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+/* specfunc/ellint.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_precision.h>
+#include <gsl/gsl_sf_ellint.h>
+
+#include "error.h"
+
+/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
+
+inline
+static double locMAX3(double x, double y, double z)
+{
+ double xy = GSL_MAX(x, y);
+ return GSL_MAX(xy, z);
+}
+
+inline
+static double locMAX4(double x, double y, double z, double w)
+{
+ double xy = GSL_MAX(x, y);
+ double xyz = GSL_MAX(xy, z);
+ return GSL_MAX(xyz, w);
+}
+
+
+/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
+
+
+/* based on Carlson's algorithms:
+ [B. C. Carlson Numer. Math. 33, 1 (1979)]
+
+ see also:
+ [B.C. Carlson, Special Functions of Applied Mathematics (1977)]
+ */
+
+/* According to Carlson's algorithm, the errtol parameter
+ typically effects the relative error in the following way:
+
+ relative error < 16 errtol^6 / (1 - 2 errtol)
+
+ errtol precision
+ ------ ----------
+ 0.001 1.0e-17
+ 0.003 2.0e-14
+ 0.01 2.0e-11
+ 0.03 2.0e-8
+ 0.1 2.0e-5
+*/
+
+
+int
+gsl_sf_ellint_RC_e(double x, double y, gsl_mode_t mode, gsl_sf_result * result)
+{
+ const double lolim = 5.0 * GSL_DBL_MIN;
+ const double uplim = 0.2 * GSL_DBL_MAX;
+ const gsl_prec_t goal = GSL_MODE_PREC(mode);
+ const double errtol = ( goal == GSL_PREC_DOUBLE ? 0.001 : 0.03 );
+ const double prec = gsl_prec_eps[goal];
+
+ if(x < 0.0 || y < 0.0 || x + y < lolim) {
+ DOMAIN_ERROR(result);
+ }
+ else if(GSL_MAX(x, y) < uplim) {
+ const double c1 = 1.0 / 7.0;
+ const double c2 = 9.0 / 22.0;
+ double xn = x;
+ double yn = y;
+ double mu, sn, lamda, s;
+ while(1) {
+ mu = (xn + yn + yn) / 3.0;
+ sn = (yn + mu) / mu - 2.0;
+ if (fabs(sn) < errtol) break;
+ lamda = 2.0 * sqrt(xn) * sqrt(yn) + yn;
+ xn = (xn + lamda) * 0.25;
+ yn = (yn + lamda) * 0.25;
+ }
+ s = sn * sn * (0.3 + sn * (c1 + sn * (0.375 + sn * c2)));
+ result->val = (1.0 + s) / sqrt(mu);
+ result->err = prec * fabs(result->val);
+ return GSL_SUCCESS;
+ }
+ else {
+ DOMAIN_ERROR(result);
+ }
+}
+
+
+int
+gsl_sf_ellint_RD_e(double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result)
+{
+ const gsl_prec_t goal = GSL_MODE_PREC(mode);
+ const double errtol = ( goal == GSL_PREC_DOUBLE ? 0.001 : 0.03 );
+ const double prec = gsl_prec_eps[goal];
+ const double lolim = 2.0/pow(GSL_DBL_MAX, 2.0/3.0);
+ const double uplim = pow(0.1*errtol/GSL_DBL_MIN, 2.0/3.0);
+
+ if(GSL_MIN(x,y) < 0.0 || GSL_MIN(x+y,z) < lolim) {
+ DOMAIN_ERROR(result);
+ }
+ else if(locMAX3(x,y,z) < uplim) {
+ const double c1 = 3.0 / 14.0;
+ const double c2 = 1.0 / 6.0;
+ const double c3 = 9.0 / 22.0;
+ const double c4 = 3.0 / 26.0;
+ double xn = x;
+ double yn = y;
+ double zn = z;
+ double sigma = 0.0;
+ double power4 = 1.0;
+ double ea, eb, ec, ed, ef, s1, s2;
+ double mu, xndev, yndev, zndev;
