newlib/libm/mathfp/ef_j0.c

   1 /* ef_j0.c -- float version of e_j0.c.
   2  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
   3  */
   4
   5 /*
   6  * ====================================================
   7  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   8  *
   9  * Developed at SunPro, a Sun Microsystems, Inc. business.
  10  * Permission to use, copy, modify, and distribute this
  11  * software is freely granted, provided that this notice
  12  * is preserved.
  13  * ====================================================
  14  */
  15
  16 #include "fdlibm.h"
  17
  18 #ifdef __STDC__
  19 static float pzerof(float), qzerof(float);
  20 #else
  21 static float pzerof(), qzerof();
  22 #endif
  23
  24 #ifdef __STDC__
  25 static const float
  26 #else
  27 static float
  28 #endif
  29 huge    = 1e30,
  30 one     = 1.0,
  31 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  32 tpi      =  6.3661974669e-01, /* 0x3f22f983 */
  33                 /* R0/S0 on [0, 2.00] */
  34 R02  =  1.5625000000e-02, /* 0x3c800000 */
  35 R03  = -1.8997929874e-04, /* 0xb947352e */
  36 R04  =  1.8295404516e-06, /* 0x35f58e88 */
  37 R05  = -4.6183270541e-09, /* 0xb19eaf3c */
  38 S01  =  1.5619102865e-02, /* 0x3c7fe744 */
  39 S02  =  1.1692678527e-04, /* 0x38f53697 */
  40 S03  =  5.1354652442e-07, /* 0x3509daa6 */
  41 S04  =  1.1661400734e-09; /* 0x30a045e8 */
  42
  43 #ifdef __STDC__
  44 static const float zero = 0.0;
  45 #else
  46 static float zero = 0.0;
  47 #endif
  48
  49 #ifdef __STDC__
  50         float j0f(float x)
  51 #else
  52         float j0f(x)
  53         float x;
  54 #endif
  55 {
  56         float z, s,c,ss,cc,r,u,v;
  57         __int32_t hx,ix;
  58
  59         GET_FLOAT_WORD(hx,x);
  60         ix = hx&0x7fffffff;
  61         if(ix>=0x7f800000) return one/(x*x);
  62         x = fabsf(x);
  63         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  64                 s = sinf(x);
  65                 c = cosf(x);
  66                 ss = s-c;
  67                 cc = s+c;
  68                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
  69                     z = -cosf(x+x);
  70                     if ((s*c)<zero) cc = z/ss;
  71                     else            ss = z/cc;
  72                 }
  73         /*
  74          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
  75          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
  76          */
  77                 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x);
  78                 else {
  79                     u = pzerof(x); v = qzerof(x);
  80                     z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
  81                 }
  82                 return z;
  83         }
  84         if(ix<0x39000000) {     /* |x| < 2**-13 */
  85             if(huge+x>one) {    /* raise inexact if x != 0 */
  86                 if(ix<0x32000000) return one;   /* |x|<2**-27 */
  87                 else          return one - (float)0.25*x*x;
  88             }
  89         }
  90         z = x*x;
  91         r =  z*(R02+z*(R03+z*(R04+z*R05)));
  92         s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
  93         if(ix < 0x3F800000) {   /* |x| < 1.00 */
  94             return one + z*((float)-0.25+(r/s));
  95         } else {
  96             u = (float)0.5*x;
  97             return((one+u)*(one-u)+z*(r/s));
  98         }
  99 }
 100
 101 #ifdef __STDC__
 102 static const float
 103 #else
 104 static float
 105 #endif
 106 u00  = -7.3804296553e-02, /* 0xbd9726b5 */
 107 u01  =  1.7666645348e-01, /* 0x3e34e80d */
 108 u02  = -1.3818567619e-02, /* 0xbc626746 */
 109 u03  =  3.4745343146e-04, /* 0x39b62a69 */
 110 u04  = -3.8140706238e-06, /* 0xb67ff53c */
 111 u05  =  1.9559013964e-08, /* 0x32a802ba */
 112 u06  = -3.9820518410e-11, /* 0xae2f21eb */
 113 v01  =  1.2730483897e-02, /* 0x3c509385 */
 114 v02  =  7.6006865129e-05, /* 0x389f65e0 */
 115 v03  =  2.5915085189e-07, /* 0x348b216c */
 116 v04  =  4.4111031494e-10; /* 0x2ff280c2 */
 117
 118 #ifdef __STDC__
 119         float y0f(float x)
 120 #else
 121         float y0f(x)
 122         float x;
 123 #endif
 124 {
 125         float z, s,c,ss,cc,u,v;
 126         __int32_t hx,ix;
 127
 128         GET_FLOAT_WORD(hx,x);
 129         ix = 0x7fffffff&hx;
 130     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
 131         if(ix>=0x7f800000) return  one/(x+x*x);
 132         if(ix==0) return -one/zero;
 133         if(hx<0) return zero/zero;
 134         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 135         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
 136          * where x0 = x-pi/4
 137          *      Better formula:
 138          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 139          *                      =  1/sqrt(2) * (sin(x) + cos(x))
 140          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 141          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 142          * To avoid cancellation, use
 143          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 144          * to compute the worse one.
