sysdeps/ieee754/dbl-64/e_j0.c

   1 /* @(#)e_j0.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/26,
  13    for performance improvement on pipelined processors.
  14 */
  15
  16 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: e_j0.c,v 1.8 1995/05/10 20:45:23 jtc Exp $";
  18 #endif
  19
  20 /* __ieee754_j0(x), __ieee754_y0(x)
  21  * Bessel function of the first and second kinds of order zero.
  22  * Method -- j0(x):
  23  *      1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
  24  *      2. Reduce x to |x| since j0(x)=j0(-x),  and
  25  *         for x in (0,2)
  26  *              j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
  27  *         (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
  28  *         for x in (2,inf)
  29  *              j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
  30  *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
  31  *         as follow:
  32  *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
  33  *                      = 1/sqrt(2) * (cos(x) + sin(x))
  34  *              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
  35  *                      = 1/sqrt(2) * (sin(x) - cos(x))
  36  *         (To avoid cancellation, use
  37  *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  38  *          to compute the worse one.)
  39  *
  40  *      3 Special cases
  41  *              j0(nan)= nan
  42  *              j0(0) = 1
  43  *              j0(inf) = 0
  44  *
  45  * Method -- y0(x):
  46  *      1. For x<2.
  47  *         Since
  48  *              y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
  49  *         therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
  50  *         We use the following function to approximate y0,
  51  *              y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
  52  *         where
  53  *              U(z) = u00 + u01*z + ... + u06*z^6
  54  *              V(z) = 1  + v01*z + ... + v04*z^4
  55  *         with absolute approximation error bounded by 2**-72.
  56  *         Note: For tiny x, U/V = u0 and j0(x)~1, hence
  57  *              y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
  58  *      2. For x>=2.
  59  *              y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
  60  *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
  61  *         by the method mentioned above.
  62  *      3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
  63  */
  64
  65 #include "math.h"
  66 #include "math_private.h"
  67
  68 #ifdef __STDC__
  69 static double pzero(double), qzero(double);
  70 #else
  71 static double pzero(), qzero();
  72 #endif
  73
  74 #ifdef __STDC__
  75 static const double
  76 #else
  77 static double
  78 #endif
  79 huge    = 1e300,
  80 one     = 1.0,
  81 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  82 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  83                 /* R0/S0 on [0, 2.00] */
  84 R[]  =  {0.0, 0.0, 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
  85  -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
  86   1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
  87  -4.61832688532103189199e-09}, /* 0xBE33D5E7, 0x73D63FCE */
  88 S[]  =  {0.0, 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
  89   1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
  90   5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
  91   1.16614003333790000205e-09}; /* 0x3E1408BC, 0xF4745D8F */
  92
  93 #ifdef __STDC__
  94 static const double zero = 0.0;
  95 #else
  96 static double zero = 0.0;
  97 #endif
  98
  99 #ifdef __STDC__
 100         double __ieee754_j0(double x)
 101 #else
 102         double __ieee754_j0(x)
 103         double x;
 104 #endif
 105 {
 106         double z, s,c,ss,cc,r,u,v,r1,r2,s1,s2,z2,z4;
 107         int32_t hx,ix;
 108
 109         GET_HIGH_WORD(hx,x);
 110         ix = hx&0x7fffffff;
 111         if(ix>=0x7ff00000) return one/(x*x);
 112         x = fabs(x);
 113         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 114                 __sincos (x, &s, &c);
 115                 ss = s-c;
 116                 cc = s+c;
 117                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
 118                     z = -__cos(x+x);
 119                     if ((s*c)<zero) cc = z/ss;
 120                     else            ss = z/cc;
 121                 }
 122         /*
 123          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 124          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 125          */
 126                 if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrt(x);
 127                 else {
 128                     u = pzero(x); v = qzero(x);
 129                     z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrt(x);
 130                 }
 131                 return z;
 132         }
 133         if(ix<0x3f200000) {     /* |x| < 2**-13 */
 134             if(huge+x>one) {    /* raise inexact if x != 0 */
 135                 if(ix<0x3e400000) return one;   /* |x|<2**-27 */
 136                 else          return one - 0.25*x*x;
 137             }
 138         }
 139         z = x*x;
 140 #ifdef DO_NOT_USE_THIS
 141         r =  z*(R02+z*(R03+z*(R04+z*R05)));
 142         s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
 143 #else
 144         r1 = z*R[2]; z2=z*z;
 145         r2 = R[3]+z*R[4]; z4=z2*z2;
 146         r  = r1 + z2*r2 + z4*R[5];
 147         s1 = one+z*S[1];
 148         s2 = S[2]+z*S[3];
 149         s = s1 + z2*s2 + z4*S[4];
 150 #endif
 151         if(ix < 0x3FF00000) {   /* |x| < 1.00 */
 152             return one + z*(-0.25+(r/s));
 153         } else {
 154             u = 0.5*x;
 155             return((one+u)*(one-u)+z*(r/s));
 156         }
 157 }
 158
 159 #ifdef __STDC__
 160 static const double
 161 #else
 162 static double
 163 #endif
 164 U[]  = {-7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
 165   1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
 166  -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
 167   3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
 168  -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
 169   1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
 170  -3.98205194132103398453e-11}, /* 0xBDC5E43D, 0x693FB3C8 */
 171 V[]  =  {1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
 172   7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
 173   2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
 174   4.41110311332675467403e-10}; /* 0x3DFE5018, 0x3BD6D9EF */
 175
 176 #ifdef __STDC__
 177         double __ieee754_y0(double x)
 178 #else
 179         double __ieee754_y0(x)
 180         double x;
 181 #endif
 182 {
 183         double z, s,c,ss,cc,u,v,z2,z4,z6,u1,u2,u3,v1,v2;
