/* * ==================================================== * 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 are Copyright (C) 2001 Stephen L. Moshier and are incorporated herein by permission of the author. The author reserves the right to distribute this material elsewhere under different copying permissions. These modifications are distributed here under the following terms: This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with this library; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ /* __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 * x); if (ix >= 0x7fff) return one / (x + x * x); if ((i0 | i1) == 0) return -HUGE_VALL + x; /* -inf and overflow exception. */ 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; }