| /* 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 |
| invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ |
| tpi = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */ |
| |
| static double common(uint32_t ix, double x, int y1, int sign) |
| { |
| double z,s,c,ss,cc; |
| |
| /* |
| * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x-3pi/4)-q1(x)*sin(x-3pi/4)) |
| * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x-3pi/4)+q1(x)*cos(x-3pi/4)) |
| * |
| * sin(x-3pi/4) = -(sin(x) + cos(x))/sqrt(2) |
| * cos(x-3pi/4) = (sin(x) - cos(x))/sqrt(2) |
| * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) |
| */ |
| s = sin(x); |
| if (y1) |
| s = -s; |
| c = cos(x); |
| cc = s-c; |
| if (ix < 0x7fe00000) { |
| /* avoid overflow in 2*x */ |
| ss = -s-c; |
| z = cos(2*x); |
| if (s*c > 0) |
| cc = z/ss; |
| else |
| ss = z/cc; |
| if (ix < 0x48000000) { |
| if (y1) |
| ss = -ss; |
| cc = pone(x)*cc-qone(x)*ss; |
| } |
| } |
| if (sign) |
| cc = -cc; |
| return invsqrtpi*cc/sqrt(x); |
| } |
| |
| /* R0/S0 on [0,2] */ |
| static const double |
| 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 */ |
| |
| double j1(double x) |
| { |
| double z,r,s; |
| uint32_t ix; |
| int sign; |
| |
| GET_HIGH_WORD(ix, x); |
| sign = ix>>31; |
| ix &= 0x7fffffff; |
| if (ix >= 0x7ff00000) |
| return 1/(x*x); |
| if (ix >= 0x40000000) /* |x| >= 2 */ |
| return common(ix, fabs(x), 0, sign); |
| if (ix >= 0x38000000) { /* |x| >= 2**-127 */ |
| z = x*x; |
| r = z*(r00+z*(r01+z*(r02+z*r03))); |
| s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); |
| z = r/s; |
| } else |
| /* avoid underflow, raise inexact if x!=0 */ |
| z = x; |
| return (0.5 + z)*x; |
| } |
| |
| 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,u,v; |
| uint32_t ix,lx; |
| |
| EXTRACT_WORDS(ix, lx, x); |
| /* y1(nan)=nan, y1(<0)=nan, y1(0)=-inf, y1(inf)=0 */ |
| if ((ix<<1 | lx) == 0) |
| return -1/0.0; |
| if (ix>>31) |
| return 0/0.0; |
| if (ix >= 0x7ff00000) |
| return 1/x; |
| |
| if (ix >= 0x40000000) /* x >= 2 */ |
| return common(ix, x, 1, 0); |
| 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 = 1+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); |
| return x*(u/v) + tpi*(j1(x)*log(x)-1/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_t z,r,s; |
| uint32_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 /*ix >= 0x40000000*/ {p = pr2; q = ps2;} |
| z = 1.0/(x*x); |
| r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
| s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); |
| return 1.0+ 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_t s,r,z; |
| uint32_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 /*ix >= 0x40000000*/ {p = qr2; q = qs2;} |
| z = 1.0/(x*x); |
| r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
| s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); |
| return (.375 + r/s)/x; |
| } |
