/* * ==================================================== * 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_j0(x), __ieee754_y0(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; * 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 * * 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 #include static long double pzero (long double), qzero (long double); static const long double huge = 1e4930L, one = 1.0L, invsqrtpi = 5.6418958354775628694807945156077258584405e-1L, tpi = 6.3661977236758134307553505349005744813784e-1L, /* J0(x) = 1 - x^2 / 4 + x^4 R0(x^2) / S0(x^2) 0 <= x <= 2 peak relative error 1.41e-22 */ R[5] = { 4.287176872744686992880841716723478740566E7L, -6.652058897474241627570911531740907185772E5L, 7.011848381719789863458364584613651091175E3L, -3.168040850193372408702135490809516253693E1L, 6.030778552661102450545394348845599300939E-2L, }, S[4] = { 2.743793198556599677955266341699130654342E9L, 3.364330079384816249840086842058954076201E7L, 1.924119649412510777584684927494642526573E5L, 6.239282256012734914211715620088714856494E2L, /* 1.000000000000000000000000000000000000000E0L,*/ }; static const long double zero = 0.0; long double __ieee754_j0l (long double x) { long double z, s, c, ss, cc, r, u, v; int32_t ix; u_int32_t se; GET_LDOUBLE_EXP (se, x); ix = se & 0x7fff; if (__glibc_unlikely (ix >= 0x7fff)) return one / (x * x); x = fabsl (x); 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; } /* * 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 (__glibc_unlikely (ix > 0x4080)) /* 2^129 */ z = (invsqrtpi * cc) / __ieee754_sqrtl (x); else { u = pzero (x); v = qzero (x); z = invsqrtpi * (u * cc - v * ss) / __ieee754_sqrtl (x); } return z; } if (__glibc_unlikely (ix < 0x3fef)) /* |x| < 2**-16 */ { /* raise inexact if x != 0 */ math_force_eval (huge + x); if (ix < 0x3fde) /* |x| < 2^-33 */ return one; else return one - 0.25 * x * x; } 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))); if (ix < 0x3fff) { /* |x| < 1.00 */ return (one - 0.25 * z + z * (r / s)); } else { u = 0.5 * x; return ((one + u) * (one - u) + z * (r / s)); } } strong_alias (__ieee754_j0l, __j0l_finite) /* y0(x) = 2/pi ln(x) J0(x) + U(x^2)/V(x^2) 0 < x <= 2 peak relative error 1.7e-21 */ static const long double U[6] = { -1.054912306975785573710813351985351350861E10L, 2.520192609749295139432773849576523636127E10L, -1.856426071075602001239955451329519093395E9L, 4.079209129698891442683267466276785956784E7L, -3.440684087134286610316661166492641011539E5L, 1.005524356159130626192144663414848383774E3L, }; static const long double V[5] = { 1.429337283720789610137291929228082613676E11L, 2.492593075325119157558811370165695013002E9L, 2.186077620785925464237324417623665138376E7L, 1.238407896366385175196515057064384929222E5L, 4.693924035211032457494368947123233101664E2L, /* 1.000000000000000000000000000000000000000E0L */ }; long double __ieee754_y0l (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; /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(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 */ /* 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. */ __sincosl (x, &s, &c); 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 (ix < 0x7ffe) { /* make sure x+x not overflow */ z = -__cosl (x + x); if ((s * c) < zero) cc = z / ss; else ss = z / cc; } if (__glibc_unlikely (ix > 0x4080)) /* 1e39 */ z = (invsqrtpi * ss) / __ieee754_sqrtl (x); else { u = pzero (x); v = qzero (x); z = invsqrtpi * (u * ss + v * cc) / __ieee754_sqrtl (x); } return z; } if (__glibc_unlikely (ix <= 0x3fde)) /* x < 2^-33 */ { z = -7.380429510868722527629822444004602747322E-2L + tpi * __ieee754_logl (x); return z; } z = x * x; u = U[0] + z * (U[1] + z * (U[2] + z * (U[3] + z * (U[4] + z * U[5])))); v = V[0] + z * (V[1] + z * (V[2] + z * (V[3] + z * (V[4] + z)))); return (u / v + tpi * (__ieee754_j0l (x) * __ieee754_logl (x))); } strong_alias (__ieee754_y0l, __y0l_finite) /* 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 + s^2 R(s^2) / S(s^2) */ static const long double pR8[7] = { /* 8 <= x <= inf Peak relative error 4.62 */ -4.094398895124198016684337960227780260127E-9L, -8.929643669432412640061946338524096893089E-7L, -6.281267456906136703868258380673108109256E-5L, -1.736902783620362966354814353559382399665E-3L, -1.831506216290984960532230842266070146847E-2L, -5.827178869301452892963280214772398135283E-2L, -2.087563267939546435460286895807046616992E-2L, }; static const long double pS8[6] = { 5.823145095287749230197031108839653988393E-8L, 1.279281986035060320477759999428992730280E-5L, 9.132668954726626677174825517150228961304E-4L, 2.606019379433060585351880541545146252534E-2L, 2.956262215119520464228467583516287175244E-1L, 1.149498145388256448535563278632697465675E0L, /* 1.000000000000000000000000000000000000000E0L, */ }; static const long double pR5[7] = { /* 4.54541015625 <= x <= 8 Peak relative error 6.51E-22 */ -2.041226787870240954326915847282179737987E-7L, -2.255373879859413325570636768224534428156E-5L, -7.957485746440825353553537274569102059990E-4L, -1.093205102486816696940149222095559439425E-2L, -5.657957849316537477657603125260701114646E-2L, -8.641175552716402616180994954177818461588E-2L, -1.354654710097134007437166939230619726157E-2L, }; static const long double pS5[6] = { 2.903078099681108697057258628212823545290E-6L, 3.253948449946735405975737677123673867321E-4L, 1.181269751723085006534147920481582279979E-2L, 1.719212057790143888884745200257619469363E-1L, 1.006306498779212467670654535430694221924E0L, 2.069568808688074324555596301126375951502E0L, /* 1.000000000000000000000000000000000000000E0L, */ }; static const long double pR3[7] = { /* 2.85711669921875 <= x <= 4.54541015625 peak relative error 5.25e-21 */ -5.755732156848468345557663552240816066802E-6L, -3.703675625855715998827966962258113034767E-4L, -7.390893350679637611641350096842846433236E-3L, -5.571922144490038765024591058478043873253E-2L, -1.531290690378157869291151002472627396088E-1L, -1.193350853469302941921647487062620011042E-1L, -8.567802507331578894302991505331963782905E-3L, }; static const long double pS3[6] = { 8.185931139070086158103309281525036712419E-5L, 5.398016943778891093520574483111255476787E-3L, 1.130589193590489566669164765853409621081E-1L, 9.358652328786413274673192987670237145071E-1L, 3.091711512598349056276917907005098085273E0L, 3.594602474737921977972586821673124231111E0L, /* 1.000000000000000000000000000000000000000E0L, */ }; static const long double pR2[7] = { /* 2 <= x <= 2.85711669921875 peak relative error 2.64e-21 */ -1.219525235804532014243621104365384992623E-4L, -4.838597135805578919601088680065298763049E-3L, -5.732223181683569266223306197751407418301E-2L, -2.472947430526425064982909699406646503758E-1L, -3.753373645974077960207588073975976327695E-1L, -1.556241316844728872406672349347137975495E-1L, -5.355423239526452209595316733635519506958E-3L, }; static const long double pS2[6] = { 1.734442793664291412489066256138894953823E-3L, 7.158111826468626405416300895617986926008E-2L, 9.153839713992138340197264669867993552641E-1L, 4.539209519433011393525841956702487797582E0L, 8.868932430625331650266067101752626253644E0L, 6.067161890196324146320763844772857713502E0L, /* 1.000000000000000000000000000000000000000E0L, */ }; static long double pzero (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) { p = pR8; q = pS8; } /* x >= 8 */ 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 