/* specfunc/gamma_inc.c

 *

 * Copyright (C) 2007 Brian Gough

 * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman

 *

 * This program is free software; you can redistribute it and/or modify

 * it under the terms of the GNU General Public License as published by

 * the Free Software Foundation; either version 3 of the License, or (at

 * your option) any later version.

 *

 * This program is distributed in the hope that it will be useful, but

 * WITHOUT ANY WARRANTY; without even the implied warranty of

 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU

 * General Public License for more details.

 *

 * You should have received a copy of the GNU General Public License

 * along with this program; if not, write to the Free Software

 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.

 */

/* Author:  G. Jungman */

#include <config.h>

#include <gsl/gsl_math.h>

#include <gsl/gsl_errno.h>

#include <gsl/gsl_sf_erf.h>

#include <gsl/gsl_sf_exp.h>

#include <gsl/gsl_sf_log.h>

#include <gsl/gsl_sf_gamma.h>

#include <gsl/gsl_sf_expint.h>

#include "error.h"

/* The dominant part,

 * D(a,x) := x^a e^(-x) / Gamma(a+1)

 */

static

int

gamma_inc_D(const double a, const double x, gsl_sf_result * result)

{

  if(a < 10.0) {

    double lnr;

    gsl_sf_result lg;

    gsl_sf_lngamma_e(a+1.0, &lg);

    lnr = a * log(x) - x - lg.val;

    result->val = exp(lnr);

    result->err = 2.0 * GSL_DBL_EPSILON * (fabs(lnr) + 1.0) * fabs(result->val);

    return GSL_SUCCESS;

  }

  else {

    gsl_sf_result gstar;

    gsl_sf_result ln_term;

    double term1;

    if (x < 0.5*a) {

      double u = x/a;   

      double ln_u = log(u);

      ln_term.val = ln_u - u + 1.0;

      ln_term.err = (fabs(ln_u) + fabs(u) + 1.0) * GSL_DBL_EPSILON;

    } else {

      double mu = (x-a)/a;

      gsl_sf_log_1plusx_mx_e(mu, &ln_term);  /* log(1+mu) - mu */

      /* Propagate cancellation error from x-a, since the absolute

         error of mu=x-a is DBL_EPSILON */

      ln_term.err += GSL_DBL_EPSILON * fabs(mu);

    };

    gsl_sf_gammastar_e(a, &gstar);

    term1 = exp(a*ln_term.val)/sqrt(2.0*M_PI*a);

    result->val  = term1/gstar.val;

    result->err  = 2.0 * GSL_DBL_EPSILON * (fabs(a*ln_term.val) + 1.0) * fabs(result->val);

    /* Include propagated error from log term */

    result->err += fabs(a) * ln_term.err * fabs(result->val);

    result->err += gstar.err/fabs(gstar.val) * fabs(result->val);

    return GSL_SUCCESS;

  }

}

/* P series representation.

 */

static

int

gamma_inc_P_series(const double a, const double x, gsl_sf_result * result)

{

  const int nmax = 10000;

  gsl_sf_result D;

  int stat_D = gamma_inc_D(a, x, &D);

  /* Approximating the terms of the series using Stirling's

     approximation gives t_n = (x/a)^n * exp(-n(n+1)/(2a)), so the

     convergence condition is n^2 / (2a) + (1-(x/a) + (1/2a)) n >>

     -log(GSL_DBL_EPS) if we want t_n < O(1e-16) t_0. The condition

     below detects cases where the minimum value of n is > 5000 */

  if (x > 0.995 * a && a > 1e5) { /* Difficult case: try continued fraction */

    gsl_sf_result cf_res;

    int status =  gsl_sf_exprel_n_CF_e(a, x, &cf_res);

    result->val = D.val * cf_res.val;

    result->err = fabs(D.val * cf_res.err) + fabs(D.err * cf_res.val);

    return status;

  }

  /* Series would require excessive number of terms */

  if (x > (a + nmax)) {

    GSL_ERROR ("gamma_inc_P_series x>>a exceeds range", GSL_EMAXITER);

  }

  /* Normal case: sum the series */

  {

    double sum  = 1.0;

    double term = 1.0;

    double remainder;

    int n;

