/* * ==================================================== * 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, see . */ /* __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 #include #include #include static long double pone (long double), qone (long double); static const long double 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, */ }; static const long double zero = 0.0; long double __ieee754_j1l (long double x) { long double z, c, r, s, ss, cc, u, v, y; int32_t ix; u_int32_t se; GET_LDOUBLE_EXP (se, x); ix = se & 0x7fff; if (__glibc_unlikely (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 (__glibc_unlikely (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 (__glibc_unlikely (ix < 0x3fde)) /* |x| < 2^-33 */ { if (huge + x > one) /* inexact if x!=0 necessary */ { long double ret = 0.5 * x; math_check_force_underflow (ret); if (ret == 0 && x != 0) __set_errno (ERANGE); return ret; } } 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); } strong_alias (__ieee754_j1l, __j1l_finite) /* Y1(x) = 2/pi * (log(x) * j1(x) - 1/x) + x R(x^2) 0 <= x <= 2 Peak relative error 2.3e-23 */ static const long double U0[6] = { -5.908077186259914699178903164682444848615E10L, 1.546219327181478013495975514375773435962E10L, -6.438303331169223128870035584107053228235E8L, 9.708540045657182600665968063824819371216E6L, -6.138043997084355564619377183564196265471E4L, 1.418503228220927321096904291501161800215E2L, }; static const long double V0[5] = { 3.013447341682896694781964795373783679861E11L, 4.669546565705981649470005402243136124523E9L, 3.595056091631351184676890179233695857260E7L, 1.761554028569108722903944659933744317994E5L, 5.668480419646516568875555062047234534863E2L, /* 1.000000000000000000000000000000000000000E0L, */ }; long double __ieee754_y1l (long double x) { 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 (__glibc_unlikely (se & 0x8000)) return zero / (zero * x); if (__glibc_unlikely (ix >= 0x7fff)) return one / (x + x * x); if (__glibc_unlikely ((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 < 0x7ffe) { /* 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 (__glibc_unlikely (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 (__glibc_unlikely (ix <= 0x3fbe)) { /* x < 2**-65 */ z = -tpi / x; if (isinf (z)) __set_errno (ERANGE); return z; } 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)); } strong_alias (__ieee754_y1l, __y1l_finite) /* 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 */ static const long double pr8[7] = { 8.402048819032978959298664869941375143163E-9L, 1.813743245316438056192649247507255996036E-6L, 1.260704554112906152344932388588243836276E-4L, 3.439294839869103014614229832700986965110E-3L, 3.576910849712074184504430254290179501209E-2L, 1.131111483254318243139953003461511308672E-1L, 4.480715825681029711521286449131671880953E-2L, }; static const long double ps8[6] = { 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 */ static const long double pr5[7] = { 4.318486887948814529950980396300969247900E-7L, 4.715341880798817230333360497524173929315E-5L, 1.642719430496086618401091544113220340094E-3L, 2.228688005300803935928733750456396149104E-2L, 1.142773760804150921573259605730018327162E-1L, 1.755576530055079253910829652698703791957E-1L, 3.218803858282095929559165965353784980613E-2L, }; static const long double ps5[6] = { 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 */ static const long double pr3[7] = { 1.265251153957366716825382654273326407972E-5L, 8.031057269201324914127680782288352574567E-4L, 1.581648121115028333661412169396282881035E-2L, 1.179534658087796321928362981518645033967E-1L, 3.227936912780465219246440724502790727866E-1L, 2.559223765418386621748404398017602935764E-1L, 2.277136933287817911091370397134882441046E-2L, }; static const long double ps3[6] = { 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 */ static const long double pr2[7] = { 2.795623248568412225239401141338714516445E-4L, 1.092578168441856711925254839815430061135E-2L, 1.278024620468953761154963591853679640560E-1L, 5.469680473691500673112904286228351988583E-1L, 8.313769490922351300461498619045639016059E-1L, 3.544176317308370086415403567097130611468E-1L, 1.604142674802373041247957048801599740644E-2L, }; static const long double ps2[6] = { 2.385605161555183386205027000675875235980E-3L, 9.616778294482695283928617708206967248579E-2L, 1.195215570959693572089824415393951258510E0L, 5.718412857897054829999458736064922974662E0L, 1.065626298505499086386584642761602177568E1L, 6.809140730053382188468983548092322151791E0L, /* 1.000000000000000000000000000000000000000E0L, */ }; static long double pone (long double x) { const long double *p, *q; long double z, r, s; int32_t ix; u_int32_t se, i0, i1; GET_LDOUBLE_WORDS (se, i0, i1, x); ix = se & 0x7fff; /* ix >= 0x4000 for all calls to this function. */ 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 /* x >= 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 */ static const long double qr8[7] = { -5.691925079044209246015366919809404457380E-10L, -1.632587664706999307871963065396218379137E-7L, -1.577424682764651970003637263552027114600E-5L, -6.377627959241053914770158336842725291713E-4L, -1.087408516779972735197277149494929568768E-2L, -6.854943629378084419631926076882330494217E-2L, -1.055448290469180032312893377152490183203E-1L, }; static const long double qs8[7] = { 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 */ static const long double qr5[7] = { -6.719134139179190546324213696633564965983E-8L, -9.467871458774950479909851595678622044140E-6L, -4.429341875348286176950914275723051452838E-4L, -8.539898021757342531563866270278505014487E-3L, -6.818691805848737010422337101409276287170E-2L, -1.964432669771684034858848142418228214855E-1L, -1.333896496989238600119596538299938520726E-1L, }; static const long double qs5[7] = { 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 */ static const long double qr3[7] = { -3.618746299358445926506719188614570588404E-6L, -2.951146018465419674063882650970344502798E-4L, -7.728518171262562194043409753656506795258E-3L, -8.058010968753999435006488158237984014883E-2L, -3.356232856677966691703904770937143483472E-1L, -4.858192581793118040782557808823460276452E-1L, -1.592399251246473643510898335746432479373E-1L, }; static const long double qs3[7] = { 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 */ static const long double qr2[7] = { -1.372751603025230017220666013816502528318E-4L, -6.879190253347766576229143006767218972834E-3L, -1.061253572090925414598304855316280077828E-1L, -6.262164224345471241219408329354943337214E-1L, -1.423149636514768476376254324731437473915E0L, -1.087955310491078933531734062917489870754E0L, -1.826821119773182847861406108689273719137E-1L, }; static const long double qs2[7] = { 1.338768933634451601814048220627185324007E-3L, 7.071099998918497559736318523932241901810E-2L, 1.200511429784048632105295629933382142221E0L, 8.327301713640367079030141077172031825276E0L, 2.468479301872299311658145549931764426840E1L, 2.961179686096262083509383820557051621644E1L, 1.201402313144305153005639494661767354977E1L, /* 1.000000000000000000000000000000000000000E0L, */ }; static long double qone (long double x) { const long double *p, *q; 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; /* ix >= 0x4000 for all calls to this function. */ 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 /* x >= 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; }