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airy.c

/* specfunc/airy.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 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_trig.h>
#include <gsl/gsl_sf_airy.h>

#include "error.h"
#include "check.h"

#include "chebyshev.h"
#include "cheb_eval_mode.c"

/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/


/* chebyshev expansions for Airy modulus and phase
   based on SLATEC r9aimp()

 Series for AM21       on the interval -1.25000D-01 to  0.
                                        with weighted error   2.89E-17
                                         log weighted error  16.54
                               significant figures required  14.15
                                    decimal places required  17.34

 Series for ATH1       on the interval -1.25000D-01 to  0.
                                        with weighted error   2.53E-17
                                         log weighted error  16.60
                               significant figures required  15.15
                                    decimal places required  17.38

 Series for AM22       on the interval -1.00000D+00 to -1.25000D-01
                                        with weighted error   2.99E-17
                                         log weighted error  16.52
                               significant figures required  14.57
                                    decimal places required  17.28

 Series for ATH2       on the interval -1.00000D+00 to -1.25000D-01
                                        with weighted error   2.57E-17
                                         log weighted error  16.59
                               significant figures required  15.07
                                    decimal places required  17.34
*/

static double am21_data[37] = {
  0.0065809191761485,
  0.0023675984685722,
  0.0001324741670371,
  0.0000157600904043,
  0.0000027529702663,
  0.0000006102679017,
  0.0000001595088468,
  0.0000000471033947,
  0.0000000152933871,
  0.0000000053590722,
  0.0000000020000910,
  0.0000000007872292,
  0.0000000003243103,
  0.0000000001390106,
  0.0000000000617011,
  0.0000000000282491,
  0.0000000000132979,
  0.0000000000064188,
  0.0000000000031697,
  0.0000000000015981,
  0.0000000000008213,
  0.0000000000004296,
  0.0000000000002284,
  0.0000000000001232,
  0.0000000000000675,
  0.0000000000000374,
  0.0000000000000210,
  0.0000000000000119,
  0.0000000000000068,
  0.0000000000000039,
  0.0000000000000023,
  0.0000000000000013,
  0.0000000000000008,
  0.0000000000000005,
  0.0000000000000003,
  0.0000000000000001,
  0.0000000000000001
};
static cheb_series am21_cs = {
  am21_data,
  36,
  -1, 1,
  20
};

static double ath1_data[36] = {
  -0.07125837815669365,
  -0.00590471979831451,
  -0.00012114544069499,
  -0.00000988608542270,
  -0.00000138084097352,
  -0.00000026142640172,
  -0.00000006050432589,
  -0.00000001618436223,
  -0.00000000483464911,
  -0.00000000157655272,
  -0.00000000055231518,
  -0.00000000020545441,
  -0.00000000008043412,
  -0.00000000003291252,
  -0.00000000001399875,
  -0.00000000000616151,
  -0.00000000000279614,
  -0.00000000000130428,
  -0.00000000000062373,
  -0.00000000000030512,
  -0.00000000000015239,
  -0.00000000000007758,
  -0.00000000000004020,
  -0.00000000000002117,
  -0.00000000000001132,
  -0.00000000000000614,
  -0.00000000000000337,
  -0.00000000000000188,
  -0.00000000000000105,
  -0.00000000000000060,
  -0.00000000000000034,
  -0.00000000000000020,
  -0.00000000000000011,
  -0.00000000000000007,
  -0.00000000000000004,
  -0.00000000000000002
};
static cheb_series ath1_cs = {
  ath1_data,
  35,
  -1, 1,
  15
};

static double am22_data[33] = {
 -0.01562844480625341,
  0.00778336445239681,
  0.00086705777047718,
  0.00015696627315611,
  0.00003563962571432,
  0.00000924598335425,
  0.00000262110161850,
  0.00000079188221651,
  0.00000025104152792,
  0.00000008265223206,
  0.00000002805711662,
  0.00000000976821090,
  0.00000000347407923,
  0.00000000125828132,
  0.00000000046298826,
  0.00000000017272825,
  0.00000000006523192,
  0.00000000002490471,
  0.00000000000960156,
  0.00000000000373448,
  0.00000000000146417,
  0.00000000000057826,
  0.00000000000022991,
  0.00000000000009197,
  0.00000000000003700,
  0.00000000000001496,
  0.00000000000000608,
  0.00000000000000248,
  0.00000000000000101,
  0.00000000000000041,
  0.00000000000000017,
  0.00000000000000007,
  0.00000000000000002
};
static cheb_series am22_cs = {
  am22_data,
  32,
  -1, 1,
  15
};