    /* Handle lower part of the series where t_n is increasing, |x| > a+n */

    int nlow = (x > a) ? (x - a): 0;

    for(n=1; n < nlow; n++) {

      term *= x/(a+n);

      sum  += term;

    }

    /* Handle upper part of the series where t_n is decreasing, |x| < a+n */

    for (/* n = previous n */ ; n

      term *= x/(a+n);

      sum  += term;

      if(fabs(term/sum) < GSL_DBL_EPSILON) break;

    }

    /*  Estimate remainder of series ~ t_(n+1)/(1-x/(a+n+1)) */

    {

      double tnp1 = (x/(a+n)) * term;

      remainder =  tnp1 / (1.0 - x/(a + n + 1.0));

    }

    result->val  = D.val * sum;

    result->err  = D.err * fabs(sum) + fabs(D.val * remainder);

    result->err += (1.0 + n) * GSL_DBL_EPSILON * fabs(result->val);

    if(n == nmax && fabs(remainder/sum) > GSL_SQRT_DBL_EPSILON)

      GSL_ERROR ("gamma_inc_P_series failed to converge", GSL_EMAXITER);

    else

      return stat_D;

  }

}

/* Q large x asymptotic

 */

static

int

gamma_inc_Q_large_x(const double a, const double x, gsl_sf_result * result)

{

  const int nmax = 5000;

  gsl_sf_result D;

  const int stat_D = gamma_inc_D(a, x, &D);

  double sum  = 1.0;

  double term = 1.0;

  double last = 1.0;

  int n;

  for(n=1; n

    term *= (a-n)/x;

    if(fabs(term/last) > 1.0) break;

    if(fabs(term/sum)  < GSL_DBL_EPSILON) break;

    sum  += term;

    last  = term;

  }

  result->val  = D.val * (a/x) * sum;

  result->err  = D.err * fabs((a/x) * sum);

  result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

  if(n == nmax)

    GSL_ERROR ("error in large x asymptotic", GSL_EMAXITER);

  else

    return stat_D;

}

/* Uniform asymptotic for x near a, a and x large.

 * See [Temme, p. 285]

 */

static

int

gamma_inc_Q_asymp_unif(const double a, const double x, gsl_sf_result * result)

{

  const double rta = sqrt(a);

  const double eps = (x-a)/a;

  gsl_sf_result ln_term;

  const int stat_ln = gsl_sf_log_1plusx_mx_e(eps, &ln_term);  /* log(1+eps) - eps */

  const double eta  = GSL_SIGN(eps) * sqrt(-2.0*ln_term.val);

  gsl_sf_result erfc;

  double R;

  double c0, c1;

  /* This used to say erfc(eta*M_SQRT2*rta), which is wrong.

   * The sqrt(2) is in the denominator. Oops.

   * Fixed: [GJ] Mon Nov 15 13:25:32 MST 2004

   */

  gsl_sf_erfc_e(eta*rta/M_SQRT2, &erfc);

  if(fabs(eps) < GSL_ROOT5_DBL_EPSILON) {

    c0 = -1.0/3.0 + eps*(1.0/12.0 - eps*(23.0/540.0 - eps*(353.0/12960.0 - eps*589.0/30240.0)));

    c1 = -1.0/540.0 - eps/288.0;

  }

  else {

    const double rt_term = sqrt(-2.0 * ln_term.val/(eps*eps));

    const double lam = x/a;

    c0 = (1.0 - 1.0/rt_term)/eps;

    c1 = -(eta*eta*eta * (lam*lam + 10.0*lam + 1.0) - 12.0 * eps*eps*eps) / (12.0 * eta*eta*eta*eps*eps*eps);

  }

  R = exp(-0.5*a*eta*eta)/(M_SQRT2*M_SQRTPI*rta) * (c0 + c1/a);

  result->val  = 0.5 * erfc.val + R;

  result->err  = GSL_DBL_EPSILON * fabs(R * 0.5 * a*eta*eta) + 0.5 * erfc.err;

  result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

  return stat_ln;

}

/* Continued fraction which occurs in evaluation

 * of Q(a,x) or Gamma(a,x).