 145          */
 146                 s = sinf(x);
 147                 c = cosf(x);
 148                 ss = s-c;
 149                 cc = s+c;
 150         /*
 151          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 152          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 153          */
 154                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
 155                     z = -cosf(x+x);
 156                     if ((s*c)<zero) cc = z/ss;
 157                     else            ss = z/cc;
 158                 }
 159                 if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x);
 160                 else {
 161                     u = pzerof(x); v = qzerof(x);
 162                     z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
 163                 }
 164                 return z;
 165         }
 166         if(ix<=0x32000000) {    /* x < 2**-27 */
 167             return(u00 + tpi*logf(x));
 168         }
 169         z = x*x;
 170         u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
 171         v = one+z*(v01+z*(v02+z*(v03+z*v04)));
 172         return(u/v + tpi*(j0f(x)*logf(x)));
 173 }
 174
 175 /* The asymptotic expansions of pzero is
 176  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
 177  * For x >= 2, We approximate pzero by
 178  *      pzero(x) = 1 + (R/S)
 179  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
 180  *        S = 1 + pS0*s^2 + ... + pS4*s^10
 181  * and
 182  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
 183  */
 184 #ifdef __STDC__
 185 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 186 #else
 187 static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 188 #endif
 189   0.0000000000e+00, /* 0x00000000 */
 190  -7.0312500000e-02, /* 0xbd900000 */
 191  -8.0816707611e+00, /* 0xc1014e86 */
 192  -2.5706311035e+02, /* 0xc3808814 */
 193  -2.4852163086e+03, /* 0xc51b5376 */
 194  -5.2530439453e+03, /* 0xc5a4285a */
 195 };
 196 #ifdef __STDC__
 197 static const float pS8[5] = {
 198 #else
 199 static float pS8[5] = {
 200 #endif
 201   1.1653436279e+02, /* 0x42e91198 */
 202   3.8337448730e+03, /* 0x456f9beb */
 203   4.0597855469e+04, /* 0x471e95db */
 204   1.1675296875e+05, /* 0x47e4087c */
 205   4.7627726562e+04, /* 0x473a0bba */
 206 };
 207 #ifdef __STDC__
 208 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 209 #else
 210 static float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 211 #endif
 212  -1.1412546255e-11, /* 0xad48c58a */
 213  -7.0312492549e-02, /* 0xbd8fffff */
 214  -4.1596107483e+00, /* 0xc0851b88 */
 215  -6.7674766541e+01, /* 0xc287597b */
 216  -3.3123129272e+02, /* 0xc3a59d9b */
 217  -3.4643338013e+02, /* 0xc3ad3779 */
 218 };
 219 #ifdef __STDC__
 220 static const float pS5[5] = {
 221 #else
 222 static float pS5[5] = {
 223 #endif
 224   6.0753936768e+01, /* 0x42730408 */
 225   1.0512523193e+03, /* 0x44836813 */
 226   5.9789707031e+03, /* 0x45bad7c4 */
 227   9.6254453125e+03, /* 0x461665c8 */
 228   2.4060581055e+03, /* 0x451660ee */
 229 };
 230
 231 #ifdef __STDC__
 232 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 233 #else
 234 static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 235 #endif
