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/* 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
*/
/*
* 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 "mconf.h"
/* S(x) for small x */
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,
};
/* C(x) for small x */
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,
};
/* Auxiliary function f(x) */
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,
};
/* Auxiliary function g(x) */
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,
};
extern double MACHEP;
int fresnl(xxa, ssa, cca)
double xxa, *ssa, *cca;
{
double f, g, cc, ss, c, s, t, u;
double x, x2;
if (cephes_isinf(xxa)) {
cc = 0.5;
ss = 0.5;
goto done;
}
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) {
/*
* http://functions.wolfram.com/GammaBetaErf/FresnelC/06/02/
* http://functions.wolfram.com/GammaBetaErf/FresnelS/06/02/
*/
cc = 0.5 + 1/(NPY_PI*x) * sin(NPY_PI*x*x/2);
ss = 0.5 - 1/(NPY_PI*x) * cos(NPY_PI*x*x/2);
goto done;
}
/* Asymptotic power series auxiliary functions
* for large argument
*/
x2 = x * x;
t = NPY_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 = NPY_PI_2 * x2;
c = cos(t);
s = sin(t);
t = NPY_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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