 *

 *              1   (1-a)/x  1/x  (2-a)/x   2/x  (3-a)/x

 *   F(a,x) =  ---- ------- ----- -------- ----- -------- ...

 *             1 +   1 +     1 +   1 +      1 +   1 +

 *

 * Hans E. Plesser, 2002-01-22 (hans dot plesser at itf dot nlh dot no).

 *

 * Split out from gamma_inc_Q_CF() by GJ [Tue Apr  1 13:16:41 MST 2003].

 * See gamma_inc_Q_CF() below.

 *

 */

static int

gamma_inc_F_CF(const double a, const double x, gsl_sf_result * result)

{

  const int    nmax  =  5000;

  const double small =  gsl_pow_3 (GSL_DBL_EPSILON);

  double hn = 1.0;           /* convergent */

  double Cn = 1.0 / small;

  double Dn = 1.0;

  int n;

  /* n == 1 has a_1, b_1, b_0 independent of a,x,

     so that has been done by hand                */

  for ( n = 2 ; n < nmax ; n++ )

  {

    double an;

    double delta;

    if(GSL_IS_ODD(n))

      an = 0.5*(n-1)/x;

    else

      an = (0.5*n-a)/x;

    Dn = 1.0 + an * Dn;

    if ( fabs(Dn) < small )

      Dn = small;

    Cn = 1.0 + an/Cn;

    if ( fabs(Cn) < small )

      Cn = small;

    Dn = 1.0 / Dn;

    delta = Cn * Dn;

    hn *= delta;

    if(fabs(delta-1.0) < GSL_DBL_EPSILON) break;

  }

  result->val = hn;

  result->err = 2.0*GSL_DBL_EPSILON * fabs(hn);

  result->err += GSL_DBL_EPSILON * (2.0 + 0.5*n) * fabs(result->val);

  if(n == nmax)

    GSL_ERROR ("error in CF for F(a,x)", GSL_EMAXITER);

  else

    return GSL_SUCCESS;

}

/* Continued fraction for Q.

 *

 * Q(a,x) = D(a,x) a/x F(a,x)

 *

 * Hans E. Plesser, 2002-01-22 (hans dot plesser at itf dot nlh dot no):

 *

 * Since the Gautschi equivalent series method for CF evaluation may lead

 * to singularities, I have replaced it with the modified Lentz algorithm

 * given in

 *

 * I J Thompson and A R Barnett

 * Coulomb and Bessel Functions of Complex Arguments and Order

 * J Computational Physics 64:490-509 (1986)

 *

 * In consequence, gamma_inc_Q_CF_protected() is now obsolete and has been

 * removed.

 *

 * Identification of terms between the above equation for F(a, x) and

 * the first equation in the appendix of Thompson&Barnett is as follows:

 *

 *    b_0 = 0, b_n = 1 for all n > 0

 *

 *    a_1 = 1

 *    a_n = (n/2-a)/x    for n even

 *    a_n = (n-1)/(2x)   for n odd

 *

 */

static

int

gamma_inc_Q_CF(const double a, const double x, gsl_sf_result * result)

{

  gsl_sf_result D;

  gsl_sf_result F;

  const int stat_D = gamma_inc_D(a, x, &D);

  const int stat_F = gamma_inc_F_CF(a, x, &F);

  result->val  = D.val * (a/x) * F.val;

  result->err  = D.err * fabs((a/x) * F.val) + fabs(D.val * a/x * F.err);

  return GSL_ERROR_SELECT_2(stat_F, stat_D);

}

/* Useful for small a and x. Handles the subtraction analytically.