+ while(1) {
+ double xnroot, ynroot, znroot, lamda;
+ double epslon;
+ mu = (xn + yn + 3.0 * zn) * 0.2;
+ xndev = (mu - xn) / mu;
+ yndev = (mu - yn) / mu;
+ zndev = (mu - zn) / mu;
+ epslon = locMAX3(fabs(xndev), fabs(yndev), fabs(zndev));
+ if (epslon < errtol) break;
+ xnroot = sqrt(xn);
+ ynroot = sqrt(yn);
+ znroot = sqrt(zn);
+ lamda = xnroot * (ynroot + znroot) + ynroot * znroot;
+ sigma += power4 / (znroot * (zn + lamda));
+ power4 *= 0.25;
+ xn = (xn + lamda) * 0.25;
+ yn = (yn + lamda) * 0.25;
+ zn = (zn + lamda) * 0.25;
+ }
+ ea = xndev * yndev;
+ eb = zndev * zndev;
+ ec = ea - eb;
+ ed = ea - 6.0 * eb;
+ ef = ed + ec + ec;
+ s1 = ed * (- c1 + 0.25 * c3 * ed - 1.5 * c4 * zndev * ef);
+ s2 = zndev * (c2 * ef + zndev * (- c3 * ec + zndev * c4 * ea));
+ result->val = 3.0 * sigma + power4 * (1.0 + s1 + s2) / (mu * sqrt(mu));
+ result->err = prec * fabs(result->val);
+ return GSL_SUCCESS;
+ }
+ else {
+ DOMAIN_ERROR(result);
+ }
+}
+
+
+int
+gsl_sf_ellint_RF_e(double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result)
+{
+ const double lolim = 5.0 * GSL_DBL_MIN;
+ const double uplim = 0.2 * GSL_DBL_MAX;
+ const gsl_prec_t goal = GSL_MODE_PREC(mode);
+ const double errtol = ( goal == GSL_PREC_DOUBLE ? 0.001 : 0.03 );
+ const double prec = gsl_prec_eps[goal];
+
+ if(x < 0.0 || y < 0.0 || z < 0.0) {
+ DOMAIN_ERROR(result);
+ }
+ else if(x+y < lolim || x+z < lolim || y+z < lolim) {
+ DOMAIN_ERROR(result);
+ }
+ else if(locMAX3(x,y,z) < uplim) {
+ const double c1 = 1.0 / 24.0;
+ const double c2 = 3.0 / 44.0;
+ const double c3 = 1.0 / 14.0;
+ double xn = x;
+ double yn = y;
+ double zn = z;
+ double mu, xndev, yndev, zndev, e2, e3, s;
+ while(1) {
+ double epslon, lamda;
+ double xnroot, ynroot, znroot;
+ mu = (xn + yn + zn) / 3.0;
+ xndev = 2.0 - (mu + xn) / mu;
+ yndev = 2.0 - (mu + yn) / mu;
+ zndev = 2.0 - (mu + zn) / mu;
+ epslon = locMAX3(fabs(xndev), fabs(yndev), fabs(zndev));
+ if (epslon < errtol) break;
+ xnroot = sqrt(xn);
+ ynroot = sqrt(yn);
+ znroot = sqrt(zn);
+ lamda = xnroot * (ynroot + znroot) + ynroot * znroot;
+ xn = (xn + lamda) * 0.25;
+ yn = (yn + lamda) * 0.25;
+ zn = (zn + lamda) * 0.25;
+ }
+ e2 = xndev * yndev - zndev * zndev;
+ e3 = xndev * yndev * zndev;
+ s = 1.0 + (c1 * e2 - 0.1 - c2 * e3) * e2 + c3 * e3;
+ result->val = s / sqrt(mu);
+ result->err = prec * fabs(result->val);
+ return GSL_SUCCESS;
+ }
+ else {
+ DOMAIN_ERROR(result);
+ }
+}
+
+
+int
+gsl_sf_ellint_RJ_e(double x, double y, double z, double p, gsl_mode_t mode, gsl_sf_result * result)
+{
+ const gsl_prec_t goal = GSL_MODE_PREC(mode);
+ const double errtol = ( goal == GSL_PREC_DOUBLE ? 0.001 : 0.03 );
+ const double prec = gsl_prec_eps[goal];
+ const double lolim = pow(5.0 * GSL_DBL_MIN, 1.0/3.0);
+ const double uplim = 0.3 * pow(0.2 * GSL_DBL_MAX, 1.0/3.0);