static double ath2_data[32] = {
   0.00440527345871877,
  -0.03042919452318455,
  -0.00138565328377179,
  -0.00018044439089549,
  -0.00003380847108327,
  -0.00000767818353522,
  -0.00000196783944371,
  -0.00000054837271158,
  -0.00000016254615505,
  -0.00000005053049981,
  -0.00000001631580701,
  -0.00000000543420411,
  -0.00000000185739855,
  -0.00000000064895120,
  -0.00000000023105948,
  -0.00000000008363282,
  -0.00000000003071196,
  -0.00000000001142367,
  -0.00000000000429811,
  -0.00000000000163389,
  -0.00000000000062693,
  -0.00000000000024260,
  -0.00000000000009461,
  -0.00000000000003716,
  -0.00000000000001469,
  -0.00000000000000584,
  -0.00000000000000233,
  -0.00000000000000093,
  -0.00000000000000037,
  -0.00000000000000015,
  -0.00000000000000006,
  -0.00000000000000002
};
static cheb_series ath2_cs = {
  ath2_data,
  31,
  -1, 1,
  16
};


/* Airy modulus and phase for x < -1 */
static
int
airy_mod_phase(const double x, gsl_mode_t mode, gsl_sf_result * mod, gsl_sf_result * phase)
{
  gsl_sf_result result_m;
  gsl_sf_result result_p;
  double m, p;
  double sqx;

  if(x < -2.0) {
    double z = 16.0/(x*x*x) + 1.0;
    cheb_eval_mode_e(&am21_cs, z, mode, &result_m);
    cheb_eval_mode_e(&ath1_cs, z, mode, &result_p);
  }
  else if(x <= -1.0) {
    double z = (16.0/(x*x*x) + 9.0)/7.0;
    cheb_eval_mode_e(&am22_cs, z, mode, &result_m);
    cheb_eval_mode_e(&ath2_cs, z, mode, &result_p);
  }
  else {
    mod->val = 0.0;
    mod->err = 0.0;
    phase->val = 0.0;
    phase->err = 0.0;
    GSL_ERROR ("x is greater than 1.0", GSL_EDOM);
  }

  m =  0.3125 + result_m.val;
  p = -0.625  + result_p.val;

  sqx = sqrt(-x);

  mod->val   = sqrt(m/sqx);
  mod->err  = fabs(mod->val) * (GSL_DBL_EPSILON + fabs(result_m.err/result_m.val));
  phase->val = M_PI_4 - x*sqx * p;
  phase->err = fabs(phase->val) * (GSL_DBL_EPSILON + fabs(result_p.err/result_p.val));

  return GSL_SUCCESS;
}



/* Chebyshev series for Ai(x) with x in [-1,1]
   based on SLATEC ai(x)

 series for aif        on the interval -1.00000d+00 to  1.00000d+00
                                   with weighted error   1.09e-19
                                    log weighted error  18.96
                          significant figures required  17.76
                               decimal places required  19.44

 series for aig        on the interval -1.00000d+00 to  1.00000d+00
                                   with weighted error   1.51e-17
                                    log weighted error  16.82
                          significant figures required  15.19
                               decimal places required  17.27
 */
static double ai_data_f[9] = {
  -0.03797135849666999750,
   0.05919188853726363857,
   0.00098629280577279975,
   0.00000684884381907656,
   0.00000002594202596219,
   0.00000000006176612774,
   0.00000000000010092454,
   0.00000000000000012014,
   0.00000000000000000010
};
static cheb_series aif_cs = {
  ai_data_f,
  8,
  -1, 1,
  8
};

static double ai_data_g[8] = {
   0.01815236558116127,
   0.02157256316601076,
   0.00025678356987483,
   0.00000142652141197,
   0.00000000457211492,
   0.00000000000952517,
   0.00000000000001392,
   0.00000000000000001
};
static cheb_series aig_cs = {
  ai_data_g,
  7,
  -1, 1,
  7
};


