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diff --git a/sysdeps/ieee754/ldbl-96/e_j1l.c b/sysdeps/ieee754/ldbl-96/e_j1l.c
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-/*
- * ====================================================
- * 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;
-}
generated by cgit v1.2.3-70-g09d2 (git 2.48.1) at 2025-04-09 14:05:15 +0000