+
+ if(x < 0.0 || y < 0.0 || z < 0.0) {
+ DOMAIN_ERROR(result);
+ }
+ else if(x + y < lolim || x + z < lolim || y + z < lolim || p < lolim) {
+ DOMAIN_ERROR(result);
+ }
+ else if(locMAX4(x,y,z,p) < uplim) {
+ const double c1 = 3.0 / 14.0;
+ const double c2 = 1.0 / 3.0;
+ const double c3 = 3.0 / 22.0;
+ const double c4 = 3.0 / 26.0;
+ double xn = x;
+ double yn = y;
+ double zn = z;
+ double pn = p;
+ double sigma = 0.0;
+ double power4 = 1.0;
+ double mu, xndev, yndev, zndev, pndev;
+ double ea, eb, ec, e2, e3, s1, s2, s3;
+ while(1) {
+ double xnroot, ynroot, znroot;
+ double lamda, alfa, beta;
+ double epslon;
+ gsl_sf_result rcresult;
+ int rcstatus;
+ mu = (xn + yn + zn + pn + pn) * 0.2;
+ xndev = (mu - xn) / mu;
+ yndev = (mu - yn) / mu;
+ zndev = (mu - zn) / mu;
+ pndev = (mu - pn) / mu;
+ epslon = locMAX4(fabs(xndev), fabs(yndev), fabs(zndev), fabs(pndev));
+ if(epslon < errtol) break;
+ xnroot = sqrt(xn);
+ ynroot = sqrt(yn);
+ znroot = sqrt(zn);
+ lamda = xnroot * (ynroot + znroot) + ynroot * znroot;
+ alfa = pn * (xnroot + ynroot + znroot) + xnroot * ynroot * znroot;
+ alfa = alfa * alfa;
+ beta = pn * (pn + lamda) * (pn + lamda);
+ rcstatus = gsl_sf_ellint_RC_e(alfa, beta, mode, &rcresult);
+ if(rcstatus != GSL_SUCCESS) {
+ result->val = 0.0;
+ result->err = 0.0;
+ return rcstatus;
+ }
+ sigma += power4 * rcresult.val;
+ power4 *= 0.25;
+ xn = (xn + lamda) * 0.25;
+ yn = (yn + lamda) * 0.25;
+ zn = (zn + lamda) * 0.25;
+ pn = (pn + lamda) * 0.25;
+ }
+
+ ea = xndev * (yndev + zndev) + yndev * zndev;
+ eb = xndev * yndev * zndev;
+ ec = pndev * pndev;
+ e2 = ea - 3.0 * ec;
+ e3 = eb + 2.0 * pndev * (ea - ec);
+ s1 = 1.0 + e2 * (- c1 + 0.75 * c3 * e2 - 1.5 * c4 * e3);
+ s2 = eb * (0.5 * c2 + pndev * (- c3 - c3 + pndev * c4));
+ s3 = pndev * ea * (c2 - pndev * c3) - c2 * pndev * ec;
+ result->val = 3.0 * sigma + power4 * (s1 + s2 + s3) / (mu * sqrt(mu));
+ result->err = prec * fabs(result->val);
+ return GSL_SUCCESS;
+ }
+ else {
+ DOMAIN_ERROR(result);
+ }
+}
+
+
+/* [Carlson, Numer. Math. 33 (1979) 1, (4.1)] */
+int
+gsl_sf_ellint_F_e(double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
+{
+ /* Angular reduction to -pi/2 < phi < pi/2 (we should really use an
+ exact reduction but this will have to do for now) BJG */
+
+ double nc = floor(phi/M_PI + 0.5);
+ double phi_red = phi - nc * M_PI;
+ phi = phi_red;
+
+ {
+ double sin_phi = sin(phi);
+ double sin2_phi = sin_phi*sin_phi;
+ double x = 1.0 - sin2_phi;
+ double y = 1.0 - k*k*sin2_phi;
+ gsl_sf_result rf;
+ int status = gsl_sf_ellint_RF_e(x, y, 1.0, mode, &rf);
+ result->val = sin_phi * rf.val;
+ result->err = GSL_DBL_EPSILON * fabs(result->val) + fabs(sin_phi*rf.err);
+ if (nc == 0) {
+ return status;
+ } else {
+ gsl_sf_result rk; /* add extra terms from periodicity */
