/* $OpenBSD: n_j1.c,v 1.7 2009/10/27 23:59:29 deraadt Exp $ */ /* $NetBSD: n_j1.c,v 1.1 1995/10/10 23:36:53 ragge Exp $ */ /*- * 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. 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. */ /* * 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 j1(double x), y1(double 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; * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) * 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 follows: * 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 * where for x in [0,2] (abs err less than 2**-65.89) * U(z) = u0 + u1*z + ... + u4*z^4 * V(z) = 1 + v1*z + ... + v5*z^5 * Note: For tiny x, 1/x dominate y1 and hence * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) * 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 #include #include #if defined(__vax__) #define _IEEE 0 #else #define _IEEE 1 #define infnan(x) (0.0) #endif static double pone(), qone(); static double huge = 1e300, zero = 0.0, one = 1.0, invsqrtpi= 5.641895835477562869480794515607725858441e-0001, tpi = 0.636619772367581343075535053490057448, /* R0/S0 on [0,2] */ r00 = -6.250000000000000020842322918309200910191e-0002, r01 = 1.407056669551897148204830386691427791200e-0003, r02 = -1.599556310840356073980727783817809847071e-0005, r03 = 4.967279996095844750387702652791615403527e-0008, s01 = 1.915375995383634614394860200531091839635e-0002, s02 = 1.859467855886309024045655476348872850396e-0004, s03 = 1.177184640426236767593432585906758230822e-0006, s04 = 5.046362570762170559046714468225101016915e-0009, s05 = 1.235422744261379203512624973117299248281e-0011; #define two_129 6.80564733841876926e+038 /* 2^129 */ #define two_m54 5.55111512312578270e-017 /* 2^-54 */ double j1(double x) { double z, s,c,ss,cc,r,u,v,y; y = fabs(x); if (!finite(x)) /* Inf or NaN */ if (isnan(x)) return(x); else return (copysign(x, zero)); y = fabs(x); if (y >= 2) /* |x| >= 2.0 */ { s = sin(y); c = cos(y); ss = -s-c; cc = s-c; if (y < .5*DBL_MAX) { /* make sure y+y not overflow */ z = cos(y+y); if ((s*c) two_129) /* x > 2^129 */ z = (invsqrtpi*cc)/sqrt(y); else #endif /* !defined(__vax__) */ { u = pone(y); v = qone(y); z = invsqrtpi*(u*cc-v*ss)/sqrt(y); } if (x < 0) return -z; else return z; } if (y < 7.450580596923828125e-009) { /* |x|<2**-27 */ if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */ } z = x*x; r = z*(r00+z*(r01+z*(r02+z*r03))); s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); r *= x; return (x*0.5+r/s); } static double u0[5] = { -1.960570906462389484206891092512047539632e-0001, 5.044387166398112572026169863174882070274e-0002, -1.912568958757635383926261729464141209569e-0003, 2.352526005616105109577368905595045204577e-0005, -9.190991580398788465315411784276789663849e-0008, }; static double v0[5] = { 1.991673182366499064031901734535479833387e-0002, 2.025525810251351806268483867032781294682e-0004, 1.356088010975162198085369545564475416398e-0006, 6.227414523646214811803898435084697863445e-0009, 1.665592462079920695971450872592458916421e-0011, }; double y1(double x) { double z, s, c, ss, cc, u, v; /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ if (!finite(x)) if (x < 0) return(zero/zero); else if (x > 0) return (0); else return(x); if (x <= 0) { if (_IEEE && x == 0) return -one/zero; else if(x == 0) return(infnan(-ERANGE)); else if(_IEEE) return (zero/zero); else return(infnan(EDOM)); } if (x >= 2) /* |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; } /* 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 (_IEEE && x>two_129) z = (invsqrtpi*ss)/sqrt(x); else { u = pone(x); v = qone(x); z = invsqrtpi*(u*ss+v*cc)/sqrt(x); } return z; } if (x <= two_m54) { /* x < 2**-54 */ return (-tpi/x); } z = x*x; u = u0[0]+z*(u0[1]+z*(u0[2]+z*(u0[3]+z*u0[4]))); v = one+z*(v0[0]+z*(v0[1]+z*(v0[2]+z*(v0[3]+z*v0[4])))); return (x*(u/v) + tpi*(j1(x)*log(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) * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 * S = 1 + ps0*s^2 + ... + ps4*s^10 * and * | pone(x)-1-R/S | <= 2 ** ( -60.06) */ static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 0.0, 1.171874999999886486643746274751925399540e-0001, 1.323948065930735690925827997575471527252e+0001, 4.120518543073785433325860184116512799375e+0002, 3.874745389139605254931106878336700275601e+0003, 7.914479540318917214253998253147871806507e+0003, }; static double ps8[5] = { 1.142073703756784104235066368252692471887e+0002, 3.650930834208534511135396060708677099382e+0003, 3.695620602690334708579444954937638371808e+0004, 9.760279359349508334916300080109196824151e+0004, 3.080427206278887984185421142572315054499e+0004, }; static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 1.319905195562435287967533851581013807103e-0011, 1.171874931906140985709584817065144884218e-0001, 6.802751278684328781830052995333841452280e+0000, 1.083081829901891089952869437126160568246e+0002, 5.176361395331997166796512844100442096318e+0002, 