/* Chebvyshev series for Bi(x) with x in [-1,1]
   based on SLATEC bi(x)

 series for bif        on the interval -1.00000d+00 to  1.00000d+00
                                        with weighted error   1.88e-19
                                         log weighted error  18.72
                               significant figures required  17.74
                                    decimal places required  19.20

 series for big        on the interval -1.00000d+00 to  1.00000d+00
                                        with weighted error   2.61e-17
                                         log weighted error  16.58
                               significant figures required  15.17
                                    decimal places required  17.03
 */
static double data_bif[9] = {
  -0.01673021647198664948,
   0.10252335834249445610,
   0.00170830925073815165,
   0.00001186254546774468,
   0.00000004493290701779,
   0.00000000010698207143,
   0.00000000000017480643,
   0.00000000000000020810,
   0.00000000000000000018
};
static cheb_series bif_cs = {
  data_bif,
  8,
  -1, 1,
  8
};

static double data_big[8] = {
   0.02246622324857452,
   0.03736477545301955,
   0.00044476218957212,
   0.00000247080756363,
   0.00000000791913533,
   0.00000000001649807,
   0.00000000000002411,
   0.00000000000000002
};
static cheb_series big_cs = {
  data_big,
  7,
  -1, 1,
  7
};


/* Chebyshev series for Bi(x) with x in [1,8]
   based on SLATEC bi(x)
 */
static double data_bif2[10] = {
  0.0998457269381604100,
  0.4786249778630055380,
  0.0251552119604330118,
  0.0005820693885232645,
  0.0000074997659644377,
  0.0000000613460287034,
  0.0000000003462753885,
  0.0000000000014288910,
  0.0000000000000044962,
  0.0000000000000000111
};
static cheb_series bif2_cs = {
  data_bif2,
  9,
  -1, 1,
  9
};

static double data_big2[10] = {
  0.033305662145514340,
  0.161309215123197068,
  0.0063190073096134286,
  0.0001187904568162517,
  0.0000013045345886200,
  0.0000000093741259955,
  0.0000000000474580188,
  0.0000000000001783107,
  0.0000000000000005167,
  0.0000000000000000011
};
static cheb_series big2_cs = {
  data_big2,
  9,
  -1, 1,
  9
};


/* chebyshev for Ai(x) asymptotic factor 
   based on SLATEC aie()

 Series for AIP        on the interval  0.          to  1.00000D+00
                   with weighted error   5.10E-17
                    log weighted error  16.29
          significant figures required  14.41
               decimal places required  17.06
               
 [GJ] Sun Apr 19 18:14:31 EDT 1998
 There was something wrong with these coefficients. I was getting
 errors after 3 or 4 digits. So I recomputed this table. Now I get
 double precision agreement with Mathematica. But it does not seem
 possible that the small differences here would account for the
 original discrepancy. There must have been something wrong with my
 original usage...
*/
static double data_aip[36] = {
 -0.0187519297793867540198,
 -0.0091443848250055004725,
  0.0009010457337825074652,
 -0.0001394184127221491507,
  0.0000273815815785209370,
 -0.0000062750421119959424,
  0.0000016064844184831521,
 -0.0000004476392158510354,
  0.0000001334635874651668,
 -0.0000000420735334263215,
  0.0000000139021990246364,
 -0.0000000047831848068048,
  0.0000000017047897907465,
 -0.0000000006268389576018,
  0.0000000002369824276612,
 -0.0000000000918641139267,
  0.0000000000364278543037,
 -0.0000000000147475551725,
  0.0000000000060851006556,
 -0.0000000000025552772234,
  0.0000000000010906187250,
 -0.0000000000004725870319,
  0.0000000000002076969064,
 -0.0000000000000924976214,
  0.0000000000000417096723,
 -0.0000000000000190299093,
  0.0000000000000087790676,
 -0.0000000000000040927557,
  0.0000000000000019271068,
 -0.0000000000000009160199,
  0.0000000000000004393567,
 -0.0000000000000002125503,
  0.0000000000000001036735,
 -0.0000000000000000509642,
  0.0000000000000000252377,
 -0.0000000000000000125793
/*
  -.0187519297793868
  -.0091443848250055,
   .0009010457337825,
  -.0001394184127221,
   .0000273815815785,
  -.0000062750421119,
   .0000016064844184,
  -.0000004476392158,
   .0000001334635874,
  -.0000000420735334,
   .0000000139021990,
  -.0000000047831848,
   .0000000017047897,
  -.0000000006268389,
   .0000000002369824,
  -.0000000000918641,
   .0000000000364278,
  -.0000000000147475,
   .0000000000060851,
  -.0000000000025552,
   .0000000000010906,
  -.0000000000004725,
   .0000000000002076,
  -.0000000000000924,
   .0000000000000417,
  -.0000000000000190,
   .0000000000000087,
  -.0000000000000040,
   .0000000000000019,
  -.0000000000000009,
   .0000000000000004,
  -.0000000000000002,
   .0000000000000001,
  -.0000000000000000
*/
};
static cheb_series aip_cs = {
  data_aip,
  35,
  -1, 1,
  17
};