+ const int rkstatus = gsl_sf_ellint_Kcomp_e(k, mode, &rk);
+ result->val += 2*nc*rk.val;
+ result->err += 2*fabs(nc)*rk.err;
+ return GSL_ERROR_SELECT_2(status, rkstatus);
+ }
+ }
+}
+
+
+/* [Carlson, Numer. Math. 33 (1979) 1, (4.2)] */
+int
+gsl_sf_ellint_E_e(double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
+{
+ /* Angular reduction to -pi/2 < phi < pi/2 (we should really use an
+ exact reduction but this will have to do for now) BJG */
+
+ double nc = floor(phi/M_PI + 0.5);
+ double phi_red = phi - nc * M_PI;
+ phi = phi_red;
+
+ {
+ const double sin_phi = sin(phi);
+ const double sin2_phi = sin_phi * sin_phi;
+ const double x = 1.0 - sin2_phi;
+ const double y = 1.0 - k*k*sin2_phi;
+
+ if(x < GSL_DBL_EPSILON) {
+ gsl_sf_result re;
+ const int status = gsl_sf_ellint_Ecomp_e(k, mode, &re);
+ /* could use A&S 17.4.14 to improve the value below */
+ result->val = 2*nc*re.val + GSL_SIGN(sin_phi) * re.val;
+ result->err = 2*fabs(nc)*re.err + re.err;
+ return status;
+ }
+ else {
+ gsl_sf_result rf, rd;
+ const double sin3_phi = sin2_phi * sin_phi;
+ const int rfstatus = gsl_sf_ellint_RF_e(x, y, 1.0, mode, &rf);
+ const int rdstatus = gsl_sf_ellint_RD_e(x, y, 1.0, mode, &rd);
+ result->val = sin_phi * rf.val - k*k/3.0 * sin3_phi * rd.val;
+ result->err = GSL_DBL_EPSILON * fabs(sin_phi * rf.val);
+ result->err += fabs(sin_phi*rf.err);
+ result->err += k*k/3.0 * GSL_DBL_EPSILON * fabs(sin3_phi * rd.val);
+ result->err += k*k/3.0 * fabs(sin3_phi*rd.err);
+ if (nc == 0) {
+ return GSL_ERROR_SELECT_2(rfstatus, rdstatus);
+ } else {
+ gsl_sf_result re; /* add extra terms from periodicity */
+ const int restatus = gsl_sf_ellint_Ecomp_e(k, mode, &re);
+ result->val += 2*nc*re.val;
+ result->err += 2*fabs(nc)*re.err;
+ return GSL_ERROR_SELECT_3(rfstatus, rdstatus, restatus);
+ }
+ }
+ }
+}
+
+
+/* [Carlson, Numer. Math. 33 (1979) 1, (4.3)] */
+int
+gsl_sf_ellint_P_e(double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
+{
+ /* Angular reduction to -pi/2 < phi < pi/2 (we should really use an
+ exact reduction but this will have to do for now) BJG */
+
+ double nc = floor(phi/M_PI + 0.5);
+ double phi_red = phi - nc * M_PI;
+ phi = phi_red;
+
+ /* FIXME: need to handle the case of small x, as for E,F */
+
+ {
+ const double sin_phi = sin(phi);
+ const double sin2_phi = sin_phi * sin_phi;
+ const double sin3_phi = sin2_phi * sin_phi;
+ const double x = 1.0 - sin2_phi;
+ const double y = 1.0 - k*k*sin2_phi;
+ gsl_sf_result rf;
+ gsl_sf_result rj;
+ const int rfstatus = gsl_sf_ellint_RF_e(x, y, 1.0, mode, &rf);
+ const int rjstatus = gsl_sf_ellint_RJ_e(x, y, 1.0, 1.0 + n*sin2_phi, mode, &rj);
+ result->val = sin_phi * rf.val - n/3.0*sin3_phi * rj.val;
+ result->err = GSL_DBL_EPSILON * fabs(sin_phi * rf.val);
+ result->err += n/3.0 * fabs(sin3_phi*rj.err);
+ if (nc == 0) {
