e_j0.c revision 279856
1141296Sdas 2141296Sdas/* @(#)e_j0.c 1.3 95/01/18 */ 32116Sjkh/* 42116Sjkh * ==================================================== 52116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 62116Sjkh * 7141296Sdas * Developed at SunSoft, a Sun Microsystems, Inc. business. 82116Sjkh * Permission to use, copy, modify, and distribute this 9141296Sdas * software is freely granted, provided that this notice 102116Sjkh * is preserved. 112116Sjkh * ==================================================== 122116Sjkh */ 132116Sjkh 14176451Sdas#include <sys/cdefs.h> 15176451Sdas__FBSDID("$FreeBSD: head/lib/msun/src/e_j0.c 279856 2015-03-10 17:10:54Z kargl $"); 162116Sjkh 172116Sjkh/* __ieee754_j0(x), __ieee754_y0(x) 182116Sjkh * Bessel function of the first and second kinds of order zero. 192116Sjkh * Method -- j0(x): 202116Sjkh * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ... 212116Sjkh * 2. Reduce x to |x| since j0(x)=j0(-x), and 222116Sjkh * for x in (0,2) 232116Sjkh * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x; 242116Sjkh * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 ) 252116Sjkh * for x in (2,inf) 262116Sjkh * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) 272116Sjkh * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) 282116Sjkh * as follow: 292116Sjkh * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 302116Sjkh * = 1/sqrt(2) * (cos(x) + sin(x)) 312116Sjkh * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) 322116Sjkh * = 1/sqrt(2) * (sin(x) - cos(x)) 332116Sjkh * (To avoid cancellation, use 342116Sjkh * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 352116Sjkh * to compute the worse one.) 36141296Sdas * 372116Sjkh * 3 Special cases 382116Sjkh * j0(nan)= nan 392116Sjkh * j0(0) = 1 402116Sjkh * j0(inf) = 0 41141296Sdas * 422116Sjkh * Method -- y0(x): 432116Sjkh * 1. For x<2. 44141296Sdas * Since 452116Sjkh * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...) 462116Sjkh * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. 472116Sjkh * We use the following function to approximate y0, 482116Sjkh * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2 49141296Sdas * where 502116Sjkh * U(z) = u00 + u01*z + ... + u06*z^6 512116Sjkh * V(z) = 1 + v01*z + ... + v04*z^4 522116Sjkh * with absolute approximation error bounded by 2**-72. 532116Sjkh * Note: For tiny x, U/V = u0 and j0(x)~1, hence 542116Sjkh * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) 552116Sjkh * 2. For x>=2. 562116Sjkh * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) 572116Sjkh * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) 582116Sjkh * by the method mentioned above. 592116Sjkh * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. 602116Sjkh */ 612116Sjkh 622116Sjkh#include "math.h" 632116Sjkh#include "math_private.h" 642116Sjkh 652116Sjkhstatic double pzero(double), qzero(double); 662116Sjkh 67279856Skarglstatic const volatile double vone = 1, vzero = 0; 68279856Skargl 698870Srgrimesstatic const double 702116Sjkhhuge = 1e300, 712116Sjkhone = 1.0, 722116Sjkhinvsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 732116Sjkhtpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ 742116Sjkh /* R0/S0 on [0, 2.00] */ 752116SjkhR02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */ 762116SjkhR03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */ 772116SjkhR04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */ 782116SjkhR05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */ 792116SjkhS01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */ 802116SjkhS02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */ 812116SjkhS03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */ 822116SjkhS04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */ 832116Sjkh 842116Sjkhstatic const double zero = 0.0; 852116Sjkh 8697413Salfreddouble 8797413Salfred__ieee754_j0(double x) 882116Sjkh{ 892116Sjkh double z, s,c,ss,cc,r,u,v; 902116Sjkh int32_t hx,ix; 912116Sjkh 922116Sjkh GET_HIGH_WORD(hx,x); 932116Sjkh ix = hx&0x7fffffff; 942116Sjkh if(ix>=0x7ff00000) return one/(x*x); 952116Sjkh x = fabs(x); 962116Sjkh if(ix >= 0x40000000) { /* |x| >= 2.0 */ 972116Sjkh s = sin(x); 982116Sjkh c = cos(x); 992116Sjkh ss = s-c; 1002116Sjkh cc = s+c; 1012116Sjkh if(ix<0x7fe00000) { /* make sure x+x not overflow */ 1022116Sjkh z = -cos(x+x); 1032116Sjkh if ((s*c)<zero) cc = z/ss; 1042116Sjkh else ss = z/cc; 1052116Sjkh } 1062116Sjkh /* 1072116Sjkh * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 1082116Sjkh * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 1092116Sjkh */ 1102116Sjkh if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x); 1112116Sjkh else { 1122116Sjkh u = pzero(x); v = qzero(x); 1132116Sjkh z = invsqrtpi*(u*cc-v*ss)/sqrt(x); 1142116Sjkh } 1152116Sjkh return z; 1162116Sjkh } 1172116Sjkh if(ix<0x3f200000) { /* |x| < 2**-13 */ 1182116Sjkh if(huge+x>one) { /* raise inexact if x != 0 */ 1192116Sjkh if(ix<0x3e400000) return one; /* |x|<2**-27 */ 120275518Skargl else return one - x*x/4; 1212116Sjkh } 1222116Sjkh } 1232116Sjkh z = x*x; 1242116Sjkh r = z*(R02+z*(R03+z*(R04+z*R05))); 1252116Sjkh s = one+z*(S01+z*(S02+z*(S03+z*S04))); 1262116Sjkh if(ix < 0x3FF00000) { /* |x| < 1.00 */ 1272116Sjkh return one + z*(-0.25+(r/s)); 1282116Sjkh } else { 1292116Sjkh u = 0.5*x; 1302116Sjkh return((one+u)*(one-u)+z*(r/s)); 1312116Sjkh } 1322116Sjkh} 1332116Sjkh 1342116Sjkhstatic const double 1352116Sjkhu00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */ 1362116Sjkhu01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */ 1372116Sjkhu02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */ 1382116Sjkhu03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */ 1392116Sjkhu04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */ 1402116Sjkhu05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */ 1412116Sjkhu06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */ 1422116Sjkhv01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */ 1432116Sjkhv02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */ 1442116Sjkhv03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */ 1452116Sjkhv04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */ 1462116Sjkh 14797413Salfreddouble 14897413Salfred__ieee754_y0(double x) 1492116Sjkh{ 1502116Sjkh double z, s,c,ss,cc,u,v; 1512116Sjkh int32_t hx,ix,lx; 1522116Sjkh 1532116Sjkh EXTRACT_WORDS(hx,lx,x); 1542116Sjkh ix = 0x7fffffff&hx; 155279856Skargl /* 156279856Skargl * y0(NaN) = NaN. 157279856Skargl * y0(Inf) = 0. 158279856Skargl * y0(-Inf) = NaN and raise invalid exception. 159279856Skargl */ 160279856Skargl if(ix>=0x7ff00000) return vone/(x+x*x); 161279856Skargl /* y0(+-0) = -inf and raise divide-by-zero exception. */ 162279856Skargl if((ix|lx)==0) return -one/vzero; 163279856Skargl /* y0(x<0) = NaN and raise invalid exception. */ 164279856Skargl if(hx<0) return vzero/vzero; 1652116Sjkh if(ix >= 0x40000000) { /* |x| >= 2.0 */ 1662116Sjkh /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 1672116Sjkh * where x0 = x-pi/4 1682116Sjkh * Better formula: 1692116Sjkh * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 1702116Sjkh * = 1/sqrt(2) * (sin(x) + cos(x)) 1712116Sjkh * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 1722116Sjkh * = 1/sqrt(2) * (sin(x) - cos(x)) 1732116Sjkh * To avoid cancellation, use 1742116Sjkh * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 1752116Sjkh * to compute the worse one. 1762116Sjkh */ 1772116Sjkh s = sin(x); 1782116Sjkh c = cos(x); 1792116Sjkh ss = s-c; 1802116Sjkh cc = s+c; 1812116Sjkh /* 1822116Sjkh * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 1832116Sjkh * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 1842116Sjkh */ 1852116Sjkh if(ix<0x7fe00000) { /* make sure x+x not overflow */ 1862116Sjkh z = -cos(x+x); 1872116Sjkh if ((s*c)<zero) cc = z/ss; 1882116Sjkh else ss = z/cc; 1892116Sjkh } 1902116Sjkh if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x); 1912116Sjkh else { 1922116Sjkh u = pzero(x); v = qzero(x); 1932116Sjkh z = invsqrtpi*(u*ss+v*cc)/sqrt(x); 1942116Sjkh } 1952116Sjkh return z; 1962116Sjkh } 1972116Sjkh if(ix<=0x3e400000) { /* x < 2**-27 */ 1982116Sjkh return(u00 + tpi*__ieee754_log(x)); 1992116Sjkh } 2002116Sjkh z = x*x; 2012116Sjkh u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 2022116Sjkh v = one+z*(v01+z*(v02+z*(v03+z*v04))); 2032116Sjkh return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x))); 2042116Sjkh} 2052116Sjkh 2062116Sjkh/* The asymptotic expansions of pzero is 2072116Sjkh * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 2082116Sjkh * For x >= 2, We approximate pzero by 2092116Sjkh * pzero(x) = 1 + (R/S) 2102116Sjkh * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 2112116Sjkh * S = 1 + pS0*s^2 + ... + pS4*s^10 2122116Sjkh * and 2132116Sjkh * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 2142116Sjkh */ 2152116Sjkhstatic const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 2162116Sjkh 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 2172116Sjkh -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */ 2182116Sjkh -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */ 2192116Sjkh -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */ 2202116Sjkh -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */ 2212116Sjkh -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */ 2222116Sjkh}; 2232116Sjkhstatic const double pS8[5] = { 2242116Sjkh 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */ 2252116Sjkh 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */ 2262116Sjkh 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */ 2272116Sjkh 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */ 2282116Sjkh 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */ 2292116Sjkh}; 2302116Sjkh 2312116Sjkhstatic const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 2322116Sjkh -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */ 2332116Sjkh -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */ 2342116Sjkh -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */ 2352116Sjkh -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */ 2362116Sjkh -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */ 2372116Sjkh -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */ 2382116Sjkh}; 2392116Sjkhstatic const double pS5[5] = { 2402116Sjkh 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */ 2412116Sjkh 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */ 2422116Sjkh 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */ 2432116Sjkh 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */ 2442116Sjkh 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */ 2452116Sjkh}; 2462116Sjkh 2472116Sjkhstatic const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 2482116Sjkh -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */ 2492116Sjkh -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */ 2502116Sjkh -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */ 2512116Sjkh -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */ 2522116Sjkh -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */ 2532116Sjkh -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */ 2542116Sjkh}; 2552116Sjkhstatic const double pS3[5] = { 2562116Sjkh 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */ 2572116Sjkh 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */ 2582116Sjkh 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */ 2592116Sjkh 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */ 2602116Sjkh 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */ 2612116Sjkh}; 2622116Sjkh 2632116Sjkhstatic const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 2642116Sjkh -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */ 2652116Sjkh -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */ 2662116Sjkh -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */ 2672116Sjkh -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */ 2682116Sjkh -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */ 2692116Sjkh -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */ 2702116Sjkh}; 2712116Sjkhstatic const double pS2[5] = { 2722116Sjkh 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */ 2732116Sjkh 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */ 2742116Sjkh 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */ 2752116Sjkh 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */ 2762116Sjkh 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */ 2772116Sjkh}; 2782116Sjkh 279279493Skarglstatic __inline double 280279493Skarglpzero(double x) 2812116Sjkh{ 2822116Sjkh const double *p,*q; 2832116Sjkh double z,r,s; 2842116Sjkh int32_t ix; 2852116Sjkh GET_HIGH_WORD(ix,x); 2862116Sjkh ix &= 0x7fffffff; 2872116Sjkh if(ix>=0x40200000) {p = pR8; q= pS8;} 2882116Sjkh else if(ix>=0x40122E8B){p = pR5; q= pS5;} 2892116Sjkh else if(ix>=0x4006DB6D){p = pR3; q= pS3;} 290279127Spfg else {p = pR2; q= pS2;} /* ix>=0x40000000 */ 2912116Sjkh z = one/(x*x); 2922116Sjkh r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 2932116Sjkh s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 2942116Sjkh return one+ r/s; 2952116Sjkh} 296141296Sdas 2972116Sjkh 2982116Sjkh/* For x >= 8, the asymptotic expansions of qzero is 2992116Sjkh * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 3002116Sjkh * We approximate pzero by 3012116Sjkh * qzero(x) = s*(-1.25 + (R/S)) 3022116Sjkh * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 3032116Sjkh * S = 1 + qS0*s^2 + ... + qS5*s^12 3042116Sjkh * and 3052116Sjkh * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 3062116Sjkh */ 3072116Sjkhstatic