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Diffstat (limited to 'sysdeps/ieee754/ldbl-96/e_j0l.c')
| -rw-r--r-- | sysdeps/ieee754/ldbl-96/e_j0l.c | 531 |
1 files changed, 0 insertions, 531 deletions
diff --git a/sysdeps/ieee754/ldbl-96/e_j0l.c b/sysdeps/ieee754/ldbl-96/e_j0l.c deleted file mode 100644 index a536054cde..0000000000 --- a/sysdeps/ieee754/ldbl-96/e_j0l.c +++ /dev/null @@ -1,531 +0,0 @@ -/* - * ==================================================== - * 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; -} |