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fresnl.c

/*                                        fresnl.c
 *
 *    Fresnel integral
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, S, C;
 * void fresnl();
 *
 * fresnl( x, _&S, _&C );
 *
 *
 * DESCRIPTION:
 *
 * Evaluates the Fresnel integrals
 *
 *           x
 *           -
 *          | |
 * C(x) =   |   cos(pi/2 t**2) dt,
 *        | |
 *         -
 *          0
 *
 *           x
 *           -
 *          | |
 * S(x) =   |   sin(pi/2 t**2) dt.
 *        | |
 *         -
 *          0
 *
 *
 * The integrals are evaluated by a power series for x < 1.
 * For x >= 1 auxiliary functions f(x) and g(x) are employed
 * such that
 *
 * C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
 * S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )
 *
 *
 *
 * ACCURACY:
 *
 *  Relative error.
 *
 * Arithmetic  function   domain     # trials      peak         rms
 *   IEEE       S(x)      0, 10       10000       2.0e-15     3.2e-16
 *   IEEE       C(x)      0, 10       10000       1.8e-15     3.3e-16
 *   DEC        S(x)      0, 10        6000       2.2e-16     3.9e-17
 *   DEC        C(x)      0, 10        5000       2.3e-16     3.9e-17
 */

/*
Cephes Math Library Release 2.1:  January, 1989
Copyright 1984, 1987, 1989 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

#include <math.h>
#include "mconf.h"
#include "cephes.h"

/* S(x) for small x */
#ifdef UNK
static double sn[6] = {
-2.99181919401019853726E3,
 7.08840045257738576863E5,
-6.29741486205862506537E7,
 2.54890880573376359104E9,
-4.42979518059697779103E10,
 3.18016297876567817986E11,
};
static double sd[6] = {
/* 1.00000000000000000000E0,*/
 2.81376268889994315696E2,
 4.55847810806532581675E4,
 5.17343888770096400730E6,
 4.19320245898111231129E8,
 2.24411795645340920940E10,
 6.07366389490084639049E11,
};
#endif
#ifdef DEC
static unsigned short sn[24] = {
0143072,0176433,0065455,0127034,
0045055,0007200,0134540,0026661,
0146560,0035061,0023667,0127545,
0050027,0166503,0002673,0153756,
0151045,0002721,0121737,0102066,
0051624,0013177,0033451,0021271,
};
static unsigned short sd[24] = {
/*0040200,0000000,0000000,0000000,*/
0042214,0130051,0112070,0101617,
0044062,0010307,0172346,0152510,
0045635,0160575,0143200,0136642,
0047307,0171215,0127457,0052361,
0050647,0031447,0032621,0013510,
0052015,0064733,0117362,0012653,
};
#endif
#ifdef IBMPC
static unsigned short sn[24] = {
0xb5c3,0x6d65,0x5fa3,0xc0a7,
0x05b6,0x172c,0xa1d0,0x4125,
0xf5ed,0x24f6,0x0746,0xc18e,
0x7afe,0x60b7,0xfda8,0x41e2,
0xf087,0x347b,0xa0ba,0xc224,
0x2457,0xe6e5,0x82cf,0x4252,
};
static unsigned short sd[24] = {
/*0x0000,0x0000,0x0000,0x3ff0,*/
0x1072,0x3287,0x9605,0x4071,
0xdaa9,0xfe9c,0x4218,0x40e6,
0x17b4,0xb8d0,0xbc2f,0x4153,
0xea9e,0xb5e5,0xfe51,0x41b8,
0x22e9,0xe6b2,0xe664,0x4214,
0x42b5,0x73de,0xad3b,0x4261,
};
#endif
#ifdef MIEEE
static unsigned short sn[24] = {
0xc0a7,0x5fa3,0x6d65,0xb5c3,
0x4125,0xa1d0,0x172c,0x05b6,
0xc18e,0x0746,0x24f6,0xf5ed,
0x41e2,0xfda8,0x60b7,0x7afe,
0xc224,0xa0ba,0x347b,0xf087,
0x4252,0x82cf,0xe6e5,0x2457,
};
static unsigned short sd[24] = {
/*0x3ff0,0x0000,0x0000,0x0000,*/
0x4071,0x9605,0x3287,0x1072,
0x40e6,0x4218,0xfe9c,0xdaa9,
0x4153,0xbc2f,0xb8d0,0x17b4,
0x41b8,0xfe51,0xb5e5,0xea9e,
0x4214,0xe664,0xe6b2,0x22e9,
0x4261,0xad3b,0x73de,0x42b5,
};
#endif

