/* origin: FreeBSD /usr/src/lib/msun/src/e_j0.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. * ==================================================== */ /* j0(x), y0(x) * Bessel function of the first and second kinds of order zero. * Method -- j0(x): * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ... * 2. Reduce x to |x| since j0(x)=j0(-x), and * for x in (0,2) * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x; * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 ) * for x in (2,inf) * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) * as follow: * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) * = 1/sqrt(2) * (cos(x) + sin(x)) * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/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 * j0(nan)= nan * j0(0) = 1 * j0(inf) = 0 * * Method -- y0(x): * 1. For x<2. * Since * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...) * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. * We use the following function to approximate y0, * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2 * where * U(z) = u00 + u01*z + ... + u06*z^6 * V(z) = 1 + v01*z + ... + v04*z^4 * with absolute approximation error bounded by 2**-72. * Note: For tiny x, U/V = u0 and j0(x)~1, hence * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) * 2. For x>=2. * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) * by the method mentioned above. * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. */ #include "libm.h" static double pzero(double), qzero(double); static const double invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ tpi = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */ /* common method when |x|>=2 */ static double common(uint32_t ix, double x, int y0) { double s,c,ss,cc,z; /* * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x-pi/4)-q0(x)*sin(x-pi/4)) * y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x-pi/4)+q0(x)*cos(x-pi/4)) * * sin(x-pi/4) = (sin(x) - cos(x))/sqrt(2) * cos(x-pi/4) = (sin(x) + cos(x))/sqrt(2) * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) */ s = sin(x); c = cos(x); if (y0) c = -c; cc = s+c; /* avoid overflow in 2*x, big ulp error when x>=0x1p1023 */ if (ix < 0x7fe00000) { ss = s-c; z = -cos(2*x); if (s*c < 0) cc = z/ss; else ss = z/cc; if (ix < 0x48000000) { if (y0) ss = -ss; cc = pzero(x)*cc-qzero(x)*ss; } } return invsqrtpi*cc/sqrt(x); } /* R0/S0 on [0, 2.00] */ static const double R02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */ R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */ R04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */ R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */ S01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */ S02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */ S03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */ S04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */ double j0(double x) { double z,r,s; uint32_t ix; GET_HIGH_WORD(ix, x); ix &= 0x7fffffff; /* j0(+-inf)=0, j0(nan)=nan */ if (ix >= 0x7ff00000) return 1/(x*x); x = fabs(x); if (ix >= 0x40000000) { /* |x| >= 2 */ /* large ulp error near zeros: 2.4, 5.52, 8.6537,.. */ return common(ix,x,0); } /* 1 - x*x/4 + x*x*R(x^2)/S(x^2) */ if (ix >= 0x3f200000) { /* |x| >= 2**-13 */ /* up to 4ulp error close to 2 */ z = x*x; r = z*(R02+z*(R03+z*(R04+z*R05))); s = 1+z*(S01+z*(S02+z*(S03+z*S04))); return (1+x/2)*(1-x/2) + z*(r/s); } /* 1 - x*x/4 */ /* prevent underflow */ /* inexact should be raised when x!=0, this is not done correctly */ if (ix >= 0x38000000) /* |x| >= 2**-127 */ x = 0.25*x*x; return 1 - x; } static const double u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */ u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */ u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */ u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */ u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */ u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */ u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */ v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */ v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */ v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */ v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */ double y0(double x) { double z,u,v; uint32_t ix,lx; EXTRACT_WORDS(ix, lx, x); /* y0(nan)=nan, y0(<0)=nan, y0(0)=-inf, y0(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 */ /* large ulp errors near zeros: 3.958, 7.086,.. */ return common(ix,x,1); } /* U(x^2)/V(x^2) + (2/pi)*j0(x)*log(x) */ if (ix >= 0x3e400000) { /* x >= 2**-27 */ /* large ulp error near the first zero, x ~= 0.89 */ z = x*x; u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); v = 1.0+z*(v01+z*(v02+z*(v03+z*v04))); return u/v + tpi*(j0(x)*log(x)); } return u00 + tpi*log(x); } /* The asymptotic expansions of pzero is * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. * For x >= 2, We approximate pzero by * pzero(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 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) */ static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */ -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */ -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */ -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */ -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */ }; static const double pS8[5] = { 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */ 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */ 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */ 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */ 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */ }; static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */ -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */ -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */ -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */ -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */ -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */ }; static const double pS5[5] = { 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */ 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */ 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */ 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */ 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */ }; static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */ -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */ -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */ -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */ -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */ -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */ }; static const double pS3[5] = { 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */ 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */ 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */ 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */ 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */ }; static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */ -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */ -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */ -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */ -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */ -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */ }; static const double pS2[5] = { 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */ 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */ 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */ 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */ 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */ }; static double pzero(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 qzero is * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. * We approximate pzero by * qzero(x) = s*(-1.25 + (R/S)) * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 * S = 1 + qS0*s^2 + ... + qS5*s^12 * and * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) */ static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */ 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */ 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */ 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */ 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */ }; static const double qS8[6] = { 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */ 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */ 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */ 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */ 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */ -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */ }; static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */ 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */ 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */ 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */ 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */ 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */ }; static const double qS5[6] = { 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */ 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */ 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */ 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */ 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */ -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */ }; static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */ 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */ 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */ 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */ 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */ 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */ }; static const double qS3[6] = { 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */ 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */ 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */ 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */ 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */ -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */ }; static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */ 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */ 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */ 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */ 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */ 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */ }; static const double qS2[6] = { 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */ 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */ 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */ 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */ 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */ -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */ }; static double qzero(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 (-.125 + r/s)/x; }