5.287152013633375676874794230748055786553e+0002, }; static double ps5[5] = { 5.928059872211313557747989128353699746120e+0001, 9.914014187336144114070148769222018425781e+0002, 5.353266952914879348427003712029704477451e+0003, 7.844690317495512717451367787640014588422e+0003, 1.504046888103610723953792002716816255382e+0003, }; static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 3.025039161373736032825049903408701962756e-0009, 1.171868655672535980750284752227495879921e-0001, 3.932977500333156527232725812363183251138e+0000, 3.511940355916369600741054592597098912682e+0001, 9.105501107507812029367749771053045219094e+0001, 4.855906851973649494139275085628195457113e+0001, }; static double ps3[5] = { 3.479130950012515114598605916318694946754e+0001, 3.367624587478257581844639171605788622549e+0002, 1.046871399757751279180649307467612538415e+0003, 8.908113463982564638443204408234739237639e+0002, 1.037879324396392739952487012284401031859e+0002, }; static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 1.077108301068737449490056513753865482831e-0007, 1.171762194626833490512746348050035171545e-0001, 2.368514966676087902251125130227221462134e+0000, 1.224261091482612280835153832574115951447e+0001, 1.769397112716877301904532320376586509782e+0001, 5.073523125888185399030700509321145995160e+0000, }; static double ps2[5] = { 2.143648593638214170243114358933327983793e+0001, 1.252902271684027493309211410842525120355e+0002, 2.322764690571628159027850677565128301361e+0002, 1.176793732871470939654351793502076106651e+0002, 8.364638933716182492500902115164881195742e+0000, }; static double pone(double x) { double *p,*q,z,r,s; if (x >= 8.0) {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 >= 2.0) */ {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 qone is * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. * We approximate pone by * qone(x) = s*(0.375 + (R/S)) * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 * S = 1 + qs1*s^2 + ... + qs6*s^12 * and * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) */ static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 0.0, -1.025390624999927207385863635575804210817e-0001, -1.627175345445899724355852152103771510209e+0001, -7.596017225139501519843072766973047217159e+0002, -1.184980667024295901645301570813228628541e+0004, -4.843851242857503225866761992518949647041e+0004, }; static double qs8[6] = { 1.613953697007229231029079421446916397904e+0002, 7.825385999233484705298782500926834217525e+0003, 1.338753362872495800748094112937868089032e+0005, 7.196577236832409151461363171617204036929e+0005, 6.666012326177764020898162762642290294625e+0005, -2.944902643038346618211973470809456636830e+0005, }; static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ -2.089799311417640889742251585097264715678e-0011, -1.025390502413754195402736294609692303708e-0001, -8.056448281239359746193011295417408828404e+0000, -1.836696074748883785606784430098756513222e+0002, -1.373193760655081612991329358017247355921e+0003, -2.612444404532156676659706427295870995743e+0003, }; static double qs5[6] = { 8.127655013843357670881559763225310973118e+0001, 1.991798734604859732508048816860471197220e+0003, 1.746848519249089131627491835267411777366e+0004, 4.985142709103522808438758919150738000353e+0004, 2.794807516389181249227113445299675335543e+0004, -4.719183547951285076111596613593553911065e+0003, }; static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ -5.078312264617665927595954813341838734288e-0009, -1.025378298208370901410560259001035577681e-0001, -4.610115811394734131557983832055607679242e+0000, -5.784722165627836421815348508816936196402e+0001, -2.282445407376317023842545937526967035712e+0002, -2.192101284789093123936441805496580237676e+0002, }; static double qs3[6] = { 4.766515503237295155392317984171640809318e+0001, 6.738651126766996691330687210949984203167e+0002, 3.380152866795263466426219644231687474174e+0003, 5.547729097207227642358288160210745890345e+0003, 1.903119193388108072238947732674639066045e+0003, -1.352011914443073322978097159157678748982e+0002, }; static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ -1.783817275109588656126772316921194887979e-0007, -1.025170426079855506812435356168903694433e-0001, -2.752205682781874520495702498875020485552e+0000, -1.966361626437037351076756351268110418862e+0001, -4.232531333728305108194363846333841480336e+0001, -2.137192117037040574661406572497288723430e+0001, }; static double qs2[6] = { 2.953336290605238495019307530224241335502e+0001, 2.529815499821905343698811319455305266409e+0002, 7.575028348686454070022561120722815892346e+0002, 7.393932053204672479746835719678434981599e+0002, 1.559490033366661142496448853793707126179e+0002, -4.959498988226281813825263003231704397158e+0000, }; static double qone(double x) { double *p,*q, s,r,z; if (x >= 8.0) {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 >= 2.0) */ {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 (.375 + r/s)/x; }