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