/* specfunc/erfc.c

 * 

 * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003 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:  J. Theiler (modifications by G. Jungman) */

/*

 * See Hart et al, Computer Approximations, John Wiley and Sons, New York (1968)

 * (This applies only to the erfc8 stuff, which is the part

 *  of the original code that survives. I have replaced much of

 *  the other stuff with Chebyshev fits. These are simpler and

 *  more precise than the original approximations. [GJ])

 */

#include <config.h>

#include <gsl/gsl_math.h>

#include <gsl/gsl_errno.h>

#include <gsl/gsl_sf_exp.h>

#include <gsl/gsl_sf_erf.h>

#include "check.h"

#include "chebyshev.h"

#include "cheb_eval.c"

#define LogRootPi_  0.57236494292470008706

static double erfc8_sum(double x)

{

  /* estimates erfc(x) valid for 8 < x < 100 */

  /* This is based on index 5725 in Hart et al */

  static double P[] = {

      2.97886562639399288862,

      7.409740605964741794425,

      6.1602098531096305440906,

      5.019049726784267463450058,

      1.275366644729965952479585264,

      0.5641895835477550741253201704

  };

  static double Q[] = {

      3.3690752069827527677,

      9.608965327192787870698,

      17.08144074746600431571095,

      12.0489519278551290360340491,

      9.396034016235054150430579648,

      2.260528520767326969591866945,

      1.0

  };

  double num=0.0, den=0.0;

  int i;

  num = P[5];

  for (i=4; i>=0; --i) {

      num = x*num + P[i];

  }

  den = Q[6];

  for (i=5; i>=0; --i) {

      den = x*den + Q[i];

  }

  return num/den;

}

inline

static double erfc8(double x)

{

  double e;

  e = erfc8_sum(x);

  e *= exp(-x*x);

  return e;

}

inline

static double log_erfc8(double x)

{

  double e;

  e = erfc8_sum(x);

  e = log(e) - x*x;

  return e;

}

#if 0

/* Abramowitz+Stegun, 7.2.14 */

static double erfcasympsum(double x)

{

  int i;

  double e = 1.;

  double coef = 1.;

  for (i=1; i<5; ++i) {

    /* coef *= -(2*i-1)/(2*x*x); ??? [GJ] */

    coef *= -(2*i+1)/(i*(4*x*x*x*x));

    e += coef;

    /*

    if (fabs(coef) < 1.0e-15) break;

    if (fabs(coef) > 1.0e10) break;

    [GJ]: These tests are not useful. This function is only

    used below. Took them out; they gum up the pipeline.

    */

  }

  return e;

}

#endif /* 0 */

/* Abramowitz+Stegun, 7.1.5 */

static int erfseries(double x, gsl_sf_result * result)

{

  double coef = x;

  double e    = coef;

  double del;

  int k;

  for (k=1; k<30; ++k) {

    coef *= -x*x/k;

    del   = coef/(2.0*k+1.0);

    e += del;

  }

  result->val = 2.0 / M_SQRTPI * e;

  result->err = 2.0 / M_SQRTPI * (fabs(del) + GSL_DBL_EPSILON);

  return GSL_SUCCESS;

}

/* Chebyshev fit for erfc((t+1)/2), -1 < t < 1

 */

static double erfc_xlt1_data[20] = {

  1.06073416421769980345174155056,

 -0.42582445804381043569204735291,

  0.04955262679620434040357683080,

  0.00449293488768382749558001242,

 -0.00129194104658496953494224761,

 -0.00001836389292149396270416979,

  0.00002211114704099526291538556,

 -5.23337485234257134673693179020e-7,

 -2.78184788833537885382530989578e-7,

  1.41158092748813114560316684249e-8,

  2.72571296330561699984539141865e-9,

 -2.06343904872070629406401492476e-10,

 -2.14273991996785367924201401812e-11,

  2.22990255539358204580285098119e-12,

  1.36250074650698280575807934155e-13,

 -1.95144010922293091898995913038e-14,

 -6.85627169231704599442806370690e-16,

  1.44506492869699938239521607493e-16,

  2.45935306460536488037576200030e-18,

 -9.29599561220523396007359328540e-19

};

static cheb_series erfc_xlt1_cs = {

  erfc_xlt1_data,

  19,

  -1, 1,

  12

};