+ return GSL_ERROR_SELECT_2(rfstatus, rjstatus);
+ } else {
+ gsl_sf_result rp; /* add extra terms from periodicity */
+ const int rpstatus = gsl_sf_ellint_Pcomp_e(k, n, mode, &rp);
+ result->val += 2*nc*rp.val;
+ result->err += 2*fabs(nc)*rp.err;
+ return GSL_ERROR_SELECT_3(rfstatus, rjstatus, rpstatus);
+ }
+ }
+}
+
+
+/* [Carlson, Numer. Math. 33 (1979) 1, (4.4)] */
+int
+gsl_sf_ellint_D_e(double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
+{
+ /* Angular reduction to -pi/2 < phi < pi/2 (we should really use an
+ exact reduction but this will have to do for now) BJG */
+
+ double nc = floor(phi/M_PI + 0.5);
+ double phi_red = phi - nc * M_PI;
+ phi = phi_red;
+
+ /* FIXME: need to handle the case of small x, as for E,F */
+ {
+ const double sin_phi = sin(phi);
+ const double sin2_phi = sin_phi * sin_phi;
+ const double sin3_phi = sin2_phi * sin_phi;
+ const double x = 1.0 - sin2_phi;
+ const double y = 1.0 - k*k*sin2_phi;
+ gsl_sf_result rd;
+ const int status = gsl_sf_ellint_RD_e(x, y, 1.0, mode, &rd);
+ result->val = sin3_phi/3.0 * rd.val;
+ result->err = GSL_DBL_EPSILON * fabs(result->val) + fabs(sin3_phi/3.0 * rd.err);
+ if (nc == 0) {
+ return status;
+ } else {
+ gsl_sf_result rd; /* add extra terms from periodicity */
+ const int rdstatus = gsl_sf_ellint_Dcomp_e(k, mode, &rd);
+ result->val += 2*nc*rd.val;
+ result->err += 2*fabs(nc)*rd.err;
+ return GSL_ERROR_SELECT_2(status, rdstatus);
+ }
+ }
+}
+
+int
+gsl_sf_ellint_Dcomp_e(double k, gsl_mode_t mode, gsl_sf_result * result)
+{
+ if(k*k >= 1.0) {
+ DOMAIN_ERROR(result);
+ } else {
+ const double y = 1.0 - k*k; /* FIXME: still need to handle k~=~1 */
+ gsl_sf_result rd;
+ const int status = gsl_sf_ellint_RD_e(0.0, y, 1.0, mode, &rd);
+ result->val = (1.0/3.0) * rd.val;
+ result->err = GSL_DBL_EPSILON * fabs(result->val) + fabs(1.0/3.0 * rd.err);
+ return status;
+ }
+}
+
+
+/* [Carlson, Numer. Math. 33 (1979) 1, (4.5)] */
+int
+gsl_sf_ellint_Kcomp_e(double k, gsl_mode_t mode, gsl_sf_result * result)
+{
+ if(k*k >= 1.0) {
+ DOMAIN_ERROR(result);
+ }
+ else if(k*k >= 1.0 - GSL_SQRT_DBL_EPSILON) {
+ /* [Abramowitz+Stegun, 17.3.33] */
+ const double y = 1.0 - k*k;
+ const double a[] = { 1.38629436112, 0.09666344259, 0.03590092383 };
+ const double b[] = { 0.5, 0.12498593597, 0.06880248576 };
+ const double ta = a[0] + y*(a[1] + y*a[2]);
+ const double tb = -log(y) * (b[0] * y*(b[1] + y*b[2]));
+ result->val = ta + tb;
+ result->err = 2.0 * GSL_DBL_EPSILON * result->val;
+ return GSL_SUCCESS;
+ }
+ else {
+ /* This was previously computed as,
+
+ return gsl_sf_ellint_RF_e(0.0, 1.0 - k*k, 1.0, mode, result);
+
+ but this underestimated the total error for small k, since the
+ argument y=1-k^2 is not exact (there is an absolute error of
+ GSL_DBL_EPSILON near y=0 due to cancellation in the subtraction).
