glibc/sysdeps/ieee754/ldbl-96/e_j1l.c

/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

/* Long double expansions contributed by
   Stephen L. Moshier <moshier@na-net.ornl.gov>  */

/* __ieee754_j1(x), __ieee754_y1(x)
 * Bessel function of the first and second kinds of order zero.
 * Method -- j1(x):
 *	1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
 *	2. Reduce x to |x| since j1(x)=-j1(-x),  and
 *	   for x in (0,2)
 *		j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
 *	   for x in (2,inf)
 * 		j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
 * 		y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
 * 	   where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
 *	   as follow:
 *		cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
 *			=  1/sqrt(2) * (sin(x) - cos(x))
 *		sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 *			= -1/sqrt(2) * (sin(x) + cos(x))
 * 	   (To avoid cancellation, use
 *		sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 * 	    to compute the worse one.)
 *
 *	3 Special cases
 *		j1(nan)= nan
 *		j1(0) = 0
 *		j1(inf) = 0
 *
 * Method -- y1(x):
 *	1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
 *	2. For x<2.
 *	   Since
 *		y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
 *	   therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
 *	   We use the following function to approximate y1,
 *		y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
 *	   Note: For tiny x, 1/x dominate y1 and hence
 *		y1(tiny) = -2/pi/tiny
 *	3. For x>=2.
 * 		y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
 * 	   where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
 *	   by method mentioned above.
 */

#include "math.h"
#include "math_private.h"

#ifdef __STDC__
static long double pone (long double), qone (long double);
#else
static long double pone (), qone ();
#endif

#ifdef __STDC__
static const long double
#else
static long double
#endif
  huge = 1e4930L,
 one = 1.0L,
 invsqrtpi = 5.6418958354775628694807945156077258584405e-1L,
  tpi =  6.3661977236758134307553505349005744813784e-1L,

  /* J1(x) = .5 x + x x^2 R(x^2) / S(x^2)
     0 <= x <= 2
     Peak relative error 4.5e-21 */
R[5] = {
    -9.647406112428107954753770469290757756814E7L,
    2.686288565865230690166454005558203955564E6L,
    -3.689682683905671185891885948692283776081E4L,
    2.195031194229176602851429567792676658146E2L,
    -5.124499848728030297902028238597308971319E-1L,
},

  S[4] =
{
  1.543584977988497274437410333029029035089E9L,
  2.133542369567701244002565983150952549520E7L,
  1.394077011298227346483732156167414670520E5L,
  5.252401789085732428842871556112108446506E2L,
  /*  1.000000000000000000000000000000000000000E0L, */
};

#ifdef __STDC__
static const long double zero = 0.0;
#else
static long double zero = 0.0;
#endif


#ifdef __STDC__
long double
__ieee754_j1l (long double x)
#else
long double
__ieee754_j1l (x)
     long double x;
#endif
{
  long double z, c, r, s, ss, cc, u, v, y;
  int32_t ix;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  if (ix >= 0x7fff)
    return one / x;
  y = fabsl (x);
  if (ix >= 0x4000)
    {				/* |x| >= 2.0 */
      __sincosl (y, &s, &c);
      ss = -s - c;
      cc = s - c;
      if (ix < 0x7ffe)
	{			/* make sure y+y not overflow */
	  z = __cosl (y + y);
	  if ((s * c) > zero)
	    cc = z / ss;
	  else
	    ss = z / cc;
	}
      /*
       * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
       * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
       */
      if (ix > 0x4080)
	z = (invsqrtpi * cc) / __ieee754_sqrtl (y);
      else
	{
	  u = pone (y);
	  v = qone (y);
	  z = invsqrtpi * (u * cc - v * ss) / __ieee754_sqrtl (y);
	}
      if (se & 0x8000)
	return -z;
      else
	return z;
    }
  if (ix < 0x3fde) /* |x| < 2^-33 */
    {
      if (huge + x > one)
	return 0.5 * x;		/* inexact if x!=0 necessary */
    }
  z = x * x;
  r = z * (R[0] + z * (R[1]+ z * (R[2] + z * (R[3] + z * R[4]))));
  s = S[0] + z * (S[1] + z * (S[2] + z * (S[3] + z)));
  r *= x;
  return (x * 0.5 + r / s);
}


/* Y1(x) = 2/pi * (log(x) * j1(x) - 1/x) + x R(x^2)
   0 <= x <= 2
   Peak relative error 2.3e-23 */
#ifdef __STDC__
static const long double U0[6] = {
#else
static long double U0[6] = {
#endif
  -5.908077186259914699178903164682444848615E10L,
  1.546219327181478013495975514375773435962E10L,
  -6.438303331169223128870035584107053228235E8L,
  9.708540045657182600665968063824819371216E6L,
  -6.138043997084355564619377183564196265471E4L,
  1.418503228220927321096904291501161800215E2L,
};
#ifdef __STDC__
static const long double V0[5] = {
#else
static long double V0[5] = {
#endif
  3.013447341682896694781964795373783679861E11L,
  4.669546565705981649470005402243136124523E9L,
  3.595056091631351184676890179233695857260E7L,
  1.761554028569108722903944659933744317994E5L,
  5.668480419646516568875555062047234534863E2L,
  /*  1.000000000000000000000000000000000000000E0L, */
};


