/*							j0l.c
 *
 *	Bessel function of order zero
 *
 *
 *
 * SYNOPSIS:
 *
 * long double x, y, j0l();
 *
 * y = j0l( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of first kind, order zero of the argument.
 *
 * The domain is divided into the intervals [0, 9] and
 * (9, infinity). In the first interval the rational approximation
 * is (x^2 - r^2) (x^2 - s^2) (x^2 - t^2) P7(x^2) / Q8(x^2),
 * where r, s, t are the first three zeros of the function.
 * In the second interval the expansion is in terms of the
 * modulus M0(x) = sqrt(J0(x)^2 + Y0(x)^2) and phase  P0(x)
 * = atan(Y0(x)/J0(x)).  M0 is approximated by sqrt(1/x)P7(1/x)/Q7(1/x).
 * The approximation to J0 is M0 * cos(x -  pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)).
 *
 *
 * ACCURACY:
 *
 *                      Absolute error:
 * arithmetic   domain      # trials      peak         rms
 *    IEEE      0, 30       10000       2.7e-19      7.3e-20
 *
 *
 */
/*							y0l.c
 *
 *	Bessel function of the second kind, order zero
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, y0l();
 *
 * y = y0l( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of the second kind, of order
 * zero, of the argument.
 *
 * The domain is divided into the intervals [0, 5>, [5,9> and
 * [9, infinity). In the first interval a rational approximation
 * R(x) is employed to compute y0(x)  = R(x) + 2/pi * log(x) * j0(x).
 *
 * In the second interval, the approximation is
 *     (x - p)(x - q)(x - r)(x - s)P7(x)/Q7(x)
 * where p, q, r, s are zeros of y0(x).
 *
 * The third interval uses the same approximations to modulus
 * and phase as j0(x), whence y0(x) = modulus * sin(phase).
 *
 * ACCURACY:
 *
 *  Absolute error, when y0(x) < 1; else relative error:
 *
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0, 30       25000       3.0e-19     7.6e-20
 *
 */

/* Copyright 1994 by Stephen L. Moshier (moshier@world.std.com).  */

#include "mathl.h"
extern long double __polevll ( long double, long double *, int );
extern long double __p1evll ( long double, long double *, int );

/*
j0(x) = (x^2-JZ1)(x^2-JZ2)(x^2-JZ3)P(x**2)/Q(x**2)
0 <= x <= 9
Relative error
n=7, d=8
Peak error =  8.49e-22
Relative error spread =  2.2e-3
*/
static long double j0n[] = {
 1.296848754518641770562068596970917461302E0L,
-3.239201943301299801017988814490822973468E3L,
 3.490002040733471400106951588156446469843E6L,
-2.076797068740966813172740536268762823178E9L,
 7.283696461857171054940603724292997493661E11L,
-1.487926133645291056388093978451001239544E14L,
 1.620335009643150402367970862340822012501E16L,
-7.173386747526788067407034559506246493841E17L
 };
static long double j0d[] = {
/* 1.000000000000000000000000000000000000000E0*/
 2.281959869176887763844919265666715670624E3L,
 2.910386840401647706984290219401654268412E6L,
 2.608400226578100610990921977837307261661E9L,
 1.752689035792859338859788285894079836315E12L,
 8.879132373286001289461351204912522718368E14L,
 3.265560832845194013668975527943185616740E17L,
 7.881340554308432241891814409857299717887E19L,
 9.466475654163919450527627558766493397282E21L
};
/*
sqrt(j0^2(1/x^2) + y0^2(1/x^2)) = z P(z**2)/Q(z**2)
z(x) = 1/sqrt(x)
Relative error
n=7, d=7
Peak error =  1.80e-20
Relative error spread =  5.1e-2
*/
static long double modulusn[] = {
 3.947542376069224461532136552047845797185E-1L,
 6.864682945702134624126371301477343341825E0L,
 1.021369773577974343843959835514109613368E1L,
 7.626141421290849630522661034590426845035E0L,
 2.842537511425216145635344663341056100390E0L,
 7.162842530423205720961523942028933535254E-1L,
 9.036664453160200052295692999857254683255E-2L,
 8.461833426898867839659448202505801120140E-3L
};
static long double modulusd[] = {
/* 1.000000000000000000000000000000000000000E0 */
 9.117176038171821115904394138261429005561E0L,
 1.301235226061478261480860548640027632230E1L,
 9.613002539386213788182145213835341179345E0L,
 3.569671060989910901903283335065262208244E0L,
 8.983920141407590632423345373045592707789E-1L,
 1.132577931332212304985636383878157848593E-1L,
 1.060533546154121770441529910796331563105E-2L
 };
/*
atan(y0(x)/j0(x)) = x - pi/4 + x P(x**2)/Q(x**2)
Absolute error
n=5, d=6
Peak error =  2.80e-21
Relative error spread =  5.5e-1
*/
static long double phasen[] = {
-7.356766355393571519038139469135140406247E-1L,
-5.001635199922493694706241245828396845424E-1L,
-7.737323518141516881715220488195636681899E-2L,
-3.998893155826990642730404602407242779985E-3L,
-7.496317036829964150970102768499885677420E-5L,
-4.290885090773112963541732090289084223939E-7L
};
static long double phased[] = {
/* 1.000000000000000000000000000000000000000E0 */
 7.377856408614376072744766256586157817057E0L,
 4.285043297797736118068767291405925283657E0L,
 6.348446472935245102890369628706013793370E-1L,
 3.229866782185025048457458026131873133385E-2L,
 6.014932317342190404134448927950191465063E-4L,
 3.432708072618490390094952090444729316515E-6L
 };