/* C(x) for small x */
#ifdef UNK
static double cn[6] = {
-4.98843114573573548651E-8,
 9.50428062829859605134E-6,
-6.45191435683965050962E-4,
 1.88843319396703850064E-2,
-2.05525900955013891793E-1,
 9.99999999999999998822E-1,
};
static double cd[7] = {
 3.99982968972495980367E-12,
 9.15439215774657478799E-10,
 1.25001862479598821474E-7,
 1.22262789024179030997E-5,
 8.68029542941784300606E-4,
 4.12142090722199792936E-2,
 1.00000000000000000118E0,
};
#endif
#ifdef DEC
static unsigned short cn[24] = {
0132126,0040141,0063733,0013231,
0034037,0072223,0010200,0075637,
0135451,0021020,0073264,0036057,
0036632,0131520,0101316,0060233,
0137522,0072541,0136124,0132202,
0040200,0000000,0000000,0000000,
};
static unsigned short cd[28] = {
0026614,0135503,0051776,0032631,
0030573,0121116,0154033,0126712,
0032406,0034100,0012442,0106212,
0034115,0017567,0150520,0164623,
0035543,0106171,0177336,0146351,
0037050,0150073,0000607,0171635,
0040200,0000000,0000000,0000000,
};
#endif
#ifdef IBMPC
static unsigned short cn[24] = {
0x62d3,0x2cfb,0xc80c,0xbe6a,
0x0f74,0x6210,0xee92,0x3ee3,
0x8786,0x0ed6,0x2442,0xbf45,
0xcc13,0x1059,0x566a,0x3f93,
0x9690,0x378a,0x4eac,0xbfca,
0x0000,0x0000,0x0000,0x3ff0,
};
static unsigned short cd[28] = {
0xc6b3,0x6a7f,0x9768,0x3d91,
0x75b9,0xdb03,0x7449,0x3e0f,
0x5191,0x02a4,0xc708,0x3e80,
0x1d32,0xfa2a,0xa3ee,0x3ee9,
0xd99d,0x3fdb,0x718f,0x3f4c,
0xfe74,0x6030,0x1a07,0x3fa5,
0x0000,0x0000,0x0000,0x3ff0,
};
#endif
#ifdef MIEEE
static unsigned short cn[24] = {
0xbe6a,0xc80c,0x2cfb,0x62d3,
0x3ee3,0xee92,0x6210,0x0f74,
0xbf45,0x2442,0x0ed6,0x8786,
0x3f93,0x566a,0x1059,0xcc13,
0xbfca,0x4eac,0x378a,0x9690,
0x3ff0,0x0000,0x0000,0x0000,
};
static unsigned short cd[28] = {
0x3d91,0x9768,0x6a7f,0xc6b3,
0x3e0f,0x7449,0xdb03,0x75b9,
0x3e80,0xc708,0x02a4,0x5191,
0x3ee9,0xa3ee,0xfa2a,0x1d32,
0x3f4c,0x718f,0x3fdb,0xd99d,
0x3fa5,0x1a07,0x6030,0xfe74,
0x3ff0,0x0000,0x0000,0x0000,
};
#endif