#ifdef __STDC__
long double
__ieee754_y1l (long double x)
#else
long double
__ieee754_y1l (x)
     long double x;
#endif
{
  long double z, s, c, ss, cc, u, v;
  int32_t ix;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
  if (se & 0x8000)
    return zero / zero;
  if (ix >= 0x7fff)
    return one / (x + x * x);
  if ((i0 | i1) == 0)
    return -one / zero;
  if (ix >= 0x4000)
    {				/* |x| >= 2.0 */
      __sincosl (x, &s, &c);
      ss = -s - c;
      cc = s - c;
      if (ix < 0x7fe00000)
	{			/* make sure x+x not overflow */
	  z = __cosl (x + x);
	  if ((s * c) > zero)
	    cc = z / ss;
	  else
	    ss = z / cc;
	}
      /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
       * where x0 = x-3pi/4
       *      Better formula:
       *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
       *                      =  1/sqrt(2) * (sin(x) - cos(x))
       *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
       *                      = -1/sqrt(2) * (cos(x) + sin(x))
       * To avoid cancellation, use
       *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
       * to compute the worse one.
       */
      if (ix > 0x4080)
	z = (invsqrtpi * ss) / __ieee754_sqrtl (x);
      else
	{
	  u = pone (x);
	  v = qone (x);
	  z = invsqrtpi * (u * ss + v * cc) / __ieee754_sqrtl (x);
	}
      return z;
    }
  if (ix <= 0x3fbe)
    {				/* x < 2**-65 */
      return (-tpi / x);
    }
  z = x * x;
 u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * (U0[4] + z * U0[5]))));
 v = V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * (V0[4] + z))));
  return (x * (u / v) +
	  tpi * (__ieee754_j1l (x) * __ieee754_logl (x) - one / x));
}


/* For x >= 8, the asymptotic expansions of pone is
 *	1 + 15/128 s^2 - 4725/2^15 s^4 - ...,	where s = 1/x.
 * We approximate pone by
 * 	pone(x) = 1 + (R/S)
 */

/* J1(x) cosX + Y1(x) sinX  =  sqrt( 2/(pi x)) P1(x)
   P1(x) = 1 + z^2 R(z^2), z=1/x
   8 <= x <= inf  (0 <= z <= 0.125)
   Peak relative error 5.2e-22  */

#ifdef __STDC__
static const long double pr8[7] = {
#else
static long double pr8[7] = {
#endif
  8.402048819032978959298664869941375143163E-9L,
  1.813743245316438056192649247507255996036E-6L,
  1.260704554112906152344932388588243836276E-4L,
  3.439294839869103014614229832700986965110E-3L,
  3.576910849712074184504430254290179501209E-2L,
  1.131111483254318243139953003461511308672E-1L,
  4.480715825681029711521286449131671880953E-2L,
};
#ifdef __STDC__
static const long double ps8[6] = {
#else
static long double ps8[6] = {
#endif
  7.169748325574809484893888315707824924354E-8L,
  1.556549720596672576431813934184403614817E-5L,
  1.094540125521337139209062035774174565882E-3L,
  3.060978962596642798560894375281428805840E-2L,
  3.374146536087205506032643098619414507024E-1L,
  1.253830208588979001991901126393231302559E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

/* J1(x) cosX + Y1(x) sinX  =  sqrt( 2/(pi x)) P1(x)
   P1(x) = 1 + z^2 R(z^2), z=1/x
   4.54541015625 <= x <= 8
   Peak relative error 7.7e-22  */
#ifdef __STDC__
static const long double pr5[7] = {
#else
static long double pr5[7] = {
#endif
  4.318486887948814529950980396300969247900E-7L,
  4.715341880798817230333360497524173929315E-5L,
  1.642719430496086618401091544113220340094E-3L,
  2.228688005300803935928733750456396149104E-2L,
  1.142773760804150921573259605730018327162E-1L,
  1.755576530055079253910829652698703791957E-1L,
  3.218803858282095929559165965353784980613E-2L,
};
#ifdef __STDC__
static const long double ps5[6] = {
#else
static long double ps5[6] = {
#endif
  3.685108812227721334719884358034713967557E-6L,
  4.069102509511177498808856515005792027639E-4L,
  1.449728676496155025507893322405597039816E-2L,
  2.058869213229520086582695850441194363103E-1L,
  1.164890985918737148968424972072751066553E0L,
  2.274776933457009446573027260373361586841E0L,
  /*  1.000000000000000000000000000000000000000E0L,*/
};