#define JZ1 5.783185962946784521175995758L
#define JZ2 30.471262343662086399078163175L
#define JZ3 7.4887006790695183444889041310136808274810908E1L

#define PIO4L 0.78539816339744830961566L

long double j0l(long double x)
{
long double xx, y, z, modulus, phase;

xx = x * x;
if( xx < 81.0L )
  {
    y = (xx - JZ1) * (xx - JZ2) * (xx -JZ3);
    y *= __polevll( xx, j0n, 7 ) / __p1evll( xx, j0d, 8 );
    return y;
  }

y = fabsl(x);
xx = 1.0/xx;
phase = __polevll( xx, phasen, 5 ) / __p1evll( xx, phased, 6 );

z = 1.0/y;
modulus = __polevll( z, modulusn, 7 ) / __p1evll( z, modulusd, 7 );

y = modulus * cosl( y -  PIO4L + z*phase) / sqrtl(y);
return y;
}


/*
y0(x) = 2/pi * log(x) * j0(x) + P(z**2)/Q(z**2)
0 <= x <= 5
Absolute error
n=7, d=7
Peak error =  8.55e-22
Relative error spread =  2.7e-1
*/
static long double y0n[] = {
 1.556909814120445353690558903926596922430E4L,
-1.464324149797947303150684329851271736158E7L,
 5.427926320587133391307283047780685085060E9L,
-9.808510181632626683952255657533575381882E11L,
 8.747842804834934784972498894571266988554E13L,
-3.461898868011666236538997563917952308530E15L,
 4.421767595991969611983299917018735928270E16L,
-1.847183690384811186958417431043844288483E16L
};
static long double y0d[] = {
/* 1.000000000000000000000000000000000000000E0 */
 1.040792201755841697888585281350416537990E3L,
 6.256391154086099882301995151394816098690E5L,
 2.686702051957904669677256742116961716912E8L,
 8.630939306572281881328204934638848292693E10L,
 2.027480766502742538762944923313331185696E13L,
 3.167750475899536301562019307293643530456E15L,
 2.502813268068711844040275540632320849671E17L
};

/*
y0(x) = (x-Y0Z1)(x-Y0Z2)(x-Y0Z3)(x-Y0Z4)P(x)/Q(x)
4.5 <= x <= 9
Absolute error
n=9, d=9
Peak error =  2.35e-20
Relative error spread =  7.8e-13
*/
static long double y059n[] = {
 2.368715538373384869795740108286354080729E-2L,
-1.472923738545276751401835498475289789724E0L,
 2.525993724177105060507377634547936257100E1L,
 7.727403527387097461580463446897153155336E1L,
-4.578271827238477598563372907206784098428E3L,
 7.051411242092171161986020461608228436219E3L,
 1.951120419910720443330833105263077652526E5L,
 6.515211089266670755621513548872432595610E5L,
-1.164760792144532266854968981115405821194E5L,
-5.566567444353735925323263816754034956079E5L
};
static long double y059d[] = {
/* 1.000000000000000000000000000000000000000E0 */
-6.235501989189125881723091161802419916592E1L,
 2.224790285641017194158141027000555212083E3L,
-5.103881883748705381186322587373400863945E4L,
 8.772616606054526158656710844751021655708E5L,
-1.096162986826467060920523203989531893118E7L,
 1.083335477747278958468238227109848215017E8L,
-7.045635226159434678832858770683824287226E8L,
 3.518324187204647941098046647734000055759E9L,
 1.173085288957116938494111248664763079367E9L
 };

#define TWOOPI 6.36619772367581343075535E-1L
#define Y0Z1 3.9576784193148578683756771869174012814186038E0L
#define Y0Z2 7.0860510603017726976236245968203524689715104E0L
#define Y0Z3 1.0222345043496417018992042276342187125994060E1L
#define Y0Z4 1.3361097473872763478267694585713786426579135E1L
#define MAXNUML 1.189731495357231765021263853E4932L

long double y0l(long double x)
{
long double xx, y, z, modulus, phase;

if( x < 0.0 )
  {
    return (-MAXNUML);
  }
xx = x * x;
if( xx < 81.0L )
  {
    if( xx < 20.25L )
      {
	y = TWOOPI * logl(x) * j0l(x);
	y += __polevll( xx, y0n, 7 ) / __p1evll( xx, y0d, 7 );
      }
    else
      {
	y = (x - Y0Z1)*(x - Y0Z2)*(x - Y0Z3)*(x - Y0Z4);
	y *= __polevll( x, y059n, 9 ) / __p1evll( x, y059d, 9 );
      }
    return y;
  }

y = fabsl(x);
xx = 1.0/xx;
phase = __polevll( xx, phasen, 5 ) / __p1evll( xx, phased, 6 );

z = 1.0/y;
modulus = __polevll( z, modulusn, 7 ) / __p1evll( z, modulusd, 7 );

y = modulus * sinl( y -  PIO4L + z*phase) / sqrtl(y);
return y;
}