b_tgamma.c

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/*  $OpenBSD: b_tgamma.c,v 1.3 2009/10/27 23:59:29 deraadt 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. 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.
 */

/*
 * 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 = NaN, -Inf */
     } else
        return (neg_gam(x));
}

/*
 * We simply call tgamma() rather than bloating the math library
 * with a float-optimized version of it.  The reason is that tgammaf()
 * is essentially useless, since the function is superexponential
 * and floats have very limited range.  -- das@freebsd.org
 */

float
tgammaf(float x)
{
    return tgamma(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));
}