sysdeps/ieee754/dbl-64/e_j1.c

   1 /* @(#)e_j1.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 /* __ieee754_j1(x), __ieee754_y1(x)
  17  * Bessel function of the first and second kinds of order zero.
  18  * Method -- j1(x):
  19  *      1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
  20  *      2. Reduce x to |x| since j1(x)=-j1(-x),  and
  21  *         for x in (0,2)
  22  *              j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
  23  *         (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
  24  *         for x in (2,inf)
  25  *              j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
  26  *              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
  27  *         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
  28  *         as follow:
  29  *              cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
  30  *                      =  1/sqrt(2) * (sin(x) - cos(x))
  31  *              sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/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  *              j1(nan)= nan
  39  *              j1(0) = 0
  40  *              j1(inf) = 0
  41  *
  42  * Method -- y1(x):
  43  *      1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
  44  *      2. For x<2.
  45  *         Since
  46  *              y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
  47  *         therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
  48  *         We use the following function to approximate y1,
  49  *              y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
  50  *         where for x in [0,2] (abs err less than 2**-65.89)
  51  *              U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
  52  *              V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
  53  *         Note: For tiny x, 1/x dominate y1 and hence
  54  *              y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
  55  *      3. For x>=2.
  56  *              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
  57  *         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
  58  *         by method mentioned above.
  59  */
  60
  61 #include <math.h>
  62 #include <math_private.h>
  63
  64 static double pone(double), qone(double);
  65
  66 static const double
  67 huge    = 1e300,
  68 one     = 1.0,
  69 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  70 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  71         /* R0/S0 on [0,2] */
  72 R[]  = {-6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
  73   1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
  74  -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
  75   4.96727999609584448412e-08}, /* 0x3E6AAAFA, 0x46CA0BD9 */
  76 S[]  =  {0.0, 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
  77   1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
  78   1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
  79   5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
  80   1.23542274426137913908e-11}; /* 0x3DAB2ACF, 0xCFB97ED8 */
  81
  82 static const double zero    = 0.0;
  83
  84 double
  85 __ieee754_j1(double x)
  86 {
  87         double z, s,c,ss,cc,r,u,v,y,r1,r2,s1,s2,s3,z2,z4;
  88         int32_t hx,ix;
  89
  90         GET_HIGH_WORD(hx,x);
  91         ix = hx&0x7fffffff;
  92         if(__builtin_expect(ix>=0x7ff00000, 0)) return one/x;
  93         y = fabs(x);
  94         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  95                 __sincos (y, &s, &c);
  96                 ss = -s-c;
  97                 cc = s-c;
  98                 if(ix<0x7fe00000) {  /* make sure y+y not overflow */
  99                     z = __cos(y+y);
 100                     if ((s*c)>zero) cc = z/ss;
 101                     else            ss = z/cc;
 102                 }
 103         /*
 104          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
 105          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
 106          */
 107                 if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrt(y);
 108                 else {
 109                     u = pone(y); v = qone(y);
 110                     z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrt(y);
 111                 }
 112                 if(hx<0) return -z;
 113                 else     return  z;
 114         }
 115         if(__builtin_expect(ix<0x3e400000, 0)) {        /* |x|<2**-27 */
 116             if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
 117         }
 118         z = x*x;
 119 #ifdef DO_NOT_USE_THIS
 120         r =  z*(r00+z*(r01+z*(r02+z*r03)));
 121         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
 122         r *= x;
 123 #else
 124         r1 = z*R[0]; z2=z*z;
 125         r2 = R[1]+z*R[2]; z4=z2*z2;
 126         r = r1 + z2*r2 + z4*R[3];
 127         r *= x;
 128         s1 = one+z*S[1];
 129         s2 = S[2]+z*S[3];
 130         s3 = S[4]+z*S[5];
 131         s = s1 + z2*s2 + z4*s3;
 132 #endif
 133         return(x*0.5+r/s);
 134 }
 135 strong_alias (__ieee754_j1, __j1_finite)
 136
 137 static const double U0[5] = {
 138  -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
 139   5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
 140  -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
 141   2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
 142  -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
 143 };
 144 static const double V0[5] = {
 145   1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
 146   2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
 147   1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
 148   6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
 149   1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
 150 };
 151
 152 double
 153 __ieee754_y1(double x)
 154 {
 155         double z, s,c,ss,cc,u,v,u1,u2,v1,v2,v3,z2,z4;
 156         int32_t hx,ix,lx;
 157
 158         EXTRACT_WORDS(hx,lx,x);
 159         ix = 0x7fffffff&hx;
 160     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
 161         if(__builtin_expect(ix>=0x7ff00000, 0)) return  one/(x+x*x);
 162         if(__builtin_expect((ix|lx)==0, 0))
 163                 return -HUGE_VAL+x; /* -inf and overflow exception.  */;
 164         if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
 165         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 166                 __sincos (x, &s, &c);
 167                 ss = -s-c;
 168                 cc = s-c;
 169                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
 170                     z = __cos(x+x);
 171                     if ((s*c)>zero) cc = z/ss;
 172                     else            ss = z/cc;
 173                 }
 174         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
 175          * where x0 = x-3pi/4
 176          *      Better formula:
 177          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
 178          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 179          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 180          *                      = -1/sqrt(2) * (cos(x) + sin(x))
 181          * To avoid cancellation, use
 182          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 183          * to compute the worse one.