 */

static

int

gamma_inc_Q_series(const double a, const double x, gsl_sf_result * result)

{

  double term1;  /* 1 - x^a/Gamma(a+1) */

  double sum;    /* 1 + (a+1)/(a+2)(-x)/2! + (a+1)/(a+3)(-x)^2/3! + ... */

  int stat_sum;

  double term2;  /* a temporary variable used at the end */

  {

    /* Evaluate series for 1 - x^a/Gamma(a+1), small a

     */

    const double pg21 = -2.404113806319188570799476;  /* PolyGamma[2,1] */

    const double lnx  = log(x);

    const double el   = M_EULER+lnx;

    const double c1 = -el;

    const double c2 = M_PI*M_PI/12.0 - 0.5*el*el;

    const double c3 = el*(M_PI*M_PI/12.0 - el*el/6.0) + pg21/6.0;

    const double c4 = -0.04166666666666666667

                       * (-1.758243446661483480 + lnx)

                       * (-0.764428657272716373 + lnx)

                       * ( 0.723980571623507657 + lnx)

                       * ( 4.107554191916823640 + lnx);

    const double c5 = -0.0083333333333333333

                       * (-2.06563396085715900 + lnx)

                       * (-1.28459889470864700 + lnx)

                       * (-0.27583535756454143 + lnx)

                       * ( 1.33677371336239618 + lnx)

                       * ( 5.17537282427561550 + lnx);

    const double c6 = -0.0013888888888888889

                       * (-2.30814336454783200 + lnx)

                       * (-1.65846557706987300 + lnx)

                       * (-0.88768082560020400 + lnx)

                       * ( 0.17043847751371778 + lnx)

                       * ( 1.92135970115863890 + lnx)

                       * ( 6.22578557795474900 + lnx);

    const double c7 = -0.00019841269841269841

                       * (-2.5078657901291800 + lnx)

                       * (-1.9478900888958200 + lnx)

                       * (-1.3194837322612730 + lnx)

                       * (-0.5281322700249279 + lnx)

                       * ( 0.5913834939078759 + lnx)

                       * ( 2.4876819633378140 + lnx)

                       * ( 7.2648160783762400 + lnx);

    const double c8 = -0.00002480158730158730

                       * (-2.677341544966400 + lnx)

                       * (-2.182810448271700 + lnx)

                       * (-1.649350342277400 + lnx)

                       * (-1.014099048290790 + lnx)

                       * (-0.191366955370652 + lnx)

                       * ( 0.995403817918724 + lnx)

                       * ( 3.041323283529310 + lnx)

                       * ( 8.295966556941250 + lnx);

    const double c9 = -2.75573192239859e-6

                       * (-2.8243487670469080 + lnx)

                       * (-2.3798494322701120 + lnx)

                       * (-1.9143674728689960 + lnx)

                       * (-1.3814529102920370 + lnx)

                       * (-0.7294312810261694 + lnx)

                       * ( 0.1299079285269565 + lnx)

                       * ( 1.3873333251885240 + lnx)

                       * ( 3.5857258865210760 + lnx)

                       * ( 9.3214237073814600 + lnx);

    const double c10 = -2.75573192239859e-7

                       * (-2.9540329644556910 + lnx)

                       * (-2.5491366926991850 + lnx)

                       * (-2.1348279229279880 + lnx)

                       * (-1.6741881076349450 + lnx)

                       * (-1.1325949616098420 + lnx)

                       * (-0.4590034650618494 + lnx)

                       * ( 0.4399352987435699 + lnx)

                       * ( 1.7702236517651670 + lnx)

                       * ( 4.1231539047474080 + lnx)

                       * ( 10.342627908148680 + lnx);

    term1 = a*(c1+a*(c2+a*(c3+a*(c4+a*(c5+a*(c6+a*(c7+a*(c8+a*(c9+a*c10)))))))));

  }

  {

    /* Evaluate the sum.

     */

    const int nmax = 5000;

    double t = 1.0;

    int n;

    sum = 1.0;

    for(n=1; n

      t *= -x/(n+1.0);

      sum += (a+1.0)/(a+n+1.0)*t;

      if(fabs(t/sum) < GSL_DBL_EPSILON) break;

    }

    if(n == nmax)

      stat_sum = GSL_EMAXITER;

    else

      stat_sum = GSL_SUCCESS;

  }

  term2 = (1.0 - term1) * a/(a+1.0) * x * sum;

  result->val  = term1 + term2;

  result->err  = GSL_DBL_EPSILON * (fabs(term1) + 2.0*fabs(term2));

  result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

  return stat_sum;

}

/* series for small a and x, but not defined for a == 0 */

static int

gamma_inc_series(double a, double x, gsl_sf_result * result)