/* Auxiliary function f(x) */
#ifdef UNK
static double fn[10] = {
  4.21543555043677546506E-1,
  1.43407919780758885261E-1,
  1.15220955073585758835E-2,
  3.45017939782574027900E-4,
  4.63613749287867322088E-6,
  3.05568983790257605827E-8,
  1.02304514164907233465E-10,
  1.72010743268161828879E-13,
  1.34283276233062758925E-16,
  3.76329711269987889006E-20,
};
static double fd[10] = {
/*  1.00000000000000000000E0,*/
  7.51586398353378947175E-1,
  1.16888925859191382142E-1,
  6.44051526508858611005E-3,
  1.55934409164153020873E-4,
  1.84627567348930545870E-6,
  1.12699224763999035261E-8,
  3.60140029589371370404E-11,
  5.88754533621578410010E-14,
  4.52001434074129701496E-17,
  1.25443237090011264384E-20,
};
#endif
#ifdef DEC
static unsigned short fn[40] = {
0037727,0152216,0106601,0016214,
0037422,0154606,0112710,0071355,
0036474,0143453,0154253,0166545,
0035264,0161606,0022250,0073743,
0033633,0110036,0024653,0136246,
0032003,0036652,0041164,0036413,
0027740,0174122,0046305,0036726,
0025501,0125270,0121317,0167667,
0023032,0150555,0076175,0047443,
0020061,0133570,0070130,0027657,
};
static unsigned short fd[40] = {
/*0040200,0000000,0000000,0000000,*/
0040100,0063767,0054413,0151452,
0037357,0061566,0007243,0065754,
0036323,0005365,0033552,0133625,
0035043,0101123,0000275,0165402,
0033367,0146614,0110623,0023647,
0031501,0116644,0125222,0144263,
0027436,0062051,0117235,0001411,
0025204,0111543,0056370,0036201,
0022520,0071351,0015227,0122144,
0017554,0172240,0112713,0005006,
};
#endif
#ifdef IBMPC
static unsigned short fn[40] = {
0x2391,0xd1b0,0xfa91,0x3fda,
0x0e5e,0xd2b9,0x5b30,0x3fc2,
0x7dad,0x7b15,0x98e5,0x3f87,
0x0efc,0xc495,0x9c70,0x3f36,
0x7795,0xc535,0x7203,0x3ed3,
0x87a1,0x484e,0x67b5,0x3e60,
0xa7bb,0x4998,0x1f0a,0x3ddc,
0xfdf7,0x1459,0x3557,0x3d48,
0xa9e4,0xaf8f,0x5a2d,0x3ca3,
0x05f6,0x0e0b,0x36ef,0x3be6,
};
static unsigned short fd[40] = {
/*0x0000,0x0000,0x0000,0x3ff0,*/
0x7a65,0xeb21,0x0cfe,0x3fe8,
0x6d7d,0xc1d4,0xec6e,0x3fbd,
0x56f3,0xa6ed,0x615e,0x3f7a,
0xbd60,0x6017,0x704a,0x3f24,
0x64f5,0x9232,0xf9b1,0x3ebe,
0x5916,0x9552,0x33b4,0x3e48,
0xa061,0x33d3,0xcc85,0x3dc3,
0x0790,0x6b9f,0x926c,0x3d30,
0xf48d,0x2352,0x0e5d,0x3c8a,
0x6141,0x12b9,0x9e94,0x3bcd,
};
#endif
#ifdef MIEEE
static unsigned short fn[40] = {
0x3fda,0xfa91,0xd1b0,0x2391,
0x3fc2,0x5b30,0xd2b9,0x0e5e,
0x3f87,0x98e5,0x7b15,0x7dad,
0x3f36,0x9c70,0xc495,0x0efc,
0x3ed3,0x7203,0xc535,0x7795,
0x3e60,0x67b5,0x484e,0x87a1,
0x3ddc,0x1f0a,0x4998,0xa7bb,
0x3d48,0x3557,0x1459,0xfdf7,
0x3ca3,0x5a2d,0xaf8f,0xa9e4,
0x3be6,0x36ef,0x0e0b,0x05f6,
};
static unsigned short fd[40] = {
/*0x3ff0,0x0000,0x0000,0x0000,*/
0x3fe8,0x0cfe,0xeb21,0x7a65,
0x3fbd,0xec6e,0xc1d4,0x6d7d,
0x3f7a,0x615e,0xa6ed,0x56f3,
0x3f24,0x704a,0x6017,0xbd60,
0x3ebe,0xf9b1,0x9232,0x64f5,
0x3e48,0x33b4,0x9552,0x5916,
0x3dc3,0xcc85,0x33d3,0xa061,
0x3d30,0x926c,0x6b9f,0x0790,
0x3c8a,0x0e5d,0x2352,0xf48d,
0x3bcd,0x9e94,0x12b9,0x6141,
};
#endif