 184          */
 185                 if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrt(x);
 186                 else {
 187                     u = pone(x); v = qone(x);
 188                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrt(x);
 189                 }
 190                 return z;
 191         }
 192         if(__builtin_expect(ix<=0x3c900000, 0)) {    /* x < 2**-54 */
 193             return(-tpi/x);
 194         }
 195         z = x*x;
 196 #ifdef DO_NOT_USE_THIS
 197         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
 198         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
 199 #else
 200         u1 = U0[0]+z*U0[1];z2=z*z;
 201         u2 = U0[2]+z*U0[3];z4=z2*z2;
 202         u  = u1 + z2*u2 + z4*U0[4];
 203         v1 = one+z*V0[0];
 204         v2 = V0[1]+z*V0[2];
 205         v3 = V0[3]+z*V0[4];
 206         v = v1 + z2*v2 + z4*v3;
 207 #endif
 208         return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
 209 }
 210 strong_alias (__ieee754_y1, __y1_finite)
 211
 212 /* For x >= 8, the asymptotic expansions of pone is
 213  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 214  * We approximate pone by
 215  *      pone(x) = 1 + (R/S)
 216  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 217  *        S = 1 + ps0*s^2 + ... + ps4*s^10
 218  * and
 219  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 220  */
 221
 222 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 223   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 224   1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
 225   1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
 226   4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
 227   3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
 228   7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
 229 };
 230 static const double ps8[5] = {
 231   1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
 232   3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
 233   3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
 234   9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
 235   3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
 236 };
 237
 238 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 239   1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
 240   1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
 241   6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
 242   1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
 243   5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
 244   5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
 245 };
 246 static const double ps5[5] = {
 247   5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
 248   9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
 249   5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
 250   7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
 251   1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
 252 };
 253
 254 static const double pr3[6] = {
 255   3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
 256   1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
 257   3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
 258   3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
 259   9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
 260   4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
 261 };
 262 static const double ps3[5] = {
 263   3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
 264   3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
 265   1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
 266   8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
 267   1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
 268 };
 269
 270 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 271   1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
 272   1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
 273   2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
 274   1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
 275   1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
 276   5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
 277 };
 278 static const double ps2[5] = {
 279   2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
 280   1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
 281   2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
 282   1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
 283   8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
 284 };
 285
 286 static double
 287 pone(double x)
 288 {
 289         const double *p,*q;
 290         double z,r,s,r1,r2,r3,s1,s2,s3,z2,z4;
 291         int32_t ix;
 292         GET_HIGH_WORD(ix,x);
 293         ix &= 0x7fffffff;
 294         if(ix>=0x40200000)     {p = pr8; q= ps8;}
 295         else if(ix>=0x40122E8B){p = pr5; q= ps5;}
 296         else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
 297         else if(ix>=0x40000000){p = pr2; q= ps2;}
 298         z = one/(x*x);
 299 #ifdef DO_NOT_USE_THIS
 300         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 301         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 302 #else
 303         r1 = p[0]+z*p[1]; z2=z*z;
 304         r2 = p[2]+z*p[3]; z4=z2*z2;
 305         r3 = p[4]+z*p[5];
 306         r = r1 + z2*r2 + z4*r3;
 307         s1 = one+z*q[0];
 308         s2 = q[1]+z*q[2];
 309         s3 = q[3]+z*q[4];
 310         s = s1 + z2*s2 + z4*s3;
 311 #endif
 312         return one+ r/s;
 313 }
 314
 315
 316 /* For x >= 8, the asymptotic expansions of qone is
 317  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 318  * We approximate pone by
 319  *      qone(x) = s*(0.375 + (R/S))
 320  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 321  *        S = 1 + qs1*s^2 + ... + qs6*s^12
 322  * and
 323  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 324  */
 325
 326 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 327   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 328  -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
 329  -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
 330  -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
 331  -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
 332  -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
 333 };
 334 static const double qs8[6] = {
 335   1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
 336   7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
 337   1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
 338   7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
 339   6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
 340  -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
 341 };
 342
 343 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 344  -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
 345  -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
 346  -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
 347  -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
 348  -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
 349  -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
 350 };
 351 static const double qs5[6] = {
 352   8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
 353   1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
 354   1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
 355   4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
 356   2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
 357  -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
 358 };
 359
 360 static const double qr3[6] = {
 361  -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
 362  -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
 363  -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
 364  -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
 365  -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
 366  -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
 367 };
 368 static const double qs3[6] = {
 369   4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
 370   6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
 371   3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
 372   5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
 373   1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
 374  -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
 375 };
 376
 377 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 378  -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
 379  -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
 380  -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
 381  -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
 382  -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
 383  -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
 384 };
 385 static const double qs2[6] = {
 386   2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
 387   2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
 388   7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
 389   7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
 390   1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
 391  -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
 392 };
 393
 394 static double
 395 qone(double x)
 396 {
 397         const double *p,*q;
 398         double  s,r,z,r1,r2,r3,s1,s2,s3,z2,z4,z6;
 399         int32_t ix;
 400         GET_HIGH_WORD(ix,x);
 401         ix &= 0x7fffffff;
 402         if(ix>=0x40200000)     {p = qr8; q= qs8;}
 403         else if(ix>=0x40122E8B){p = qr5; q= qs5;}
 404         else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
 405         else if(ix>=0x40000000){p = qr2; q= qs2;}
 406         z = one/(x*x);
 407 #ifdef DO_NOT_USE_THIS
 408         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 409         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 410 #else
 411         r1 = p[0]+z*p[1]; z2=z*z;
 412         r2 = p[2]+z*p[3]; z4=z2*z2;
 413         r3 = p[4]+z*p[5]; z6=z4*z2;
 414         r = r1 + z2*r2 + z4*r3;
 415         s1 = one+z*q[0];
 416         s2 = q[1]+z*q[2];
 417         s3 = q[3]+z*q[4];
 418         s = s1 + z2*s2 + z4*s3 + z6*q[5];
 419 #endif
 420         return (.375 + r/s)/x;
 421 }