/* Chebyshev fit for erfc(x) exp(x^2), 1 < x < 5, x = 2t + 3, -1 < t < 1

 */

static double erfc_x15_data[25] = {

  0.44045832024338111077637466616,

 -0.143958836762168335790826895326,

  0.044786499817939267247056666937,

 -0.013343124200271211203618353102,

  0.003824682739750469767692372556,

 -0.001058699227195126547306482530,

  0.000283859419210073742736310108,

 -0.000073906170662206760483959432,

  0.000018725312521489179015872934,

 -4.62530981164919445131297264430e-6,

  1.11558657244432857487884006422e-6,

 -2.63098662650834130067808832725e-7,

  6.07462122724551777372119408710e-8,

 -1.37460865539865444777251011793e-8,

  3.05157051905475145520096717210e-9,

 -6.65174789720310713757307724790e-10,

  1.42483346273207784489792999706e-10,

 -3.00141127395323902092018744545e-11,

  6.22171792645348091472914001250e-12,

 -1.26994639225668496876152836555e-12,

  2.55385883033257575402681845385e-13,

 -5.06258237507038698392265499770e-14,

  9.89705409478327321641264227110e-15,

 -1.90685978789192181051961024995e-15,

  3.50826648032737849245113757340e-16

};

static cheb_series erfc_x15_cs = {

  erfc_x15_data,

  24,

  -1, 1,

  16

};

/* Chebyshev fit for erfc(x) x exp(x^2), 5 < x < 10, x = (5t + 15)/2, -1 < t < 1

 */

static double erfc_x510_data[20] = {

  1.11684990123545698684297865808,

  0.003736240359381998520654927536,

 -0.000916623948045470238763619870,

  0.000199094325044940833965078819,

 -0.000040276384918650072591781859,

  7.76515264697061049477127605790e-6,

 -1.44464794206689070402099225301e-6,

  2.61311930343463958393485241947e-7,

 -4.61833026634844152345304095560e-8,

  8.00253111512943601598732144340e-9,

 -1.36291114862793031395712122089e-9,

  2.28570483090160869607683087722e-10,

 -3.78022521563251805044056974560e-11,

  6.17253683874528285729910462130e-12,

 -9.96019290955316888445830597430e-13,

  1.58953143706980770269506726000e-13,

 -2.51045971047162509999527428316e-14,

  3.92607828989125810013581287560e-15,

 -6.07970619384160374392535453420e-16,

  9.12600607264794717315507477670e-17

};

static cheb_series erfc_x510_cs = {

  erfc_x510_data,

  19,

  -1, 1,

  12

};

#if 0

inline

static double

erfc_asymptotic(double x)

{

  return exp(-x*x)/x * erfcasympsum(x) / M_SQRTPI;

}

inline

static double

log_erfc_asymptotic(double x)

{

  return log(erfcasympsum(x)/x) - x*x - LogRootPi_;

}

#endif /* 0 */

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

int gsl_sf_erfc_e(double x, gsl_sf_result * result)

{

  const double ax = fabs(x);

  double e_val, e_err;

  /* CHECK_POINTER(result) */

  if(ax <= 1.0) {

    double t = 2.0*ax - 1.0;

    gsl_sf_result c;

    cheb_eval_e(&erfc_xlt1_cs, t, &c);

    e_val = c.val;

    e_err = c.err;

  }

  else if(ax <= 5.0) {

    double ex2 = exp(-x*x);

    double t = 0.5*(ax-3.0);

    gsl_sf_result c;

    cheb_eval_e(&erfc_x15_cs, t, &c);

    e_val = ex2 * c.val;

    e_err = ex2 * (c.err + 2.0*fabs(x)*GSL_DBL_EPSILON);

  }

  else if(ax < 10.0) {

    double exterm = exp(-x*x) / ax;

    double t = (2.0*ax - 15.0)/5.0;

    gsl_sf_result c;

    cheb_eval_e(&erfc_x510_cs, t, &c);

    e_val = exterm * c.val;

    e_err = exterm * (c.err + 2.0*fabs(x)*GSL_DBL_EPSILON + GSL_DBL_EPSILON);

  }

  else {

    e_val = erfc8(ax);

    e_err = (x*x + 1.0) * GSL_DBL_EPSILON * fabs(e_val);

  }

  if(x < 0.0) {

    result->val  = 2.0 - e_val;

    result->err  = e_err;

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

  }

  else {

    result->val  = e_val;

    result->err  = e_err;

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

  }

  return GSL_SUCCESS;

}

int gsl_sf_log_erfc_e(double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(x*x < 10.0*GSL_ROOT6_DBL_EPSILON) {

    const double y = x / M_SQRTPI;

