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Diffstat (limited to 'sysdeps/ieee754/dbl-64/e_j1.c')
-rw-r--r-- | sysdeps/ieee754/dbl-64/e_j1.c | 466 |
1 files changed, 0 insertions, 466 deletions
diff --git a/sysdeps/ieee754/dbl-64/e_j1.c b/sysdeps/ieee754/dbl-64/e_j1.c deleted file mode 100644 index eb446fd102..0000000000 --- a/sysdeps/ieee754/dbl-64/e_j1.c +++ /dev/null @@ -1,466 +0,0 @@ -/* @(#)e_j1.c 5.1 93/09/24 */ -/* - * ==================================================== - * 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. - * ==================================================== - */ -/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/26, - for performance improvement on pipelined processors. - */ - -/* __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; - * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) - * 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 - * where for x in [0,2] (abs err less than 2**-65.89) - * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4 - * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5 - * Note: For tiny x, 1/x dominate y1 and hence - * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) - * 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 double pone (double), qone (double); - -static const double - huge = 1e300, - one = 1.0, - invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ - tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ -/* R0/S0 on [0,2] */ - R[] = { -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */ - 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */ - -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */ - 4.96727999609584448412e-08 }, /* 0x3E6AAAFA, 0x46CA0BD9 */ - S[] = { 0.0, 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */ - 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */ - 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */ - 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */ - 1.23542274426137913908e-11 }; /* 0x3DAB2ACF, 0xCFB97ED8 */ - -static const double zero = 0.0; - -double -__ieee754_j1 (double x) -{ - double z, s, c, ss, cc, r, u, v, y, r1, r2, s1, s2, s3, z2, z4; - int32_t hx, ix; - - GET_HIGH_WORD (hx, x); - ix = hx & 0x7fffffff; - if (__glibc_unlikely (ix >= 0x7ff00000)) - return one / x; - y = fabs (x); - if (ix >= 0x40000000) /* |x| >= 2.0 */ - { - __sincos (y, &s, &c); - ss = -s - c; - cc = s - c; - if (ix < 0x7fe00000) /* make sure y+y not overflow */ - { - z = __cos (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 > 0x48000000) - z = (invsqrtpi * cc) / __ieee754_sqrt (y); - else - { - u = pone (y); v = qone (y); - z = invsqrtpi * (u * cc - v * ss) / __ieee754_sqrt (y); - } - if (hx < 0) - return -z; - else - return z; - } - if (__glibc_unlikely (ix < 0x3e400000)) /* |x|<2**-27 */ - { - if (huge + x > one) /* inexact if x!=0 necessary */ - { - double ret = math_narrow_eval (0.5 * x); - math_check_force_underflow (ret); - if (ret == 0 && x != 0) - __set_errno (ERANGE); - return ret; - } - } - z = x * x; - r1 = z * R[0]; z2 = z * z; - r2 = R[1] + z * R[2]; z4 = z2 * z2; - r = r1 + z2 * r2 + z4 * R[3]; - r *= x; - s1 = one + z * S[1]; - s2 = S[2] + z * S[3]; - s3 = S[4] + z * S[5]; - s = s1 + z2 * s2 + z4 * s3; - return (x * 0.5 + r / s); -} -strong_alias (__ieee754_j1, __j1_finite) - -static const double U0[5] = { - -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */ - 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */ - -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */ - 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */ - -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */ -}; -static const double V0[5] = { - 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */ - 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */ - 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */ - 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */ - 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */ -}; - -double -__ieee754_y1 (double x) -{ - double z, s, c, ss, cc, u, v, u1, u2, v1, v2, v3, z2, z4; - int32_t hx, ix, lx; - - EXTRACT_WORDS (hx, lx, x); - ix = 0x7fffffff & hx; - /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ - if (__glibc_unlikely (ix >= 0x7ff00000)) - return one / (x + x * x); - if (__glibc_unlikely ((ix | lx) == 0)) - return -1 / zero; /* -inf and divide by zero exception. */ - /* -inf and overflow exception. */; - if (__glibc_unlikely (hx < 0)) - return zero / (zero * x); - if (ix >= 0x40000000) /* |x| >= 2.0 */ - { - __sincos (x, &s, &c); - ss = -s - c; - cc = s - c; - if (ix < 0x7fe00000) /* make sure x+x not overflow */ - { - z = __cos (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 > 0x48000000) - z = (invsqrtpi * ss) / __ieee754_sqrt (x); - else - { - u = pone (x); v = qone (x); - z = invsqrtpi * (u * ss + v * cc) / __ieee754_sqrt (x); - } - return z; - } - if (__glibc_unlikely (ix <= 0x3c900000)) /* x < 2**-54 */ - { - z = -tpi / x; - if (isinf (z)) - __set_errno (ERANGE); - return z; - } - z = x * x; - u1 = U0[0] + z * U0[1]; z2 = z * z; - u2 = U0[2] + z * U0[3]; z4 = z2 * z2; - u = u1 + z2 * u2 + z4 * U0[4]; - v1 = one + z * V0[0]; - v2 = V0[1] + z * V0[2]; - v3 = V0[3] + z * V0[4]; - v = v1 + z2 * v2 + z4 * v3; - return (x * (u / v) + tpi * (__ieee754_j1 (x) * __ieee754_log (x) - one / x)); -} -strong_alias (__ieee754_y1, __y1_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) - * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 - * S = 1 + ps0*s^2 + ... + ps4*s^10 - * and - * | pone(x)-1-R/S | <= 2 ** ( -60.06) - */ - -static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ - 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ - 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */ - 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */ - 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */ - 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */ - 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */ -}; -static const double ps8[5] = { - 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */ - 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */ - 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */ - 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */ - 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */ -}; - -static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ - 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */ - 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */ - 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */ - 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */ - 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */ - 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */ -}; -static const double