/* Auxiliary function g(x) */
#ifdef UNK
static double gn[11] = {
  5.04442073643383265887E-1,
  1.97102833525523411709E-1,
  1.87648584092575249293E-2,
  6.84079380915393090172E-4,
  1.15138826111884280931E-5,
  9.82852443688422223854E-8,
  4.45344415861750144738E-10,
  1.08268041139020870318E-12,
  1.37555460633261799868E-15,
  8.36354435630677421531E-19,
  1.86958710162783235106E-22,
};
static double gd[11] = {
/*  1.00000000000000000000E0,*/
  1.47495759925128324529E0,
  3.37748989120019970451E-1,
  2.53603741420338795122E-2,
  8.14679107184306179049E-4,
  1.27545075667729118702E-5,
  1.04314589657571990585E-7,
  4.60680728146520428211E-10,
  1.10273215066240270757E-12,
  1.38796531259578871258E-15,
  8.39158816283118707363E-19,
  1.86958710162783236342E-22,
};
#endif
#ifdef DEC
static unsigned short gn[44] = {
0040001,0021435,0120406,0053123,
0037511,0152523,0037703,0122011,
0036631,0134302,0122721,0110235,
0035463,0051712,0043215,0114732,
0034101,0025677,0147725,0057630,
0032323,0010342,0067523,0002206,
0030364,0152247,0110007,0054107,
0026230,0057654,0035464,0047124,
0023706,0036401,0167705,0045440,
0021166,0154447,0105632,0142461,
0016142,0002353,0011175,0170530,
};
static unsigned short gd[44] = {
/*0040200,0000000,0000000,0000000,*/
0040274,0145551,0016742,0127005,
0037654,0166557,0076416,0015165,
0036717,0140217,0030675,0050111,
0035525,0110060,0076405,0070502,
0034125,0176061,0060120,0031730,
0032340,0001615,0054343,0120501,
0030375,0041414,0070747,0107060,
0026233,0031034,0160757,0074526,
0023710,0003341,0137100,0144664,
0021167,0126414,0023774,0015435,
0016142,0002353,0011175,0170530,
};
#endif
#ifdef IBMPC
static unsigned short gn[44] = {
0xcaca,0xb420,0x2463,0x3fe0,
0x7481,0x67f8,0x3aaa,0x3fc9,
0x3214,0x54ba,0x3718,0x3f93,
0xb33b,0x48d1,0x6a79,0x3f46,
0xabf3,0xf9fa,0x2577,0x3ee8,
0x6091,0x4dea,0x621c,0x3e7a,
0xeb09,0xf200,0x9a94,0x3dfe,
0x89cb,0x8766,0x0bf5,0x3d73,
0xa964,0x3df8,0xc7a0,0x3cd8,
0x58a6,0xf173,0xdb24,0x3c2e,
0xbe2b,0x624f,0x409d,0x3b6c,
};
static unsigned short gd[44] = {
/*0x0000,0x0000,0x0000,0x3ff0,*/
0x55c1,0x23bc,0x996d,0x3ff7,
0xc34f,0xefa1,0x9dad,0x3fd5,
0xaa09,0xe637,0xf811,0x3f99,
0xae28,0x0fa0,0xb206,0x3f4a,
0x067b,0x2c0a,0xbf86,0x3eea,
0x7428,0xab1c,0x0071,0x3e7c,
0xf1c6,0x8e3c,0xa861,0x3dff,
0xef2b,0x9c3d,0x6643,0x3d73,
0x1936,0x37c8,0x00dc,0x3cd9,
0x8364,0x84ff,0xf5a1,0x3c2e,
0xbe2b,0x624f,0x409d,0x3b6c,
};
#endif
#ifdef MIEEE
static unsigned short gn[44] = {
0x3fe0,0x2463,0xb420,0xcaca,
0x3fc9,0x3aaa,0x67f8,0x7481,
0x3f93,0x3718,0x54ba,0x3214,
0x3f46,0x6a79,0x48d1,0xb33b,
0x3ee8,0x2577,0xf9fa,0xabf3,
0x3e7a,0x621c,0x4dea,0x6091,
0x3dfe,0x9a94,0xf200,0xeb09,
0x3d73,0x0bf5,0x8766,0x89cb,
0x3cd8,0xc7a0,0x3df8,0xa964,
0x3c2e,0xdb24,0xf173,0x58a6,
0x3b6c,0x409d,0x624f,0xbe2b,
};
static unsigned short gd[44] = {
/*0x3ff0,0x0000,0x0000,0x0000,*/
0x3ff7,0x996d,0x23bc,0x55c1,
0x3fd5,0x9dad,0xefa1,0xc34f,
0x3f99,0xf811,0xe637,0xaa09,
0x3f4a,0xb206,0x0fa0,0xae28,
0x3eea,0xbf86,0x2c0a,0x067b,
0x3e7c,0x0071,0xab1c,0x7428,
0x3dff,0xa861,0x8e3c,0xf1c6,
0x3d73,0x6643,0x9c3d,0xef2b,
0x3cd9,0x00dc,0x37c8,0x1936,
0x3c2e,0xf5a1,0x84ff,0x8364,
0x3b6c,0x409d,0x624f,0xbe2b,
};
#endif

extern double PI, PIO2, MACHEP;

int fresnl(double xxa,double* ssa,double* cca )
{
double f, g, cc, ss, c, s, t, u;
double x, x2;

x = fabs(xxa);
x2 = x * x;
if( x2 < 2.5625 )
      {
      t = x2 * x2;
      ss = x * x2 * polevl( t, sn, 5)/p1evl( t, sd, 6 );
      cc = x * polevl( t, cn, 5)/polevl(t, cd, 6 );
      goto done;
      }






if( x > 36974.0 )
      {
      cc = 0.5;
      ss = 0.5;
      goto done;
      }


/*          Asymptotic power series auxiliary functions
 *          for large argument
 */
      x2 = x * x;
      t = PI * x2;
      u = 1.0/(t * t);
      t = 1.0/t;
      f = 1.0 - u * polevl( u, fn, 9)/p1evl(u, fd, 10);
      g = t * polevl( u, gn, 10)/p1evl(u, gd, 11);

      t = PIO2 * x2;
      c = cos(t);
      s = sin(t);
      t = PI * x;
      cc = 0.5  +  (f * s  -  g * c)/t;
      ss = 0.5  -  (f * c  +  g * s)/t;

done:
if( xxa < 0.0 )
      {
      cc = -cc;
      ss = -ss;
      }

*cca = cc;
*ssa = ss;
return(0);
}

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