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Diffstat (limited to 'REORG.TODO/sysdeps/ieee754/ldbl-96/e_j0l.c')
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diff --git a/REORG.TODO/sysdeps/ieee754/ldbl-96/e_j0l.c b/REORG.TODO/sysdeps/ieee754/ldbl-96/e_j0l.c new file mode 100644 index 0000000000..a536054cde --- /dev/null +++ b/REORG.TODO/sysdeps/ieee754/ldbl-96/e_j0l.c @@ -0,0 +1,531 @@ +/* + * ==================================================== + * 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 <moshier@na-net.ornl.gov> + 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 + <http://www.gnu.org/licenses/>. */ + +/* __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 <math.h> +#include <math_private.h> + +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; +} |