ps5[5] = { - 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */ - 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */ - 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */ - 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */ - 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */ -}; - -static const double pr3[6] = { - 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */ - 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */ - 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */ - 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */ - 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */ - 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */ -}; -static const double ps3[5] = { - 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */ - 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */ - 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */ - 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */ - 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */ -}; - -static const double pr2[6] = { /* for x in [2.8570,2]=1/[0.3499,0.5] */ - 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */ - 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */ - 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */ - 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */ - 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */ - 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */ -}; -static const double ps2[5] = { - 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */ - 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */ - 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */ - 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */ - 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */ -}; - -static double -pone (double x) -{ - const double *p, *q; - double z, r, s, r1, r2, r3, s1, s2, s3, z2, z4; - int32_t ix; - GET_HIGH_WORD (ix, x); - ix &= 0x7fffffff; - /* ix >= 0x40000000 for all calls to this function. */ - if (ix >= 0x41b00000) - { - return one; - } - else if (ix >= 0x40200000) - { - p = pr8; q = ps8; - } - else if (ix >= 0x40122E8B) - { - p = pr5; q = ps5; - } - else if (ix >= 0x4006DB6D) - { - p = pr3; q = ps3; - } - else - { - p = pr2; q = ps2; - } - z = one / (x * x); - r1 = p[0] + z * p[1]; z2 = z * z; - r2 = p[2] + z * p[3]; z4 = z2 * z2; - r3 = p[4] + z * p[5]; - r = r1 + z2 * r2 + z4 * r3; - s1 = one + z * q[0]; - s2 = q[1] + z * q[2]; - s3 = q[3] + z * q[4]; - s = s1 + z2 * s2 + z4 * s3; - return one + 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)) - * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 - * S = 1 + qs1*s^2 + ... + qs6*s^12 - * and - * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) - */ - -static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ - 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ - -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */ - -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */ - -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */ - -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */ - -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */ -}; -static const double qs8[6] = { - 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */ - 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */ - 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */ - 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */ - 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */ - -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */ -}; - -static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ - -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */ - -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */ - -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */ - -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */ - -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */ - -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */ -}; -static const double qs5[6] = { - 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */ - 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */ - 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */ - 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */ - 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */ - -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */ -}; - -static const double qr3[6] = { - -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */ - -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */ - -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */ - -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */ - -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */ - -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */ -}; -static const double qs3[6] = { - 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */ - 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */ - 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */ - 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */ - 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */ - -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */ -}; - -static const double qr2[6] = { /* for x in [2.8570,2]=1/[0.3499,0.5] */ - -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */ - -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */ - -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */ - -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */ - -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */ - -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */ -}; -static const double qs2[6] = { - 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */ - 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */ - 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */ - 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */ - 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */ - -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */ -}; - -static double -qone (double x) -{ - const double *p, *q; - double s, r, z, r1, r2, r3, s1, s2, s3, z2, z4, z6; - int32_t ix; - GET_HIGH_WORD (ix, x); - ix &= 0x7fffffff; - /* ix >= 0x40000000 for all calls to this function. */ - if (ix >= 0x41b00000) - { - return .375 / x; - } - else if (ix >= 0x40200000) - { - p = qr8; q = qs8; - } - else if (ix >= 0x40122E8B) - { - p = qr5; q = qs5; - } - else if (ix >= 0x4006DB6D) - { - p = qr3; q = qs3; - } - else - { - p = qr2; q = qs2; - } - z = one / (x * x); - r1 = p[0] + z * p[1]; z2 = z * z; - r2 = p[2] + z * p[3]; z4 = z2 * z2; - r3 = p[4] + z * p[5]; z6 = z4 * z2; - r = r1 + z2 * r2 + z4 * r3; - s1 = one + z * q[0]; - s2 = q[1] + z * q[2]; - s3 = q[3] + z * q[4]; - s = s1 + z2 * s2 + z4 * s3 + z6 * q[5]; - return (.375 + r / s) / x; -} |