 236  -2.5470459075e-09, /* 0xb12f081b */
 237  -7.0311963558e-02, /* 0xbd8fffb8 */
 238  -2.4090321064e+00, /* 0xc01a2d95 */
 239  -2.1965976715e+01, /* 0xc1afba52 */
 240  -5.8079170227e+01, /* 0xc2685112 */
 241  -3.1447946548e+01, /* 0xc1fb9565 */
 242 };
 243 #ifdef __STDC__
 244 static const float pS3[5] = {
 245 #else
 246 static float pS3[5] = {
 247 #endif
 248   3.5856033325e+01, /* 0x420f6c94 */
 249   3.6151397705e+02, /* 0x43b4c1ca */
 250   1.1936077881e+03, /* 0x44953373 */
 251   1.1279968262e+03, /* 0x448cffe6 */
 252   1.7358093262e+02, /* 0x432d94b8 */
 253 };
 254
 255 #ifdef __STDC__
 256 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 257 #else
 258 static float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 259 #endif
 260  -8.8753431271e-08, /* 0xb3be98b7 */
 261  -7.0303097367e-02, /* 0xbd8ffb12 */
 262  -1.4507384300e+00, /* 0xbfb9b1cc */
 263  -7.6356959343e+00, /* 0xc0f4579f */
 264  -1.1193166733e+01, /* 0xc1331736 */
 265  -3.2336456776e+00, /* 0xc04ef40d */
 266 };
 267 #ifdef __STDC__
 268 static const float pS2[5] = {
 269 #else
 270 static float pS2[5] = {
 271 #endif
 272   2.2220300674e+01, /* 0x41b1c32d */
 273   1.3620678711e+02, /* 0x430834f0 */
 274   2.7047027588e+02, /* 0x43873c32 */
 275   1.5387539673e+02, /* 0x4319e01a */
 276   1.4657617569e+01, /* 0x416a859a */
 277 };
 278
 279 #ifdef __STDC__
 280         static float pzerof(float x)
 281 #else
 282         static float pzerof(x)
 283         float x;
 284 #endif
 285 {
 286 #ifdef __STDC__
 287         const float *p,*q;
 288 #else
 289         float *p,*q;
 290 #endif
 291         float z,r,s;
 292         __int32_t ix;
 293         GET_FLOAT_WORD(ix,x);
 294         ix &= 0x7fffffff;
 295         if(ix>=0x41000000)     {p = pR8; q= pS8;}
 296         else if(ix>=0x40f71c58){p = pR5; q= pS5;}
 297         else if(ix>=0x4036db68){p = pR3; q= pS3;}
 298       else {p = pR2; q= pS2;}
 299         z = one/(x*x);
 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         return one+ r/s;
 303 }
 304
 305
 306 /* For x >= 8, the asymptotic expansions of qzero is
 307  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 308  * We approximate qzero by
 309  *      qzero(x) = s*(-1.25 + (R/S))
 310  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
 311  *        S = 1 + qS0*s^2 + ... + qS5*s^12
 312  * and
 313  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 314  */
 315 #ifdef __STDC__
 316 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 317 #else
 318 static float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 319 #endif
 320   0.0000000000e+00, /* 0x00000000 */
 321   7.3242187500e-02, /* 0x3d960000 */
 322   1.1768206596e+01, /* 0x413c4a93 */
 323   5.5767340088e+02, /* 0x440b6b19 */
 324   8.8591972656e+03, /* 0x460a6cca */
 325   3.7014625000e+04, /* 0x471096a0 */
 326 };
 327 #ifdef __STDC__
 328 static const float qS8[6] = {
 329 #else
 330 static float qS8[6] = {
 331 #endif
 332   1.6377603149e+02, /* 0x4323c6aa */
 333   8.0983447266e+03, /* 0x45fd12c2 */
 334   1.4253829688e+05, /* 0x480b3293 */
 335   8.0330925000e+05, /* 0x49441ed4 */
 336   8.4050156250e+05, /* 0x494d3359 */
 337  -3.4389928125e+05, /* 0xc8a7eb69 */