/* chebyshev for Bi(x) asymptotic factor 
   based on SLATEC bie()

 Series for BIP        on the interval  1.25000D-01 to  3.53553D-01
                   with weighted error   1.91E-17
                    log weighted error  16.72
          significant figures required  15.35
               decimal places required  17.41

 Series for BIP2       on the interval  0.          to  1.25000D-01
                   with weighted error   1.05E-18
                    log weighted error  17.98
          significant figures required  16.74
               decimal places required  18.71
*/
static double data_bip[24] = {
  -0.08322047477943447,
   0.01146118927371174,
   0.00042896440718911,
  -0.00014906639379950,
  -0.00001307659726787,
   0.00000632759839610,
  -0.00000042226696982,
  -0.00000019147186298,
   0.00000006453106284,
  -0.00000000784485467,
  -0.00000000096077216,
   0.00000000070004713,
  -0.00000000017731789,
   0.00000000002272089,
   0.00000000000165404,
  -0.00000000000185171,
   0.00000000000059576,
  -0.00000000000012194,
   0.00000000000001334,
   0.00000000000000172,
  -0.00000000000000145,
   0.00000000000000049,
  -0.00000000000000011,
   0.00000000000000001
};
static cheb_series bip_cs = {
  data_bip,
  23,
  -1, 1,
  14
};

static double data_bip2[29] = {    
  -0.113596737585988679,
   0.0041381473947881595,
   0.0001353470622119332,
   0.0000104273166530153,
   0.0000013474954767849,
   0.0000001696537405438,
  -0.0000000100965008656,
  -0.0000000167291194937,
  -0.0000000045815364485,
   0.0000000003736681366,
   0.0000000005766930320,
   0.0000000000621812650,
  -0.0000000000632941202,
  -0.0000000000149150479,
   0.0000000000078896213,
   0.0000000000024960513,
  -0.0000000000012130075,
  -0.0000000000003740493,
   0.0000000000002237727,
   0.0000000000000474902,
  -0.0000000000000452616,
  -0.0000000000000030172,
   0.0000000000000091058,
  -0.0000000000000009814,
  -0.0000000000000016429,
   0.0000000000000005533,
   0.0000000000000002175,
  -0.0000000000000001737,
  -0.0000000000000000010
};
static cheb_series bip2_cs = {
  data_bip2,
  28,
  -1, 1,
  10
};


/* assumes x >= 1.0 */
inline static int 
airy_aie(const double x, gsl_mode_t mode, gsl_sf_result * result)
{
  double sqx = sqrt(x);
  double z   = 2.0/(x*sqx) - 1.0;
  double y   = sqrt(sqx);
  gsl_sf_result result_c;
  cheb_eval_mode_e(&aip_cs, z, mode, &result_c);
  result->val = (0.28125 + result_c.val)/y;
  result->err = result_c.err/y + GSL_DBL_EPSILON * fabs(result->val);
  return GSL_SUCCESS;
}

/* assumes x >= 2.0 */
static int airy_bie(const double x, gsl_mode_t mode, gsl_sf_result * result)
{
  const double ATR =  8.7506905708484345;
  const double BTR = -2.0938363213560543;

  if(x < 4.0) {
    double sqx = sqrt(x);
    double z   = ATR/(x*sqx) + BTR;
    double y   = sqrt(sqx);
    gsl_sf_result result_c;
    cheb_eval_mode_e(&bip_cs, z, mode, &result_c);
    result->val = (0.625 + result_c.val)/y;
    result->err = result_c.err/y + GSL_DBL_EPSILON * fabs(result->val);
  }
  else {
    double sqx = sqrt(x);
    double z   = 16.0/(x*sqx) - 1.0;
    double y   = sqrt(sqx);
    gsl_sf_result result_c;
    cheb_eval_mode_e(&bip2_cs, z, mode, &result_c);
    result->val = (0.625 + result_c.val)/y;
    result->err = result_c.err/y + GSL_DBL_EPSILON * fabs(result->val);
  }