+ Taking the singular behavior of -log(y) above gives an error
+ of 0.5*epsilon/y near y=0. (BJG) */
+
+ double y = 1.0 - k*k;
+ int status = gsl_sf_ellint_RF_e(0.0, y, 1.0, mode, result);
+ result->err += 0.5 * GSL_DBL_EPSILON / y;
+ return status ;
+ }
+}
+
+
+/* [Carlson, Numer. Math. 33 (1979) 1, (4.6)] */
+int
+gsl_sf_ellint_Ecomp_e(double k, gsl_mode_t mode, gsl_sf_result * result)
+{
+ if(k*k >= 1.0) {
+ DOMAIN_ERROR(result);
+ }
+ else if(k*k >= 1.0 - GSL_SQRT_DBL_EPSILON) {
+ /* [Abramowitz+Stegun, 17.3.36] */
+ const double y = 1.0 - k*k;
+ const double a[] = { 0.44325141463, 0.06260601220, 0.04757383546 };
+ const double b[] = { 0.24998368310, 0.09200180037, 0.04069697526 };
+ const double ta = 1.0 + y*(a[0] + y*(a[1] + a[2]*y));
+ const double tb = -y*log(y) * (b[0] + y*(b[1] + b[2]*y));
+ result->val = ta + tb;
+ result->err = 2.0 * GSL_DBL_EPSILON * result->val;
+ return GSL_SUCCESS;
+ }
+ else {
+ gsl_sf_result rf;
+ gsl_sf_result rd;
+ const double y = 1.0 - k*k;
+ const int rfstatus = gsl_sf_ellint_RF_e(0.0, y, 1.0, mode, &rf);
+ const int rdstatus = gsl_sf_ellint_RD_e(0.0, y, 1.0, mode, &rd);
+ result->val = rf.val - k*k/3.0 * rd.val;
+ result->err = rf.err + k*k/3.0 * rd.err;
+ return GSL_ERROR_SELECT_2(rfstatus, rdstatus);
+ }
+}
+
+/* [Carlson, Numer. Math. 33 (1979) 1, (4.6)] */
+int
+gsl_sf_ellint_Pcomp_e(double k, double n, gsl_mode_t mode, gsl_sf_result * result)
+{
+ if(k*k >= 1.0 || n >= 1.0) {
+ DOMAIN_ERROR(result);
+ }
+ /* FIXME: need to handle k ~=~ 1 cancellations */
+ else {
+ gsl_sf_result rf;
+ gsl_sf_result rj;
+ const double y = 1.0 - k*k;
+ const int rfstatus = gsl_sf_ellint_RF_e(0.0, y, 1.0, mode, &rf);
+ const int rjstatus = gsl_sf_ellint_RJ_e(0.0, y, 1.0, 1.0 + n, mode, &rj);
+ result->val = rf.val - (n/3.0) * rj.val;
+ result->err = rf.err + fabs(n/3.0) * rj.err;
+ return GSL_ERROR_SELECT_2(rfstatus, rjstatus);
+ }
+}
+
+
+
+/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
+
+#include "eval.h"
+
+double gsl_sf_ellint_Kcomp(double k, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_Kcomp_e(k, mode, &result));
+}
+
+double gsl_sf_ellint_Ecomp(double k, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_Ecomp_e(k, mode, &result));
+}
+
+double gsl_sf_ellint_Pcomp(double k, double n, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_Pcomp_e(k, n, mode, &result));
+}
+
+double gsl_sf_ellint_Dcomp(double k, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_Dcomp_e(k, mode, &result));
+}
+
+double gsl_sf_ellint_F(double phi, double k, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_F_e(phi, k, mode, &result));
+}
+
+double gsl_sf_ellint_E(double phi, double k, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_E_e(phi, k, mode, &result));
+}
+
+double gsl_sf_ellint_P(double phi, double k, double n, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_P_e(phi, k, n, mode, &result));
+}
+
+double gsl_sf_ellint_D(double phi, double k, double n, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_D_e(phi, k, n, mode, &result));
+}
+
+double gsl_sf_ellint_RC(double x, double y, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_RC_e(x, y, mode, &result));
+}
+
+double gsl_sf_ellint_RD(double x, double y, double z, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_RD_e(x, y, z, mode, &result));
+}
+
+double gsl_sf_ellint_RF(double x, double y, double z, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_RF_e(x, y, z, mode, &result));
+}
+
+double gsl_sf_ellint_RJ(double x, double y, double z, double p, gsl_mode_t mode)
+{
+ EVAL_RESULT(gsl_sf_ellint_RJ_e(x, y, z, p, mode, &result));
+}
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