
      double y, z, one = 1.0, zero = 0.0;
      extern double gamma();

      /* avoid destructive cancellation as much as possible */
      if (x > -170) {
            xi = (int)x;
            if (xi == x)
                  if (_IEEE)
                        return(one/zero);
                  else
                        return(infnan(ERANGE));
            y = gamma(x);
            if (y < 0)
                  y = -y, signgam = -1;
            return (log(y));
      }
      z = floor(x + .5);
      if (z == x) {           /* convention: G(-(integer)) -> +Inf */
            if (_IEEE)
                  return (one/zero);
            else
                  return (infnan(ERANGE));
      }
      y = .5*ceil(x);
      if (y == ceil(y))
            signgam = -1;
      x = -x;
      z = fabs(x + z);  /* 0 < z <= .5 */
      if (z < .25)
            z = sin(M_PI*z);
      else
            z = cos(M_PI*(0.5-z));
      z = log(M_PI/(z*x));
      y = large_lgam(x);
      return (z - y);
}

#endif

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