 184         int32_t hx,ix,lx;
 185
 186         EXTRACT_WORDS(hx,lx,x);
 187         ix = 0x7fffffff&hx;
 188     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf.  */
 189         if(ix>=0x7ff00000) return  one/(x+x*x);
 190         if((ix|lx)==0) return -HUGE_VAL+x; /* -inf and overflow exception.  */
 191         if(hx<0) return zero/(zero*x);
 192         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 193         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
 194          * where x0 = x-pi/4
 195          *      Better formula:
 196          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 197          *                      =  1/sqrt(2) * (sin(x) + cos(x))
 198          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 199          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 200          * To avoid cancellation, use
 201          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 202          * to compute the worse one.
 203          */
 204                 __sincos (x, &s, &c);
 205                 ss = s-c;
 206                 cc = s+c;
 207         /*
 208          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 209          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 210          */
 211                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
 212                     z = -__cos(x+x);
 213                     if ((s*c)<zero) cc = z/ss;
 214                     else            ss = z/cc;
 215                 }
 216                 if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrt(x);
 217                 else {
 218                     u = pzero(x); v = qzero(x);
 219                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrt(x);
 220                 }
 221                 return z;
 222         }
 223         if(ix<=0x3e400000) {    /* x < 2**-27 */
 224             return(U[0] + tpi*__ieee754_log(x));
 225         }
 226         z = x*x;
 227 #ifdef DO_NOT_USE_THIS
 228         u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
 229         v = one+z*(v01+z*(v02+z*(v03+z*v04)));
 230 #else
 231         u1 = U[0]+z*U[1]; z2=z*z;
 232         u2 = U[2]+z*U[3]; z4=z2*z2;
 233         u3 = U[4]+z*U[5]; z6=z4*z2;
 234         u = u1 + z2*u2 + z4*u3 + z6*U[6];
 235         v1 = one+z*V[0];
 236         v2 = V[1]+z*V[2];
 237         v = v1 + z2*v2 + z4*V[3];
 238 #endif
 239         return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
 240 }
 241
 242 /* The asymptotic expansions of pzero is
 243  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
 244  * For x >= 2, We approximate pzero by
 245  *      pzero(x) = 1 + (R/S)
 246  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
 247  *        S = 1 + pS0*s^2 + ... + pS4*s^10
 248  * and
 249  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
 250  */
 251 #ifdef __STDC__
 252 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 253 #else
 254 static double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 255 #endif
 256   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 257  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
 258  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
 259  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
 260  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
 261  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
 262 };
 263 #ifdef __STDC__
 264 static const double pS8[5] = {
 265 #else
 266 static double pS8[5] = {
 267 #endif
 268   1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
 269   3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
 270   4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
 271   1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
 272   4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
 273 };
 274
 275 #ifdef __STDC__
 276 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 277 #else
 278 static double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 279 #endif
 280  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
 281  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
 282  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
 283  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
 284  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
 285  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
 286 };
 287 #ifdef __STDC__
 288 static const double pS5[5] = {
 289 #else
 290 static double pS5[5] = {
 291 #endif
 292   6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
 293   1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
 294   5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
 295   9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
 296   2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
 297 };
 298
 299 #ifdef __STDC__
 300 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 301 #else
 302 static double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 303 #endif
 304  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
 305  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
 306  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
 307  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
 308  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
 309  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
 310 };
 311 #ifdef __STDC__
 312 static const double pS3[5] = {
 313 #else
 314 static double pS3[5] = {
 315 #endif
 316   3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
 317   3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
 318   1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
 319   1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
 320   1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
 321 };
 322
 323 #ifdef __STDC__
 324 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 325 #else
 326 static double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 327 #endif
 328  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
 329  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
 330  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
 331  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
 332  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
 333  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
 334 };
 335 #ifdef __STDC__
 336 static const double pS2[5] = {
 337 #else
 338 static double pS2[5] = {
 339 #endif
 340   2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
 341   1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
 342   2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
 343   1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
 344   1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
 345 };
 346
 347 #ifdef __STDC__
 348         static double pzero(double x)
 349 #else
