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+/*-
+ * 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 8.2 (Berkeley) 11/30/93";
+#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;
+ /* 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;
+}
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