/* origin: FreeBSD /usr/src/lib/msun/src/e_j1.c */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* j1(x), 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 "libm.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] */ r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */ r01 = 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */ r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */ r03 = 4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */ s01 = 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */ s02 = 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */ s03 = 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */ s04 = 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */ s05 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */ static const double zero = 0.0; double j1(double x) { double z,s,c,ss,cc,r,u,v,y; int32_t hx,ix; GET_HIGH_WORD(hx, x); ix = hx & 0x7fffffff; if (ix >= 0x7ff00000) return one/x; y = fabs(x); if (ix >= 0x40000000) { /* |x| >= 2.0 */ s = sin(y); c = cos(y); 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)/sqrt(y); else { u = pone(y); v = qone(y); z = invsqrtpi*(u*cc-v*ss)/sqrt(y); } if (hx < 0) return -z; else return z; } if (ix < 0x3e400000) { /* |x| < 2**-27 */ /* raise inexact if x!=0 */ if (huge+x > one) return 0.5*x; } z = x*x; r = z*(r00+z*(r01+z*(r02+z*r03))); s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); r *= x; return x*0.5 + r/s; } 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 y1(double x) { double z,s,c,ss,cc,u,v; 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 (ix >= 0x7ff00000) return one/(x+x*x); if ((ix|lx) == 0) return -one/zero; if (hx < 0) return zero/zero; if (ix >= 0x40000000) { /* |x| >= 2.0 */ s = sin(x); c = cos(x); 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)/sqrt(x); else { u = pone(x); v = qone(x); z = invsqrtpi*(u*ss+v*cc)/sqrt(x); } return z; } if (ix <= 0x3c900000) /* x < 2**-54 */ return -tpi/x; z = x*x; u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); return x*(u/v) + tpi*(j1(x)*log(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) * 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; int32_t ix; GET_HIGH_WORD(ix, x); ix &= 0x7fffffff; if (ix >= 0x40200000){p = pr8; q = ps8;} else if (ix >= 0x40122E8B){p = pr5; q = ps5;} else if (ix >= 0x4006DB6D){p = pr3; q = ps3;} else if (ix >= 0x40000000){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])))); s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 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; int32_t ix; GET_HIGH_WORD(ix, x); ix &= 0x7fffffff; if (ix >= 0x40200000){p = qr8; q = qs8;} else if (ix >= 0x40122E8B){p = qr5; q = qs5;} else if (ix >= 0x4006DB6D){p = qr3; q = qs3;} else if (ix >= 0x40000000){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])))); s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); return (.375 + r/s)/x; }