/* J1(x) cosX + Y1(x) sinX  =  sqrt( 2/(pi x)) P1(x)
   P1(x) = 1 + z^2 R(z^2), z=1/x
   2.85711669921875 <= x <= 4.54541015625
   Peak relative error 6.5e-21  */
#ifdef __STDC__
static const long double pr3[7] = {
#else
static long double pr3[7] = {
#endif
  1.265251153957366716825382654273326407972E-5L,
  8.031057269201324914127680782288352574567E-4L,
  1.581648121115028333661412169396282881035E-2L,
  1.179534658087796321928362981518645033967E-1L,
  3.227936912780465219246440724502790727866E-1L,
  2.559223765418386621748404398017602935764E-1L,
  2.277136933287817911091370397134882441046E-2L,
};
#ifdef __STDC__
static const long double ps3[6] = {
#else
static long double ps3[6] = {
#endif
  1.079681071833391818661952793568345057548E-4L,
  6.986017817100477138417481463810841529026E-3L,
  1.429403701146942509913198539100230540503E-1L,
  1.148392024337075609460312658938700765074E0L,
  3.643663015091248720208251490291968840882E0L,
  3.990702269032018282145100741746633960737E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

/* J1(x) cosX + Y1(x) sinX  =  sqrt( 2/(pi x)) P1(x)
   P1(x) = 1 + z^2 R(z^2), z=1/x
   2 <= x <= 2.85711669921875
   Peak relative error 3.5e-21  */
#ifdef __STDC__
static const long double pr2[7] = {
#else
static long double pr2[7] = {
#endif
  2.795623248568412225239401141338714516445E-4L,
  1.092578168441856711925254839815430061135E-2L,
  1.278024620468953761154963591853679640560E-1L,
  5.469680473691500673112904286228351988583E-1L,
  8.313769490922351300461498619045639016059E-1L,
  3.544176317308370086415403567097130611468E-1L,
  1.604142674802373041247957048801599740644E-2L,
};
#ifdef __STDC__
static const long double ps2[6] = {
#else
static long double ps2[6] = {
#endif
  2.385605161555183386205027000675875235980E-3L,
  9.616778294482695283928617708206967248579E-2L,
  1.195215570959693572089824415393951258510E0L,
  5.718412857897054829999458736064922974662E0L,
  1.065626298505499086386584642761602177568E1L,
  6.809140730053382188468983548092322151791E0L,
 /* 1.000000000000000000000000000000000000000E0L, */
};


#ifdef __STDC__
static long double
pone (long double x)
#else
static long double
pone (x)
     long double x;
#endif
{
#ifdef __STDC__
  const long double *p, *q;
#else
  long double *p, *q;
#endif
  long double z, r, s;
  int32_t ix;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  if (ix >= 0x4002) /* x >= 8 */
    {
      p = pr8;
      q = ps8;
    }
  else
    {
      i1 = (ix << 16) | (i0 >> 16);
      if (i1 >= 0x40019174)	/* x >= 4.54541015625 */
	{
	  p = pr5;
	  q = ps5;
	}
      else if (i1 >= 0x4000b6db)	/* x >= 2.85711669921875 */
	{
	  p = pr3;
	  q = ps3;
	}
      else if (ix >= 0x4000)	/* x better be >= 2 */
	{
	  p = pr2;
	  q = ps2;
	}
    }
  z = one / (x * x);
 r = p[0] + z * (p[1] +
		 z * (p[2] + z * (p[3] + z * (p[4] + z * (p[5] + z * p[6])))));
 s = q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * (q[5] + z)))));
  return one + z * r / s;
}


/* For x >= 8, the asymptotic expansions of qone is
 *	3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 * We approximate pone by
 * 	qone(x) = s*(0.375 + (R/S))
 */

/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
   Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
   8 <= x <= inf
   Peak relative error 8.3e-22 */