  return GSL_SUCCESS;
}


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

int
gsl_sf_airy_Ai_e(const double x, const gsl_mode_t mode, gsl_sf_result * result)
{
  /* CHECK_POINTER(result) */

  if(x < -1.0) {
    gsl_sf_result mod;
    gsl_sf_result theta;
    gsl_sf_result cos_result;
    int stat_mp  = airy_mod_phase(x, mode, &mod, &theta);
    int stat_cos = gsl_sf_cos_err_e(theta.val, theta.err, &cos_result);
    result->val  = mod.val * cos_result.val;
    result->err  = fabs(mod.val * cos_result.err) + fabs(cos_result.val * mod.err);
    result->err += GSL_DBL_EPSILON * fabs(result->val);
    return GSL_ERROR_SELECT_2(stat_mp, stat_cos);
  }
  else if(x <= 1.0) {
    const double z = x*x*x;
    gsl_sf_result result_c0;
    gsl_sf_result result_c1;
    cheb_eval_mode_e(&aif_cs, z, mode, &result_c0);
    cheb_eval_mode_e(&aig_cs, z, mode, &result_c1);
    result->val  = 0.375 + (result_c0.val - x*(0.25 + result_c1.val));
    result->err  = result_c0.err + fabs(x*result_c1.err);
    result->err += GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else {
    double x32 = x * sqrt(x);
    double s   = exp(-2.0*x32/3.0);
    gsl_sf_result result_aie;
    int stat_aie = airy_aie(x, mode, &result_aie);
    result->val  = result_aie.val * s;
    result->err  = result_aie.err * s + result->val * x32 * GSL_DBL_EPSILON;
    result->err += GSL_DBL_EPSILON * fabs(result->val);
    CHECK_UNDERFLOW(result);
    return stat_aie;
  }
}


int
gsl_sf_airy_Ai_scaled_e(const double x, gsl_mode_t mode, gsl_sf_result * result)
{
  /* CHECK_POINTER(result) */

  if(x < -1.0) {
    gsl_sf_result mod;
    gsl_sf_result theta;
    gsl_sf_result cos_result;
    int stat_mp  = airy_mod_phase(x, mode, &mod, &theta);
    int stat_cos = gsl_sf_cos_err_e(theta.val, theta.err, &cos_result);
    result->val  = mod.val * cos_result.val;
    result->err  = fabs(mod.val * cos_result.err) + fabs(cos_result.val * mod.err);
    result->err += GSL_DBL_EPSILON * fabs(result->val);
    return GSL_ERROR_SELECT_2(stat_mp, stat_cos);
  }
  else if(x <= 1.0) {
    const double z = x*x*x;
    gsl_sf_result result_c0;
    gsl_sf_result result_c1;
    cheb_eval_mode_e(&aif_cs, z, mode, &result_c0);
    cheb_eval_mode_e(&aig_cs, z, mode, &result_c1);
    result->val  = 0.375 + (result_c0.val - x*(0.25 + result_c1.val));
    result->err  = result_c0.err + fabs(x*result_c1.err);
    result->err += GSL_DBL_EPSILON * fabs(result->val);

    if(x > 0.0) {
      const double scale = exp(2.0/3.0 * sqrt(z));
      result->val *= scale;
      result->err *= scale;
    }

    return GSL_SUCCESS;
  }
  else {
    return airy_aie(x, mode, result);
  }
}


int gsl_sf_airy_Bi_e(const double x, gsl_mode_t mode, gsl_sf_result * result)
{
  /* CHECK_POINTER(result) */
  if(x < -1.0) {
    gsl_sf_result mod;
    gsl_sf_result theta;
    gsl_sf_result sin_result;
    int stat_mp  = airy_mod_phase(x, mode, &mod, &theta);
    int stat_sin = gsl_sf_sin_err_e(theta.val, theta.err, &sin_result);
    result->val  = mod.val * sin_result.val;
    result->err  = fabs(mod.val * sin_result.err) + fabs(sin_result.val * mod.err);
    result->err += GSL_DBL_EPSILON * fabs(result->val);
    return GSL_ERROR_SELECT_2(stat_mp, stat_sin);
  }
  else if(x < 1.0) {
    const double z = x*x*x;
    gsl_sf_result result_c0;
    gsl_sf_result result_c1;
    cheb_eval_mode_e(&bif_cs, z, mode, &result_c0);
    cheb_eval_mode_e(&big_cs, z, mode, &result_c1);
    result->val  = 0.625 + result_c0.val + x*(0.4375 + result_c1.val);
    result->err  = result_c0.err + fabs(x * result_c1.err);
    result->err += GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else if(x <= 2.0) {
    const double z = (2.0*x*x*x - 9.0)/7.0;
    gsl_sf_result result_c0;
    gsl_sf_result result_c1;
    cheb_eval_mode_e(&bif2_cs, z, mode, &result_c0);
    cheb_eval_mode_e(&big2_cs, z, mode, &result_c1);
    result->val  = 1.125 + result_c0.val + x*(0.625 + result_c1.val);
    result->err  = result_c0.err + fabs(x * result_c1.err);
    result->err += GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else {
    const double y = 2.0*x*sqrt(x)/3.0;
    const double s = exp(y);