const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 3082116Sjkh 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 3092116Sjkh 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */ 3102116Sjkh 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */ 3112116Sjkh 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */ 3122116Sjkh 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */ 3132116Sjkh 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */ 3142116Sjkh}; 3152116Sjkhstatic const double qS8[6] = { 3162116Sjkh 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */ 3172116Sjkh 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */ 3182116Sjkh 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */ 3192116Sjkh 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */ 3202116Sjkh 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */ 3212116Sjkh -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */ 3222116Sjkh}; 3232116Sjkh 3242116Sjkhstatic const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 3252116Sjkh 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */ 3262116Sjkh 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */ 3272116Sjkh 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */ 3282116Sjkh 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */ 3292116Sjkh 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */ 3302116Sjkh 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */ 3312116Sjkh}; 3322116Sjkhstatic const double qS5[6] = { 3332116Sjkh 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */ 3342116Sjkh 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */ 3352116Sjkh 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */ 3362116Sjkh 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */ 3372116Sjkh 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */ 3382116Sjkh -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */ 3392116Sjkh}; 3402116Sjkh 3412116Sjkhstatic const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 3422116Sjkh 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */ 3432116Sjkh 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */ 3442116Sjkh 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */ 3452116Sjkh 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */ 3462116Sjkh 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */ 3472116Sjkh 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */ 3482116Sjkh}; 3492116Sjkhstatic const double qS3[6] = { 3502116Sjkh 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */ 3512116Sjkh 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */ 3522116Sjkh 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */ 3532116Sjkh 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */ 3542116Sjkh 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */ 3552116Sjkh -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */ 3562116Sjkh}; 3572116Sjkh 3582116Sjkhstatic const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 3592116Sjkh 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */ 3602116Sjkh 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */ 3612116Sjkh 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */ 3622116Sjkh 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */ 3632116Sjkh 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */ 3642116Sjkh 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */ 3652116Sjkh}; 3662116Sjkhstatic const double qS2[6] = { 3672116Sjkh 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */ 3682116Sjkh 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */ 3692116Sjkh 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */ 3702116Sjkh 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */ 3712116Sjkh 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */ 3722116Sjkh -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */ 3732116Sjkh}; 3742116Sjkh 375279493Skarglstatic __inline double 376279493Skarglqzero(double x) 3772116Sjkh{ 3782116Sjkh const double *p,*q; 3792116Sjkh double s,r,z; 3802116Sjkh int32_t ix; 3812116Sjkh GET_HIGH_WORD(ix,x); 3822116Sjkh ix &= 0x7fffffff; 3832116Sjkh if(ix>=0x40200000) {p = qR8; q= qS8;} 3842116Sjkh else if(ix>=0x40122E8B){p = qR5; q= qS5;} 3852116Sjkh else if(ix>=0x4006DB6D){p = qR3; q= qS3;} 386279240Spfg else {p = qR2; q= qS2;} /* ix>=0x40000000 */ 3872116Sjkh z = one/(x*x); 3882116Sjkh r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 3892116Sjkh s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 3902116Sjkh return (-.125 + r/s)/x; 3912116Sjkh} 392