{

  gsl_sf_result Q;

  gsl_sf_result G;

  const int stat_Q = gamma_inc_Q_series(a, x, &Q);

  const int stat_G = gsl_sf_gamma_e(a, &G);

  result->val = Q.val * G.val;

  result->err = fabs(Q.val * G.err) + fabs(Q.err * G.val);

  result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

  return GSL_ERROR_SELECT_2(stat_Q, stat_G);

}

static int

gamma_inc_a_gt_0(double a, double x, gsl_sf_result * result)

{

  /* x > 0 and a > 0; use result for Q */

  gsl_sf_result Q;

  gsl_sf_result G;

  const int stat_Q = gsl_sf_gamma_inc_Q_e(a, x, &Q);

  const int stat_G = gsl_sf_gamma_e(a, &G);

  result->val = G.val * Q.val;

  result->err = fabs(G.val * Q.err) + fabs(G.err * Q.val);

  result->err += 2.0*GSL_DBL_EPSILON * fabs(result->val);

  return GSL_ERROR_SELECT_2(stat_G, stat_Q);

}

static int

gamma_inc_CF(double a, double x, gsl_sf_result * result)

{

  gsl_sf_result F;

  gsl_sf_result pre;

  const double am1lgx = (a-1.0)*log(x);

  const int stat_F = gamma_inc_F_CF(a, x, &F);

  const int stat_E = gsl_sf_exp_err_e(am1lgx - x, GSL_DBL_EPSILON*fabs(am1lgx), &pre);

  result->val = F.val * pre.val;

  result->err = fabs(F.err * pre.val) + fabs(F.val * pre.err);

  result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

  return GSL_ERROR_SELECT_2(stat_F, stat_E);

}

/* evaluate Gamma(0,x), x > 0 */

#define GAMMA_INC_A_0(x, result) gsl_sf_expint_E1_e(x, result)

/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/

int

gsl_sf_gamma_inc_Q_e(const double a, const double x, gsl_sf_result * result)

{

  if(a < 0.0 || x < 0.0) {

    DOMAIN_ERROR(result);

  }

  else if(x == 0.0) {

    result->val = 1.0;

    result->err = 0.0;

    return GSL_SUCCESS;

  }

  else if(a == 0.0)

  {

    result->val = 0.0;

    result->err = 0.0;

    return GSL_SUCCESS;

  }

  else if(x <= 0.5*a) {

    /* If the series is quick, do that. It is

     * robust and simple.

     */

    gsl_sf_result P;

    int stat_P = gamma_inc_P_series(a, x, &P);

    result->val  = 1.0 - P.val;

    result->err  = P.err;

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return stat_P;

  }

  else if(a >= 1.0e+06 && (x-a)*(x-a) < a) {

    /* Then try the difficult asymptotic regime.

     * This is the only way to do this region.

     */

    return gamma_inc_Q_asymp_unif(a, x, result);

  }

  else if(a < 0.2 && x < 5.0) {

    /* Cancellations at small a must be handled

     * analytically; x should not be too big

     * either since the series terms grow

     * with x and log(x).

     */

    return gamma_inc_Q_series(a, x, result);

  }

  else if(a <= x) {

    if(x <= 1.0e+06) {

      /* Continued fraction is excellent for x >~ a.

       * We do not let x be too large when x > a since

       * it is somewhat pointless to try this there;

       * the function is rapidly decreasing for

       * x large and x > a, and it will just

       * underflow in that region anyway. We

       * catch that case in the standard

       * large-x method.

       */

      return gamma_inc_Q_CF(a, x, result);

    }

    else {

      return gamma_inc_Q_large_x(a, x, result);

    }

  }

  else {

    if(x > a - sqrt(a)) {

      /* Continued fraction again. The convergence

       * is a little slower here, but that is fine.

       * We have to trade that off against the slow

       * convergence of the series, which is the

       * only other option.