 338 };
 339
 340 #ifdef __STDC__
 341 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 342 #else
 343 static float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 344 #endif
 345   1.8408595828e-11, /* 0x2da1ec79 */
 346   7.3242180049e-02, /* 0x3d95ffff */
 347   5.8356351852e+00, /* 0x40babd86 */
 348   1.3511157227e+02, /* 0x43071c90 */
 349   1.0272437744e+03, /* 0x448067cd */
 350   1.9899779053e+03, /* 0x44f8bf4b */
 351 };
 352 #ifdef __STDC__
 353 static const float qS5[6] = {
 354 #else
 355 static float qS5[6] = {
 356 #endif
 357   8.2776611328e+01, /* 0x42a58da0 */
 358   2.0778142090e+03, /* 0x4501dd07 */
 359   1.8847289062e+04, /* 0x46933e94 */
 360   5.6751113281e+04, /* 0x475daf1d */
 361   3.5976753906e+04, /* 0x470c88c1 */
 362  -5.3543427734e+03, /* 0xc5a752be */
 363 };
 364
 365 #ifdef __STDC__
 366 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 367 #else
 368 static float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 369 #endif
 370   4.3774099900e-09, /* 0x3196681b */
 371   7.3241114616e-02, /* 0x3d95ff70 */
 372   3.3442313671e+00, /* 0x405607e3 */
 373   4.2621845245e+01, /* 0x422a7cc5 */
 374   1.7080809021e+02, /* 0x432acedf */
 375   1.6673394775e+02, /* 0x4326bbe4 */
 376 };
 377 #ifdef __STDC__
 378 static const float qS3[6] = {
 379 #else
 380 static float qS3[6] = {
 381 #endif
 382   4.8758872986e+01, /* 0x42430916 */
 383   7.0968920898e+02, /* 0x44316c1c */
 384   3.7041481934e+03, /* 0x4567825f */
 385   6.4604252930e+03, /* 0x45c9e367 */
 386   2.5163337402e+03, /* 0x451d4557 */
 387  -1.4924745178e+02, /* 0xc3153f59 */
 388 };
 389
 390 #ifdef __STDC__
 391 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 392 #else
 393 static float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 394 #endif
 395   1.5044444979e-07, /* 0x342189db */
 396   7.3223426938e-02, /* 0x3d95f62a */
 397   1.9981917143e+00, /* 0x3fffc4bf */
 398   1.4495602608e+01, /* 0x4167edfd */
 399   3.1666231155e+01, /* 0x41fd5471 */
 400   1.6252708435e+01, /* 0x4182058c */
 401 };
 402 #ifdef __STDC__
 403 static const float qS2[6] = {
 404 #else
 405 static float qS2[6] = {
 406 #endif
 407   3.0365585327e+01, /* 0x41f2ecb8 */
 408   2.6934811401e+02, /* 0x4386ac8f */
 409   8.4478375244e+02, /* 0x44533229 */
 410   8.8293585205e+02, /* 0x445cbbe5 */
 411   2.1266638184e+02, /* 0x4354aa98 */
 412  -5.3109550476e+00, /* 0xc0a9f358 */
 413 };
 414
 415 #ifdef __STDC__
 416         static float qzerof(float x)
 417 #else
 418         static float qzerof(x)
 419         float x;
 420 #endif
 421 {
 422 #ifdef __STDC__
 423         const float *p,*q;
 424 #else
 425         float *p,*q;
 426 #endif
 427         float s,r,z;
 428         __int32_t ix;
 429         GET_FLOAT_WORD(ix,x);
 430         ix &= 0x7fffffff;
 431         if(ix>=0x41000000)     {p = qR8; q= qS8;}
 432         else if(ix>=0x40f71c58){p = qR5; q= qS5;}
 433         else if(ix>=0x4036db68){p = qR3; q= qS3;}
 434       else {p = qR2; q= qS2;}
 435         z = one/(x*x);
 436         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 437         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 438         return (-(float).125 + r/s)/x;
 439 }