 350         static double pzero(x)
 351         double x;
 352 #endif
 353 {
 354 #ifdef __STDC__
 355         const double *p,*q;
 356 #else
 357         double *p,*q;
 358 #endif
 359         double z,r,s,z2,z4,r1,r2,r3,s1,s2,s3;
 360         int32_t ix;
 361         GET_HIGH_WORD(ix,x);
 362         ix &= 0x7fffffff;
 363         if(ix>=0x40200000)     {p = pR8; q= pS8;}
 364         else if(ix>=0x40122E8B){p = pR5; q= pS5;}
 365         else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
 366         else if(ix>=0x40000000){p = pR2; q= pS2;}
 367         z = one/(x*x);
 368 #ifdef DO_NOT_USE_THIS
 369         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 370         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 371 #else
 372         r1 = p[0]+z*p[1]; z2=z*z;
 373         r2 = p[2]+z*p[3]; z4=z2*z2;
 374         r3 = p[4]+z*p[5];
 375         r = r1 + z2*r2 + z4*r3;
 376         s1 = one+z*q[0];
 377         s2 = q[1]+z*q[2];
 378         s3 = q[3]+z*q[4];
 379         s = s1 + z2*s2 + z4*s3;
 380 #endif
 381         return one+ r/s;
 382 }
 383
 384
 385 /* For x >= 8, the asymptotic expansions of qzero is
 386  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 387  * We approximate pzero by
 388  *      qzero(x) = s*(-1.25 + (R/S))
 389  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
 390  *        S = 1 + qS0*s^2 + ... + qS5*s^12
 391  * and
 392  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 393  */
 394 #ifdef __STDC__
 395 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 396 #else
 397 static double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 398 #endif
 399   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 400   7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
 401   1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
 402   5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
 403   8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
 404   3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
 405 };
 406 #ifdef __STDC__
 407 static const double qS8[6] = {
 408 #else
 409 static double qS8[6] = {
 410 #endif
 411   1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
 412   8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
 413   1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
 414   8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
 415   8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
 416  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
 417 };
 418
 419 #ifdef __STDC__
 420 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 421 #else
 422 static double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 423 #endif
 424   1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
 425   7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
 426   5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
 427   1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
 428   1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
 429   1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
 430 };
 431 #ifdef __STDC__
 432 static const double qS5[6] = {
 433 #else
 434 static double qS5[6] = {
 435 #endif
 436   8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
 437   2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
 438   1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
 439   5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
 440   3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
 441  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
 442 };
 443
 444 #ifdef __STDC__
 445 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 446 #else
 447 static double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 448 #endif
 449   4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
 450   7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
 451   3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
 452   4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
 453   1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
 454   1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
 455 };
 456 #ifdef __STDC__
 457 static const double qS3[6] = {
 458 #else
 459 static double qS3[6] = {
 460 #endif
 461   4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
 462   7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
 463   3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
 464   6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
 465   2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
 466  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
 467 };
 468
 469 #ifdef __STDC__
 470 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 471 #else
 472 static double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 473 #endif
 474   1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
 475   7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
 476   1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
 477   1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
 478   3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
 479   1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
 480 };
 481 #ifdef __STDC__
 482 static const double qS2[6] = {
 483 #else
 484 static double qS2[6] = {
 485 #endif
 486   3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
 487   2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
 488   8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
 489   8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
 490   2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
 491  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
 492 };
 493
 494 #ifdef __STDC__
 495         static double qzero(double x)
 496 #else
 497         static double qzero(x)
 498         double x;
 499 #endif
 500 {
 501 #ifdef __STDC__
 502         const double *p,*q;
 503 #else
 504         double *p,*q;
 505 #endif
 506         double s,r,z,z2,z4,z6,r1,r2,r3,s1,s2,s3;
 507         int32_t ix;
 508         GET_HIGH_WORD(ix,x);
 509         ix &= 0x7fffffff;
 510         if(ix>=0x40200000)     {p = qR8; q= qS8;}
 511         else if(ix>=0x40122E8B){p = qR5; q= qS5;}
 512         else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
 513         else if(ix>=0x40000000){p = qR2; q= qS2;}
 514         z = one/(x*x);
 515 #ifdef DO_NOT_USE_THIS
 516         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 517         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 518 #else
 519         r1 = p[0]+z*p[1]; z2=z*z;
 520         r2 = p[2]+z*p[3]; z4=z2*z2;
 521         r3 = p[4]+z*p[5]; z6=z4*z2;
 522         r= r1 + z2*r2 + z4*r3;
 523         s1 = one+z*q[0];
 524         s2 = q[1]+z*q[2];
 525         s3 = q[3]+z*q[4];
 526         s = s1 + z2*s2 + z4*s3 +z6*q[5];
 527 #endif
 528         return (-.125 + r/s)/x;
 529 }