    if(y > GSL_LOG_DBL_MAX - 1.0) {
      OVERFLOW_ERROR(result);
    }
    else {
      gsl_sf_result result_bie;
      int stat_bie = airy_bie(x, mode, &result_bie);
      result->val  = result_bie.val * s;
      result->err  = result_bie.err * s + fabs(1.5*y * (GSL_DBL_EPSILON * result->val));
      result->err += GSL_DBL_EPSILON * fabs(result->val);
      return stat_bie;
    }
  }
}


int
gsl_sf_airy_Bi_scaled_e(const double x, gsl_mode_t mode, gsl_sf_result * result)
{
  /* CHECK_POINTER(result) */

  if(x < -1.0) {
    gsl_sf_result mod;
    gsl_sf_result theta;
    gsl_sf_result sin_result;
    int stat_mp  = airy_mod_phase(x, mode, &mod, &theta);
    int stat_sin = gsl_sf_sin_err_e(theta.val, theta.err, &sin_result);
    result->val  = mod.val * sin_result.val;
    result->err  = fabs(mod.val * sin_result.err) + fabs(sin_result.val * mod.err);
    result->err += GSL_DBL_EPSILON * fabs(result->val);
    return GSL_ERROR_SELECT_2(stat_mp, stat_sin);
  }
  else if(x < 1.0) {
    const double z = x*x*x;
    gsl_sf_result result_c0;
    gsl_sf_result result_c1;
    cheb_eval_mode_e(&bif_cs, z, mode, &result_c0);
    cheb_eval_mode_e(&big_cs, z, mode, &result_c1);
    result->val  = 0.625 + result_c0.val + x*(0.4375 + result_c1.val);
    result->err  = result_c0.err + fabs(x * result_c1.err);
    result->err += GSL_DBL_EPSILON * fabs(result->val);
    if(x > 0.0) {
      const double scale = exp(-2.0/3.0 * sqrt(z));
      result->val *= scale;
      result->err *= scale;
    }
    return GSL_SUCCESS;
  }
  else if(x <= 2.0) {
    const double x3 = x*x*x;
    const double z  = (2.0*x3 - 9.0)/7.0;
    const double s  = exp(-2.0/3.0 * sqrt(x3));
    gsl_sf_result result_c0;
    gsl_sf_result result_c1;
    cheb_eval_mode_e(&bif2_cs, z, mode, &result_c0);
    cheb_eval_mode_e(&big2_cs, z, mode, &result_c1);
    result->val  = s * (1.125 + result_c0.val + x*(0.625 + result_c1.val));
    result->err  = s * (result_c0.err + fabs(x * result_c1.err));
    result->err += GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else {
    return airy_bie(x, mode, result);
  }
}


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

#include "eval.h"

double gsl_sf_airy_Ai(const double x, gsl_mode_t mode)
{
  EVAL_RESULT(gsl_sf_airy_Ai_e(x, mode, &result));
}

double gsl_sf_airy_Ai_scaled(const double x, gsl_mode_t mode)
{
  EVAL_RESULT(gsl_sf_airy_Ai_scaled_e(x, mode, &result));
}

double gsl_sf_airy_Bi(const double x, gsl_mode_t mode)
{
  EVAL_RESULT(gsl_sf_airy_Bi_e(x, mode, &result));
}

double gsl_sf_airy_Bi_scaled(const double x, gsl_mode_t mode)
{
  EVAL_RESULT(gsl_sf_airy_Bi_scaled_e(x, mode, &result));
}



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