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glibc/sysdeps/ieee754/ldbl-96/e_j1l.c
Andreas Jaeger 68b0e1db30 * sysdeps/ieee754/ldbl-96/e_j1l.c (__ieee754_y1l): Use sincos 
instead of separate sin and cos. 
(__ieee754_j1l): Likewise.
2001-04-07 18:31:54 +00:00

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/*
* ====================================================
* 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;
}
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