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+/* specfunc/gamma_inc.c
+ *
+ * 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 2 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 */
+ };
+ 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);
+ 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 = 5000;
+
+ gsl_sf_result D;
+ int stat_D = gamma_inc_D(a, x, &D);
+
+ double sum = 1.0;
+ double term = 1.0;
+ int n;
+ for(n=1; n<nmax; n++) {
+ term *= x/(a+n);
+ sum += term;
+ if(fabs(term/sum) < GSL_DBL_EPSILON) break;
+ }
+
+ result->val = D.val * sum;
+ result->err = D.err * fabs(sum);
+ result->err += (1.0 + n) * GSL_DBL_EPSILON * fabs(result->val);
+
+ if(n == nmax)
+ GSL_ERROR ("error", 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<nmax; 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<nmax; 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 int stat_F = gamma_inc_F_CF(a, x, &F);
+ const int stat_E = gsl_sf_exp_e((a-1.0)*log(x) - x, &pre);
+
+ result->val = F.val * pre.val;
+ result->err = fabs(F.err * pre.val) + fabs(F.val * pre.err);
+ result->err += (2.0 + fabs(a)) * 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));
+}
generated by cgit v1.2.3 (git 2.25.1) at 2024-05-01 02:50:26 +0000