4.4BSD/usr/src/lib/libm/common_source/j0.c
/*-
* Copyright (c) 1992, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#ifndef lint
static char sccsid[] = "@(#)j0.c 8.1 (Berkeley) 6/4/93";
#endif /* not lint */
/*
* 16 December 1992
* Minor modifications by Peter McIlroy to adapt non-IEEE architecture.
*/
/*
* ====================================================
* Copyright (C) 1992 by Sun Microsystems, Inc.
*
* 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.
* ====================================================
*
* ******************* WARNING ********************
* This is an alpha version of SunPro's FDLIBM (Freely
* Distributable Math Library) for IEEE double precision
* arithmetic. FDLIBM is a basic math library written
* in C that runs on machines that conform to IEEE
* Standard 754/854. This alpha version is distributed
* for testing purpose. Those who use this software
* should report any bugs to
*
* fdlibm-comments@sunpro.eng.sun.com
*
* -- K.C. Ng, Oct 12, 1992
* ************************************************
*/
/* double j0(double x), y0(double x)
* Bessel function of the first and second kinds of order zero.
* Method -- j0(x):
* 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
* 2. Reduce x to |x| since j0(x)=j0(-x), and
* for x in (0,2)
* j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
* (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
* for x in (2,inf)
* j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
* as follow:
* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
* = 1/sqrt(2) * (cos(x) + sin(x))
* sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/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
* j0(nan)= nan
* j0(0) = 1
* j0(inf) = 0
*
* Method -- y0(x):
* 1. For x<2.
* Since
* y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
* therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
* We use the following function to approximate y0,
* y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
* where
* U(z) = u0 + u1*z + ... + u6*z^6
* V(z) = 1 + v1*z + ... + v4*z^4
* with absolute approximation error bounded by 2**-72.
* Note: For tiny x, U/V = u0 and j0(x)~1, hence
* y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
* 2. For x>=2.
* y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
* by the method mentioned above.
* 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
*/
#include <math.h>
#include <float.h>
#if defined(vax) || defined(tahoe)
#define _IEEE 0
#else
#define _IEEE 1
#define infnan(x) (0.0)
#endif
static double pzero __P((double)), qzero __P((double));
static double
huge = 1e300,
zero = 0.0,
one = 1.0,
invsqrtpi= 5.641895835477562869480794515607725858441e-0001,
tpi = 0.636619772367581343075535053490057448,
/* R0/S0 on [0, 2.00] */
r02 = 1.562499999999999408594634421055018003102e-0002,
r03 = -1.899792942388547334476601771991800712355e-0004,
r04 = 1.829540495327006565964161150603950916854e-0006,
r05 = -4.618326885321032060803075217804816988758e-0009,
s01 = 1.561910294648900170180789369288114642057e-0002,
s02 = 1.169267846633374484918570613449245536323e-0004,
s03 = 5.135465502073181376284426245689510134134e-0007,
s04 = 1.166140033337900097836930825478674320464e-0009;
double
j0(x)
double x;
{
double z, s,c,ss,cc,r,u,v;
if (!finite(x))
if (_IEEE) return one/(x*x);
else return (0);
x = fabs(x);
if (x >= 2.0) { /* |x| >= 2.0 */
s = sin(x);
c = cos(x);
ss = s-c;
cc = s+c;
if (x < .5 * DBL_MAX) { /* make sure x+x not overflow */
z = -cos(x+x);
if ((s*c)<zero) cc = z/ss;
else ss = z/cc;
}
/*
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
*/
if (_IEEE && x> 6.80564733841876927e+38) /* 2^129 */
z = (invsqrtpi*cc)/sqrt(x);
else {
u = pzero(x); v = qzero(x);
z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
}
return z;
}
if (x < 1.220703125e-004) { /* |x| < 2**-13 */
if (huge+x > one) { /* raise inexact if x != 0 */
if (x < 7.450580596923828125e-009) /* |x|<2**-27 */
return one;
else return (one - 0.25*x*x);
}
}
z = x*x;
r = z*(r02+z*(r03+z*(r04+z*r05)));
s = one+z*(s01+z*(s02+z*(s03+z*s04)));
if (x < one) { /* |x| < 1.00 */
return (one + z*(-0.25+(r/s)));
} else {
u = 0.5*x;
return ((one+u)*(one-u)+z*(r/s));
}