    /* series for -1/2 Log[Erfc[Sqrt[Pi] y]] */

    const double c3 = (4.0 - M_PI)/3.0;

    const double c4 = 2.0*(1.0 - M_PI/3.0);

    const double c5 = -0.001829764677455021;  /* (96.0 - 40.0*M_PI + 3.0*M_PI*M_PI)/30.0  */

    const double c6 =  0.02629651521057465;   /* 2.0*(120.0 - 60.0*M_PI + 7.0*M_PI*M_PI)/45.0 */

    const double c7 = -0.01621575378835404;

    const double c8 =  0.00125993961762116;

    const double c9 =  0.00556964649138;

    const double c10 = -0.0045563339802;

    const double c11 =  0.0009461589032;

    const double c12 =  0.0013200243174;

    const double c13 = -0.00142906;

    const double c14 =  0.00048204;

    double series = c8 + y*(c9 + y*(c10 + y*(c11 + y*(c12 + y*(c13 + c14*y)))));

    series = y*(1.0 + y*(1.0 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*(c7 + y*series)))))));

    result->val = -2.0 * series;

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

    return GSL_SUCCESS;

  }

  /*

  don't like use of log1p(); added above series stuff for small x instead, should be ok [GJ]

  else if (fabs(x) < 1.0) {

    gsl_sf_result result_erf;

    gsl_sf_erf_e(x, &result_erf);

    result->val  = log1p(-result_erf.val);

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

    return GSL_SUCCESS;

  }

  */

  else if(x > 8.0) {

    result->val = log_erfc8(x);

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

    return GSL_SUCCESS;

  }

  else {

    gsl_sf_result result_erfc;

    gsl_sf_erfc_e(x, &result_erfc);

    result->val  = log(result_erfc.val);

    result->err  = fabs(result_erfc.err / result_erfc.val);

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

    return GSL_SUCCESS;

  }

}

int gsl_sf_erf_e(double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(fabs(x) < 1.0) {

    return erfseries(x, result);

  }

  else {

    gsl_sf_result result_erfc;

    gsl_sf_erfc_e(x, &result_erfc);

    result->val  = 1.0 - result_erfc.val;

    result->err  = result_erfc.err;

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

    return GSL_SUCCESS;

  }

}

int gsl_sf_erf_Z_e(double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  {

    const double ex2 = exp(-x*x/2.0);

    result->val  = ex2 / (M_SQRT2 * M_SQRTPI);

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

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

    CHECK_UNDERFLOW(result);

    return GSL_SUCCESS;

  }

}

int gsl_sf_erf_Q_e(double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  {

    gsl_sf_result result_erfc;

    int stat = gsl_sf_erfc_e(x/M_SQRT2, &result_erfc);

    result->val  = 0.5 * result_erfc.val;

    result->err  = 0.5 * result_erfc.err;

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

    return stat;

  }

}

int gsl_sf_hazard_e(double x, gsl_sf_result * result)

{

  if(x < 25.0)

  {

    gsl_sf_result result_ln_erfc;

    const int stat_l = gsl_sf_log_erfc_e(x/M_SQRT2, &result_ln_erfc);

    const double lnc = -0.22579135264472743236; /* ln(sqrt(2/pi)) */

    const double arg = lnc - 0.5*x*x - result_ln_erfc.val;

    const int stat_e = gsl_sf_exp_e(arg, result);

    result->err += 3.0 * (1.0 + fabs(x)) * GSL_DBL_EPSILON * fabs(result->val);

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

    return GSL_ERROR_SELECT_2(stat_l, stat_e);

  }

  else

  {

    const double ix2 = 1.0/(x*x);

    const double corrB = 1.0 - 9.0*ix2 * (1.0 - 11.0*ix2);

    const double corrM = 1.0 - 5.0*ix2 * (1.0 - 7.0*ix2 * corrB);

    const double corrT = 1.0 - ix2 * (1.0 - 3.0*ix2*corrM);

    result->val = x / corrT;

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

    return GSL_SUCCESS;

  }

}

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

#include "eval.h"

double gsl_sf_erfc(double x)

{

  EVAL_RESULT(gsl_sf_erfc_e(x, &result));

}

double gsl_sf_log_erfc(double x)

{

  EVAL_RESULT(gsl_sf_log_erfc_e(x, &result));

}

double gsl_sf_erf(double x)

{

  EVAL_RESULT(gsl_sf_erf_e(x, &result));

}

double gsl_sf_erf_Z(double x)

{

  EVAL_RESULT(gsl_sf_erf_Z_e(x, &result));

}

double gsl_sf_erf_Q(double x)

{

  EVAL_RESULT(gsl_sf_erf_Q_e(x, &result));

}

double gsl_sf_hazard(double x)

{

  EVAL_RESULT(gsl_sf_hazard_e(x, &result));

}