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