}
static double
u00 = -7.380429510868722527422411862872999615628e-0002,
u01 = 1.766664525091811069896442906220827182707e-0001,
u02 = -1.381856719455968955440002438182885835344e-0002,
u03 = 3.474534320936836562092566861515617053954e-0004,
u04 = -3.814070537243641752631729276103284491172e-0006,
u05 = 1.955901370350229170025509706510038090009e-0008,
u06 = -3.982051941321034108350630097330144576337e-0011,
v01 = 1.273048348341237002944554656529224780561e-0002,
v02 = 7.600686273503532807462101309675806839635e-0005,
v03 = 2.591508518404578033173189144579208685163e-0007,
v04 = 4.411103113326754838596529339004302243157e-0010;
double
y0(x)
double x;
{
double z, s,c,ss,cc,u,v,j0();
/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
if (!finite(x))
if (_IEEE)
return (one/(x+x*x));
else
return (0);
if (x == 0)
if (_IEEE) return (-one/zero);
else return(infnan(-ERANGE));
if (x<0)
if (_IEEE) return (zero/zero);
else return (infnan(EDOM));
if (x >= 2.00) { /* |x| >= 2.0 */
/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
* where x0 = x-pi/4
* Better formula:
* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
* = 1/sqrt(2) * (sin(x) + cos(x))
* sin(x0) = 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.
*/
s = sin(x);
c = cos(x);
ss = s-c;
cc = s+c;
/*
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
*/
if (x < .5 * DBL_MAX) { /* make sure x+x not overflow */
z = -cos(x+x);
if ((s*c)<zero) cc = z/ss;
else ss = z/cc;
}
if (_IEEE && x > 6.80564733841876927e+38) /* > 2^129 */
z = (invsqrtpi*ss)/sqrt(x);
else {
u = pzero(x); v = qzero(x);
z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
}
return z;
}
if (x <= 7.450580596923828125e-009) { /* x < 2**-27 */
return (u00 + tpi*log(x));
}
z = x*x;
u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
v = one+z*(v01+z*(v02+z*(v03+z*v04)));
return (u/v + tpi*(j0(x)*log(x)));
}
/* The asymptotic expansions of pzero is
* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
* For x >= 2, We approximate pzero by
* pzero(x) = 1 + (R/S)
* where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
* S = 1 + ps0*s^2 + ... + ps4*s^10
* and
* | pzero(x)-1-R/S | <= 2 ** ( -60.26)
*/
static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
0.0,
-7.031249999999003994151563066182798210142e-0002,
-8.081670412753498508883963849859423939871e+0000,
-2.570631056797048755890526455854482662510e+0002,
-2.485216410094288379417154382189125598962e+0003,
-5.253043804907295692946647153614119665649e+0003,
};
static double ps8[5] = {
1.165343646196681758075176077627332052048e+0002,
3.833744753641218451213253490882686307027e+0003,
4.059785726484725470626341023967186966531e+0004,
1.167529725643759169416844015694440325519e+0005,
4.762772841467309430100106254805711722972e+0004,
};
static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
-1.141254646918944974922813501362824060117e-0011,
-7.031249408735992804117367183001996028304e-0002,
-4.159610644705877925119684455252125760478e+0000,
-6.767476522651671942610538094335912346253e+0001,
-3.312312996491729755731871867397057689078e+0002,
-3.464333883656048910814187305901796723256e+0002,
};
static double ps5[5] = {
6.075393826923003305967637195319271932944e+0001,
1.051252305957045869801410979087427910437e+0003,
5.978970943338558182743915287887408780344e+0003,
9.625445143577745335793221135208591603029e+0003,
2.406058159229391070820491174867406875471e+0003,
};
static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
-2.547046017719519317420607587742992297519e-0009,
-7.031196163814817199050629727406231152464e-0002,
-2.409032215495295917537157371488126555072e+0000,
-2.196597747348830936268718293366935843223e+0001,
-5.807917047017375458527187341817239891940e+0001,
-3.144794705948885090518775074177485744176e+0001,
};
static double ps3[5] = {
3.585603380552097167919946472266854507059e+0001,
3.615139830503038919981567245265266294189e+0002,
1.193607837921115243628631691509851364715e+0003,
1.127996798569074250675414186814529958010e+0003,
1.735809308133357510239737333055228118910e+0002,
};
static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
-8.875343330325263874525704514800809730145e-0008,
-7.030309954836247756556445443331044338352e-0002,