#ifdef __STDC__
static const long double qr8[7] = {
#else
static long double qr8[7] = {
#endif
  -5.691925079044209246015366919809404457380E-10L,
  -1.632587664706999307871963065396218379137E-7L,
  -1.577424682764651970003637263552027114600E-5L,
  -6.377627959241053914770158336842725291713E-4L,
  -1.087408516779972735197277149494929568768E-2L,
  -6.854943629378084419631926076882330494217E-2L,
  -1.055448290469180032312893377152490183203E-1L,
};
#ifdef __STDC__
static const long double qs8[7] = {
#else
static long double qs8[7] = {
#endif
  5.550982172325019811119223916998393907513E-9L,
  1.607188366646736068460131091130644192244E-6L,
  1.580792530091386496626494138334505893599E-4L,
  6.617859900815747303032860443855006056595E-3L,
  1.212840547336984859952597488863037659161E-1L,
  9.017885953937234900458186716154005541075E-1L,
  2.201114489712243262000939120146436167178E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
   Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
   4.54541015625 <= x <= 8
   Peak relative error 4.1e-22 */
#ifdef __STDC__
static const long double qr5[7] = {
#else
static long double qr5[7] = {
#endif
  -6.719134139179190546324213696633564965983E-8L,
  -9.467871458774950479909851595678622044140E-6L,
  -4.429341875348286176950914275723051452838E-4L,
  -8.539898021757342531563866270278505014487E-3L,
  -6.818691805848737010422337101409276287170E-2L,
  -1.964432669771684034858848142418228214855E-1L,
  -1.333896496989238600119596538299938520726E-1L,
};
#ifdef __STDC__
static const long double qs5[7] = {
#else
static long double qs5[7] = {
#endif
  6.552755584474634766937589285426911075101E-7L,
  9.410814032118155978663509073200494000589E-5L,
  4.561677087286518359461609153655021253238E-3L,
  9.397742096177905170800336715661091535805E-2L,
  8.518538116671013902180962914473967738771E-1L,
  3.177729183645800174212539541058292579009E0L,
  4.006745668510308096259753538973038902990E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
   Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
   2.85711669921875 <= x <= 4.54541015625
   Peak relative error 2.2e-21 */
#ifdef __STDC__
static const long double qr3[7] = {
#else
static long double qr3[7] = {
#endif
  -3.618746299358445926506719188614570588404E-6L,
  -2.951146018465419674063882650970344502798E-4L,
  -7.728518171262562194043409753656506795258E-3L,
  -8.058010968753999435006488158237984014883E-2L,
  -3.356232856677966691703904770937143483472E-1L,
  -4.858192581793118040782557808823460276452E-1L,
  -1.592399251246473643510898335746432479373E-1L,
};
#ifdef __STDC__
static const long double qs3[7] = {
#else
static long double qs3[7] = {
#endif
  3.529139957987837084554591421329876744262E-5L,
  2.973602667215766676998703687065066180115E-3L,
  8.273534546240864308494062287908662592100E-2L,
  9.613359842126507198241321110649974032726E-1L,
  4.853923697093974370118387947065402707519E0L,
  1.002671608961669247462020977417828796933E1L,
  7.028927383922483728931327850683151410267E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
   Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
   2 <= x <= 2.85711669921875
   Peak relative error 6.9e-22 */
#ifdef __STDC__
static const long double qr2[7] = {
#else
static long double qr2[7] = {
#endif
  -1.372751603025230017220666013816502528318E-4L,
  -6.879190253347766576229143006767218972834E-3L,
  -1.061253572090925414598304855316280077828E-1L,
  -6.262164224345471241219408329354943337214E-1L,
  -1.423149636514768476376254324731437473915E0L,
  -1.087955310491078933531734062917489870754E0L,
  -1.826821119773182847861406108689273719137E-1L,
};
#ifdef __STDC__
static const long double qs2[7] = {
#else
static long double qs2[7] = {
#endif
  1.338768933634451601814048220627185324007E-3L,
  7.071099998918497559736318523932241901810E-2L,
  1.200511429784048632105295629933382142221E0L,
  8.327301713640367079030141077172031825276E0L,
  2.468479301872299311658145549931764426840E1L,
  2.961179686096262083509383820557051621644E1L,
  1.201402313144305153005639494661767354977E1L,
 /* 1.000000000000000000000000000000000000000E0L, */
};


#ifdef __STDC__
static long double
qone (long double x)
#else
static long double
qone (x)
     long double x;
#endif
{
#ifdef __STDC__
  const long double *p, *q;
#else
  long double *p, *q;
#endif
  static long double s, r, z;
  int32_t ix;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  if (ix >= 0x4002)		/* x >= 8 */
    {
      p = qr8;
      q = qs8;
    }
  else
    {
      i1 = (ix << 16) | (i0 >> 16);
      if (i1 >= 0x40019174)	/* x >= 4.54541015625 */
	{
	  p = qr5;
	  q = qs5;
	}
      else if (i1 >= 0x4000b6db)	/* x >= 2.85711669921875 */
	{
	  p = qr3;
	  q = qs3;
	}
      else if (ix >= 0x4000)	/* x better be >= 2 */
	{
	  p = qr2;
	  q = qs2;
	}
    }
  z = one / (x * x);
  r =
    p[0] + z * (p[1] +
		z * (p[2] + z * (p[3] + z * (p[4] + z * (p[5] + z * p[6])))));
  s =
    q[0] + z * (q[1] +
		z * (q[2] +
		     z * (q[3] + z * (q[4] + z * (q[5] + z * (q[6] + z))))));
  return (.375 + z * r / s) / x;
}