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Diffstat (limited to 'sysdeps/ieee754')
| -rw-r--r-- | sysdeps/ieee754/ldbl-96/e_j1l.c | 635 |
1 files changed, 635 insertions, 0 deletions
diff --git a/sysdeps/ieee754/ldbl-96/e_j1l.c b/sysdeps/ieee754/ldbl-96/e_j1l.c new file mode 100644 index 0000000000..67d4cc8b4e --- /dev/null +++ b/sysdeps/ieee754/ldbl-96/e_j1l.c @@ -0,0 +1,635 @@ +/* + * ==================================================== + * 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 contributed by + Stephen L. Moshier <moshier@na-net.ornl.gov> */ + +/* __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 "math.h" +#include "math_private.h" + +#ifdef __STDC__ +static long double pone (long double), qone (long double); +#else +static long double pone (), qone (); +#endif + +#ifdef __STDC__ +static const long double +#else +static long double +#endif + 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, */ +}; + +#ifdef __STDC__ +static const long double zero = 0.0; +#else +static long double zero = 0.0; +#endif + + +#ifdef __STDC__ +long double +__ieee754_j1l (long double x) +#else +long double +__ieee754_j1l (x) + long double x; +#endif +{ + long double z, c, r, s, ss, cc, u, v, y; + int32_t ix; + u_int32_t se, i0, i1; + + GET_LDOUBLE_WORDS (se, i0, i1, x); + ix = se & 0x7fff; + if (ix >= 0x7fff) + return one / x; + y = fabsl (x); + if (ix >= 0x4000) + { /* |x| >= 2.0 */ + s = __sinl (y); + c = __cosl (y); + 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 (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 (ix < 0x3fde) /* |x| < 2^-33 */ + { + if (huge + x > one) + return 0.5 * x; /* inexact if x!=0 necessary */ + } + 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); +} + + +/* Y1(x) = 2/pi * (log(x) * j1(x) - 1/x) + x R(x^2) + 0 <= x <= 2 + Peak relative error 2.3e-23 */ +#ifdef __STDC__ +static const long double U0[6] = { +#else +static long double U0[6] = { +#endif + -5.908077186259914699178903164682444848615E10L, + 1.546219327181478013495975514375773435962E10L, + -6.438303331169223128870035584107053228235E8L, + 9.708540045657182600665968063824819371216E6L, + -6.138043997084355564619377183564196265471E4L, + 1.418503228220927321096904291501161800215E2L, +}; +#ifdef __STDC__ +static const long double V0[5] = { +#else +static long double V0[5] = { +#endif + 3.013447341682896694781964795373783679861E11L, + 4.669546565705981649470005402243136124523E9L, + 3.595056091631351184676890179233695857260E7L, + 1.761554028569108722903944659933744317994E5L, + 5.668480419646516568875555062047234534863E2L, + /* 1.000000000000000000000000000000000000000E0L, */ +}; + + +#ifdef __STDC__ +long double +__ieee754_y1l (long double x) +#else +long double +__ieee754_y1l (x) + long double x; +#endif +{ + 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 (se & 0x8000) + return zero / zero; + if (ix >= 0x7fff) + return one / (x + x * x); + if ((i0 | i1) == 0) + return -one / zero; + if (ix >= 0x4000) + { /* |x| >= 2.0 */ + s = __sinl (x); + c = __cosl (x); + ss = -s - c; + cc = s - c; + if (ix < 0x7fe00000) + { /* 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 (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 (ix <= 0x3fbe) + { /* x < 2**-65 */ + return (-tpi / x); + } + 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)); +} + + +/* 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 */ + +#ifdef __STDC__ +static const long double pr8[7] = { +#else +static long double pr8[7] = { +#endif + 8.402048819032978959298664869941375143163E-9L, + 1.813743245316438056192649247507255996036E-6L, + 1.260704554112906152344932388588243836276E-4L, + 3.439294839869103014614229832700986965110E-3L, + 3.576910849712074184504430254290179501209E-2L, + 1.131111483254318243139953003461511308672E-1L, + 4.480715825681029711521286449131671880953E-2L, +}; +#ifdef __STDC__ +static const long double ps8[6] = { +#else +static long double ps8[6] = { +#endif + 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 */ +#ifdef __STDC__ +static const long double pr5[7] = { +#else +static long double pr5[7] = { +#endif + 4.318486887948814529950980396300969247900E-7L, + 4.715341880798817230333360497524173929315E-5L, + 1.642719430496086618401091544113220340094E-3L, + 2.228688005300803935928733750456396149104E-2L, + 1.142773760804150921573259605730018327162E-1L, + 1.755576530055079253910829652698703791957E-1L, + 3.218803858282095929559165965353784980613E-2L, +}; +#ifdef __STDC__ +static const long double ps5[6] = { +#else +static long double ps5[6] = { +#endif + 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 */ +#ifdef __STDC__ +static const long double pr3[7] = { +#else +static long double pr3[7] = { +#endif + 1.265251153957366716825382654273326407972E-5L, + 8.031057269201324914127680782288352574567E-4L, + 1.581648121115028333661412169396282881035E-2L, + 1.179534658087796321928362981518645033967E-1L, + 