-1.450738467809529910662233622603401167409e+0000,
-7.635696138235277739186371273434739292491e+0000,
-1.119316688603567398846655082201614524650e+0001,
-3.233645793513353260006821113608134669030e+0000,
};
static double ps2[5] = {
2.222029975320888079364901247548798910952e+0001,
1.362067942182152109590340823043813120940e+0002,
2.704702786580835044524562897256790293238e+0002,
1.538753942083203315263554770476850028583e+0002,
1.465761769482561965099880599279699314477e+0001,
};
static double pzero(x)
double x;
{
double *p,*q,z,r,s;
if (x >= 8.00) {p = pr8; q= ps8;}
else if (x >= 4.54545211791992188) {p = pr5; q= ps5;}
else if (x >= 2.85714149475097656) {p = pr3; q= ps3;}
else if (x >= 2.00) {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]))));
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
return one+ r/s;
}
/* For x >= 8, the asymptotic expansions of qzero is
* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
* We approximate pzero by
* qzero(x) = s*(-1.25 + (R/S))
* where R = qr0 + qr1*s^2 + qr2*s^4 + ... + qr5*s^10
* S = 1 + qs0*s^2 + ... + qs5*s^12
* and
* | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
*/
static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
0.0,
7.324218749999350414479738504551775297096e-0002,
1.176820646822526933903301695932765232456e+0001,
5.576733802564018422407734683549251364365e+0002,
8.859197207564685717547076568608235802317e+0003,
3.701462677768878501173055581933725704809e+0004,
};
static double qs8[6] = {
1.637760268956898345680262381842235272369e+0002,
8.098344946564498460163123708054674227492e+0003,
1.425382914191204905277585267143216379136e+0005,
8.033092571195144136565231198526081387047e+0005,
8.405015798190605130722042369969184811488e+0005,
-3.438992935378666373204500729736454421006e+0005,
};
static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
1.840859635945155400568380711372759921179e-0011,
7.324217666126847411304688081129741939255e-0002,
5.835635089620569401157245917610984757296e+0000,
1.351115772864498375785526599119895942361e+0002,
1.027243765961641042977177679021711341529e+0003,
1.989977858646053872589042328678602481924e+0003,
};
static double qs5[6] = {
8.277661022365377058749454444343415524509e+0001,
2.077814164213929827140178285401017305309e+0003,
1.884728877857180787101956800212453218179e+0004,
5.675111228949473657576693406600265778689e+0004,
3.597675384251145011342454247417399490174e+0004,
-5.354342756019447546671440667961399442388e+0003,
};
static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
4.377410140897386263955149197672576223054e-0009,
7.324111800429115152536250525131924283018e-0002,
3.344231375161707158666412987337679317358e+0000,
4.262184407454126175974453269277100206290e+0001,
1.708080913405656078640701512007621675724e+0002,
1.667339486966511691019925923456050558293e+0002,
};
static double qs3[6] = {
4.875887297245871932865584382810260676713e+0001,
7.096892210566060535416958362640184894280e+0002,
3.704148226201113687434290319905207398682e+0003,
6.460425167525689088321109036469797462086e+0003,
2.516333689203689683999196167394889715078e+0003,
-1.492474518361563818275130131510339371048e+0002,
};
static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
1.504444448869832780257436041633206366087e-0007,
7.322342659630792930894554535717104926902e-0002,
1.998191740938159956838594407540292600331e+0000,
1.449560293478857407645853071687125850962e+0001,
3.166623175047815297062638132537957315395e+0001,
1.625270757109292688799540258329430963726e+0001,
};
static double qs2[6] = {
3.036558483552191922522729838478169383969e+0001,
2.693481186080498724211751445725708524507e+0002,
8.447837575953201460013136756723746023736e+0002,
8.829358451124885811233995083187666981299e+0002,
2.126663885117988324180482985363624996652e+0002,
-5.310954938826669402431816125780738924463e+0000,
};
static double qzero(x)
double x;
{
double *p,*q, s,r,z;
if (x >= 8.00) {p = qr8; q= qs8;}
else if (x >= 4.54545211791992188) {p = qr5; q= qs5;}
else if (x >= 2.85714149475097656) {p = qr3; q= qs3;}
else if (x >= 2.00) {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]))));
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
return (-.125 + r/s)/x;
}