/*      $OpenBSD: b_tgamma.c,v 1.10 2016/09/12 19:47:02 guenther 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 <float.h>
#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 (!isfinite(x)) {
                return (x - x);                 /* x = NaN, -Inf */
         } else
                return (neg_gam(x));
}
DEF_STD(tgamma);
LDBL_MAYBE_UNUSED_CLONE(tgamma);

/*
 * 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));
}