3.227936912780465219246440724502790727866E-1L, + 2.559223765418386621748404398017602935764E-1L, + 2.277136933287817911091370397134882441046E-2L, +}; +#ifdef __STDC__ +static const long double ps3[6] = { +#else +static long double ps3[6] = { +#endif + 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 */ +#ifdef __STDC__ +static const long double pr2[7] = { +#else +static long double pr2[7] = { +#endif + 2.795623248568412225239401141338714516445E-4L, + 1.092578168441856711925254839815430061135E-2L, + 1.278024620468953761154963591853679640560E-1L, + 5.469680473691500673112904286228351988583E-1L, + 8.313769490922351300461498619045639016059E-1L, + 3.544176317308370086415403567097130611468E-1L, + 1.604142674802373041247957048801599740644E-2L, +}; +#ifdef __STDC__ +static const long double ps2[6] = { +#else +static long double ps2[6] = { +#endif + 2.385605161555183386205027000675875235980E-3L, + 9.616778294482695283928617708206967248579E-2L, + 1.195215570959693572089824415393951258510E0L, + 5.718412857897054829999458736064922974662E0L, + 1.065626298505499086386584642761602177568E1L, + 6.809140730053382188468983548092322151791E0L, + /* 1.000000000000000000000000000000000000000E0L, */ +}; + + +#ifdef __STDC__ +static long double +pone (long double x) +#else +static long double +pone (x) + long double x; +#endif +{ +#ifdef __STDC__ + const long double *p, *q; +#else + long double *p, *q; +#endif + long double z, r, s; + int32_t ix; + u_int32_t se, i0, i1; + + GET_LDOUBLE_WORDS (se, i0, i1, x); + ix = se & 0x7fff; + 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 if (ix >= 0x4000) /* x better be >= 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 */ + +#ifdef __STDC__ +static const long double qr8[7] = { +#else +static long double qr8[7] = { +#endif + -5.691925079044209246015366919809404457380E-10L, + -1.632587664706999307871963065396218379137E-7L, + -1.577424682764651970003637263552027114600E-5L, + -6.377627959241053914770158336842725291713E-4L, + -1.087408516779972735197277149494929568768E-2L, + -6.854943629378084419631926076882330494217E-2L, + -1.055448290469180032312893377152490183203E-1L, +}; +#ifdef __STDC__ +static const long double qs8[7] = { +#else +static long double qs8[7] = { +#endif + 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 */ +#ifdef __STDC__ +static const long double qr5[7] = { +#else +static long double qr5[7] = { +#endif + -6.719134139179190546324213696633564965983E-8L, + -9.467871458774950479909851595678622044140E-6L, + -4.429341875348286176950914275723051452838E-4L, + -8.539898021757342531563866270278505014487E-3L, + -6.818691805848737010422337101409276287170E-2L, + -1.964432669771684034858848142418228214855E-1L, + -1.333896496989238600119596538299938520726E-1L, +}; +#ifdef __STDC__ +static const long double qs5[7] = { +#else +static long double qs5[7] = { +#endif + 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 */ +#ifdef __STDC__ +static const long double qr3[7] = { +#else +static long double qr3[7] = { +#endif + -3.618746299358445926506719188614570588404E-6L, + -2.951146018465419674063882650970344502798E-4L, + -7.728518171262562194043409753656506795258E-3L, + -8.058010968753999435006488158237984014883E-2L, + -3.356232856677966691703904770937143483472E-1L, + -4.858192581793118040782557808823460276452E-1L, + -1.592399251246473643510898335746432479373E-1L, +}; +#ifdef __STDC__ +static const long double qs3[7] = { +#else +static long double qs3[7] = { +#endif + 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 */ +#ifdef __STDC__ +static const long double qr2[7] = { +#else +static long double qr2[7] = { +#endif + -1.372751603025230017220666013816502528318E-4L, + -6.879190253347766576229143006767218972834E-3L, + -1.061253572090925414598304855316280077828E-1L, + -6.262164224345471241219408329354943337214E-1L, + -1.423149636514768476376254324731437473915E0L, + -1.087955310491078933531734062917489870754E0L, + -1.826821119773182847861406108689273719137E-1L, +}; +#ifdef __STDC__ +static const long double qs2[7] = { +#else +static long double qs2[7] = { +#endif + 1.338768933634451601814048220627185324007E-3L, + 7.071099998918497559736318523932241901810E-2L, + 1.200511429784048632105295629933382142221E0L, + 8.327301713640367079030141077172031825276E0L, + 2.468479301872299311658145549931764426840E1L, + 2.961179686096262083509383820557051621644E1L, + 1.201402313144305153005639494661767354977E1L, + /* 1.000000000000000000000000000000000000000E0L, */ +}; + + +#ifdef __STDC__ +static long double +qone (long double x) +#else +static long double +qone (x) + long double x; +#endif +{ +#ifdef __STDC__ + const long double *p, *q; +#else + long double *p, *q; +#endif + 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; + 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 if (ix >= 0x4000) /* x better be >= 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; +} |