       */

      return gamma_inc_Q_CF(a, x, result);

    }

    else {

      gsl_sf_result P;

      int stat_P = gamma_inc_P_series(a, x, &P);

      result->val  = 1.0 - P.val;

      result->err  = P.err;

      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

      return stat_P;

    }

  }

}

int

gsl_sf_gamma_inc_P_e(const double a, const double x, gsl_sf_result * result)

{

  if(a <= 0.0 || x < 0.0) {

    DOMAIN_ERROR(result);

  }

  else if(x == 0.0) {

    result->val = 0.0;

    result->err = 0.0;

    return GSL_SUCCESS;

  }

  else if(x < 20.0 || x < 0.5*a) {

    /* Do the easy series cases. Robust and quick.

     */

    return gamma_inc_P_series(a, x, result);

  }

  else if(a > 1.0e+06 && (x-a)*(x-a) < a) {

    /* Crossover region. Note that Q and P are

     * roughly the same order of magnitude here,

     * so the subtraction is stable.

     */

    gsl_sf_result Q;

    int stat_Q = gamma_inc_Q_asymp_unif(a, x, &Q);

    result->val  = 1.0 - Q.val;

    result->err  = Q.err;

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return stat_Q;

  }

  else if(a <= x) {

    /* Q <~ P in this area, so the

     * subtractions are stable.

     */

    gsl_sf_result Q;

    int stat_Q;

    if(a > 0.2*x) {

      stat_Q = gamma_inc_Q_CF(a, x, &Q);

    }

    else {

      stat_Q = gamma_inc_Q_large_x(a, x, &Q);

    }

    result->val  = 1.0 - Q.val;

    result->err  = Q.err;

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return stat_Q;

  }

  else {

    if((x-a)*(x-a) < a) {

      /* This condition is meant to insure

       * that Q is not very close to 1,

       * so the subtraction is stable.

       */

      gsl_sf_result Q;

      int stat_Q = gamma_inc_Q_CF(a, x, &Q);

      result->val  = 1.0 - Q.val;

      result->err  = Q.err;

      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

      return stat_Q;

    }

    else {

      return gamma_inc_P_series(a, x, result);

    }

  }

}

int

gsl_sf_gamma_inc_e(const double a, const double x, gsl_sf_result * result)

{

  if(x < 0.0) {

    DOMAIN_ERROR(result);

  }

  else if(x == 0.0) {

    return gsl_sf_gamma_e(a, result);

  }

  else if(a == 0.0)

  {

    return GAMMA_INC_A_0(x, result);

  }

  else if(a > 0.0)

  {

    return gamma_inc_a_gt_0(a, x, result);

  }

  else if(x > 0.25)

  {

    /* continued fraction seems to fail for x too small; otherwise

       it is ok, independent of the value of |x/a|, because of the

       non-oscillation in the expansion, i.e. the CF is

       un-conditionally convergent for a < 0 and x > 0

     */

    return gamma_inc_CF(a, x, result);

  }

  else if(fabs(a) < 0.5)

  {

    return gamma_inc_series(a, x, result);

  }

  else

  {

    /* a = fa + da; da >= 0 */

    const double fa = floor(a);

    const double da = a - fa;

    gsl_sf_result g_da;

    const int stat_g_da = ( da > 0.0 ? gamma_inc_a_gt_0(da, x, &g_da)

                                     : GAMMA_INC_A_0(x, &g_da));

    double alpha = da;

    double gax = g_da.val;

    /* Gamma(alpha-1,x) = 1/(alpha-1) (Gamma(a,x) - x^(alpha-1) e^-x) */

    do

    {

      const double shift = exp(-x + (alpha-1.0)*log(x));

      gax = (gax - shift) / (alpha - 1.0);

      alpha -= 1.0;

    } while(alpha > a);

    result->val = gax;

    result->err = 2.0*(1.0 + fabs(a))*GSL_DBL_EPSILON*fabs(gax);

    return stat_g_da;

  }

}

/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/

#include "eval.h"

double gsl_sf_gamma_inc_P(const double a, const double x)

{

  EVAL_RESULT(gsl_sf_gamma_inc_P_e(a, x, &result));

}

double gsl_sf_gamma_inc_Q(const double a, const double x)

{

  EVAL_RESULT(gsl_sf_gamma_inc_Q_e(a, x, &result));

}

double gsl_sf_gamma_inc(const double a, const double x)

{

   EVAL_RESULT(gsl_sf_gamma_inc_e(a, x, &result));

}