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Diffstat (limited to 'sysdeps/ieee754/ldbl-96/e_j1l.c')
-rw-r--r-- | sysdeps/ieee754/ldbl-96/e_j1l.c | 550 |
1 files changed, 0 insertions, 550 deletions
diff --git a/sysdeps/ieee754/ldbl-96/e_j1l.c b/sysdeps/ieee754/ldbl-96/e_j1l.c deleted file mode 100644 index e8a7349cf4..0000000000 --- a/sysdeps/ieee754/ldbl-96/e_j1l.c +++ /dev/null @@ -1,550 +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_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 <errno.h> -#include <float.h> -#include <math.h> -#include <math_private.h> - -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; -} |