qzero is * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. * We approximate qzero by * qzero(x) = s*(-.125 + R(s^2) / S(s^2)) */ static const long double qR8[7] = { /* 8 <= x <= inf peak relative error 2.23e-21 */ 3.001267180483191397885272640777189348008E-10L, 8.693186311430836495238494289942413810121E-8L, 8.496875536711266039522937037850596580686E-6L, 3.482702869915288984296602449543513958409E-4L, 6.036378380706107692863811938221290851352E-3L, 3.881970028476167836382607922840452192636E-2L, 6.132191514516237371140841765561219149638E-2L, }; static const long double qS8[7] = { 4.097730123753051126914971174076227600212E-9L, 1.199615869122646109596153392152131139306E-6L, 1.196337580514532207793107149088168946451E-4L, 5.099074440112045094341500497767181211104E-3L, 9.577420799632372483249761659674764460583E-2L, 7.385243015344292267061953461563695918646E-1L, 1.917266424391428937962682301561699055943E0L, /* 1.000000000000000000000000000000000000000E0L, */ }; static const long double qR5[7] = { /* 4.54541015625 <= x <= 8 peak relative error 1.03e-21 */ 3.406256556438974327309660241748106352137E-8L, 4.855492710552705436943630087976121021980E-6L, 2.301011739663737780613356017352912281980E-4L, 4.500470249273129953870234803596619899226E-3L, 3.651376459725695502726921248173637054828E-2L, 1.071578819056574524416060138514508609805E-1L, 7.458950172851611673015774675225656063757E-2L, }; static const long double qS5[7] = { 4.650675622764245276538207123618745150785E-7L, 6.773573292521412265840260065635377164455E-5L, 3.340711249876192721980146877577806687714E-3L, 7.036218046856839214741678375536970613501E-2L, 6.569599559163872573895171876511377891143E-1L, 2.557525022583599204591036677199171155186E0L, 3.457237396120935674982927714210361269133E0L, /* 1.000000000000000000000000000000000000000E0L,*/ }; static const long double qR3[7] = { /* 2.85711669921875 <= x <= 4.54541015625 peak relative error 5.24e-21 */ 1.749459596550816915639829017724249805242E-6L, 1.446252487543383683621692672078376929437E-4L, 3.842084087362410664036704812125005761859E-3L, 4.066369994699462547896426554180954233581E-2L, 1.721093619117980251295234795188992722447E-1L, 2.538595333972857367655146949093055405072E-1L, 8.560591367256769038905328596020118877936E-2L, }; static const long double qS3[7] = { 2.388596091707517488372313710647510488042E-5L, 2.048679968058758616370095132104333998147E-3L, 5.824663198201417760864458765259945181513E-2L, 6.953906394693328750931617748038994763958E-1L, 3.638186936390881159685868764832961092476E0L, 7.900169524705757837298990558459547842607E0L, 5.992718532451026507552820701127504582907E0L, /* 1.000000000000000000000000000000000000000E0L, */ }; static const long double qR2[7] = { /* 2 <= x <= 2.85711669921875 peak relative error 1.58e-21 */ 6.306524405520048545426928892276696949540E-5L, 3.209606155709930950935893996591576624054E-3L, 5.027828775702022732912321378866797059604E-2L, 3.012705561838718956481911477587757845163E-1L, 6.960544893905752937420734884995688523815E-1L, 5.431871999743531634887107835372232030655E-1L, 9.447736151202905471899259026430157211949E-2L, }; static const long double qS2[7] = { 8.610579901936193494609755345106129102676E-4L, 4.649054352710496997203474853066665869047E-2L, 8.104282924459837407218042945106320388339E-1L, 5.807730930825886427048038146088828206852E0L, 1.795310145936848873627710102199881642939E1L, 2.281313316875375733663657188888110605044E1L, 1.011242067883822301487154844458322200143E1L, /* 1.000000000000000000000000000000000000000E0L, */ }; static long double qzero (long double x) { const long double *p, *q; 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 (-.125 + z * r / s) / x; }