sysdeps/libm-ieee754/e_j0f.c

   1 /* e_j0f.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 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: e_j0f.c,v 1.4 1995/05/10 20:45:25 jtc Exp $";
  18 #endif
  19
  20 #include "math.h"
  21 #include "math_private.h"
  22
  23 #ifdef __STDC__
  24 static float pzerof(float), qzerof(float);
  25 #else
  26 static float pzerof(), qzerof();
  27 #endif
  28
  29 #ifdef __STDC__
  30 static const float
  31 #else
  32 static float
  33 #endif
  34 huge    = 1e30,
  35 one     = 1.0,
  36 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  37 tpi      =  6.3661974669e-01, /* 0x3f22f983 */
  38                 /* R0/S0 on [0, 2.00] */
  39 R02  =  1.5625000000e-02, /* 0x3c800000 */
  40 R03  = -1.8997929874e-04, /* 0xb947352e */
  41 R04  =  1.8295404516e-06, /* 0x35f58e88 */
  42 R05  = -4.6183270541e-09, /* 0xb19eaf3c */
  43 S01  =  1.5619102865e-02, /* 0x3c7fe744 */
  44 S02  =  1.1692678527e-04, /* 0x38f53697 */
  45 S03  =  5.1354652442e-07, /* 0x3509daa6 */
  46 S04  =  1.1661400734e-09; /* 0x30a045e8 */
  47
  48 #ifdef __STDC__
  49 static const float zero = 0.0;
  50 #else
  51 static float zero = 0.0;
  52 #endif
  53
  54 #ifdef __STDC__
  55         float __ieee754_j0f(float x)
  56 #else
  57         float __ieee754_j0f(x)
  58         float x;
  59 #endif
  60 {
  61         float z, s,c,ss,cc,r,u,v;
  62         int32_t hx,ix;
  63
  64         GET_FLOAT_WORD(hx,x);
  65         ix = hx&0x7fffffff;
  66         if(ix>=0x7f800000) return one/(x*x);
  67         x = fabsf(x);
  68         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  69                 s = __sinf(x);
  70                 c = __cosf(x);
  71                 ss = s-c;
  72                 cc = s+c;
  73                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
  74                     z = -__cosf(x+x);
  75                     if ((s*c)<zero) cc = z/ss;
  76                     else            ss = z/cc;
  77                 }
  78         /*
  79          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
  80          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
  81          */
  82                 if(ix>0x48000000) z = (invsqrtpi*cc)/__sqrtf(x);
  83                 else {
  84                     u = pzerof(x); v = qzerof(x);
  85                     z = invsqrtpi*(u*cc-v*ss)/__sqrtf(x);
  86                 }
  87                 return z;
  88         }
  89         if(ix<0x39000000) {     /* |x| < 2**-13 */
  90             if(huge+x>one) {    /* raise inexact if x != 0 */
  91                 if(ix<0x32000000) return one;   /* |x|<2**-27 */
  92                 else          return one - (float)0.25*x*x;
  93             }
  94         }
  95         z = x*x;
  96         r =  z*(R02+z*(R03+z*(R04+z*R05)));
  97         s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
  98         if(ix < 0x3F800000) {   /* |x| < 1.00 */
  99             return one + z*((float)-0.25+(r/s));
 100         } else {
 101             u = (float)0.5*x;
 102             return((one+u)*(one-u)+z*(r/s));
 103         }
 104 }
 105
 106 #ifdef __STDC__
 107 static const float
 108 #else
 109 static float
 110 #endif
 111 u00  = -7.3804296553e-02, /* 0xbd9726b5 */
 112 u01  =  1.7666645348e-01, /* 0x3e34e80d */
 113 u02  = -1.3818567619e-02, /* 0xbc626746 */
 114 u03  =  3.4745343146e-04, /* 0x39b62a69 */
 115 u04  = -3.8140706238e-06, /* 0xb67ff53c */
 116 u05  =  1.9559013964e-08, /* 0x32a802ba */
 117 u06  = -3.9820518410e-11, /* 0xae2f21eb */
 118 v01  =  1.2730483897e-02, /* 0x3c509385 */
 119 v02  =  7.6006865129e-05, /* 0x389f65e0 */
 120 v03  =  2.5915085189e-07, /* 0x348b216c */
 121 v04  =  4.4111031494e-10; /* 0x2ff280c2 */
 122
 123 #ifdef __STDC__
 124         float __ieee754_y0f(float x)
 125 #else
 126         float __ieee754_y0f(x)
 127         float x;
 128 #endif
 129 {
 130         float z, s,c,ss,cc,u,v;
 131         int32_t hx,ix;
 132
 133         GET_FLOAT_WORD(hx,x);
 134         ix = 0x7fffffff&hx;
 135     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
 136         if(ix>=0x7f800000) return  one/(x+x*x);
 137         if(ix==0) return -one/zero;
 138         if(hx<0) return zero/zero;
 139         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 140         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
 141          * where x0 = x-pi/4
 142          *      Better formula:
 143          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 144          *                      =  1/sqrt(2) * (sin(x) + cos(x))
 145          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 146          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 147          * To avoid cancellation, use
 148          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 149          * to compute the worse one.
 150          */
 151                 s = __sinf(x);
 152                 c = __cosf(x);
 153                 ss = s-c;
 154                 cc = s+c;
 155         /*
 156          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 157          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 158          */
 159                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
 160                     z = -__cosf(x+x);
 161                     if ((s*c)<zero) cc = z/ss;
 162                     else            ss = z/cc;
 163                 }
 164                 if(ix>0x48000000) z = (invsqrtpi*ss)/__sqrtf(x);
 165                 else {
 166                     u = pzerof(x); v = qzerof(x);
 167                     z = invsqrtpi*(u*ss+v*cc)/__sqrtf(x);
 168                 }
 169                 return z;
 170         }
 171         if(ix<=0x32000000) {    /* x < 2**-27 */
 172             return(u00 + tpi*__ieee754_logf(x));
 173         }
 174         z = x*x;
 175         u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
 176         v = one+z*(v01+z*(v02+z*(v03+z*v04)));
 177         return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
 178 }
 179
 180 /* The asymptotic expansions of pzero is
 181  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
 182  * For x >= 2, We approximate pzero by
 183  *      pzero(x) = 1 + (R/S)
 184  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
 185  *        S = 1 + pS0*s^2 + ... + pS4*s^10
 186  * and
 187  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
 188  */
 189 #ifdef __STDC__
 190 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 191 #else
 192 static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 193 #endif
 194   0.0000000000e+00, /* 0x00000000 */
 195  -7.0312500000e-02, /* 0xbd900000 */
 196  -8.0816707611e+00, /* 0xc1014e86 */
 197  -2.5706311035e+02, /* 0xc3808814 */
 198  -2.4852163086e+03, /* 0xc51b5376 */
 199  -5.2530439453e+03, /* 0xc5a4285a */
 200 };
 201 #ifdef __STDC__
 202 static const float pS8[5] = {
 203 #else
 204 static float pS8[5] = {
 205 #endif
 206   1.1653436279e+02, /* 0x42e91198 */
 207   3.8337448730e+03, /* 0x456f9beb */
 208   4.0597855469e+04, /* 0x471e95db */
 209   1.1675296875e+05, /* 0x47e4087c */
 210   4.7627726562e+04, /* 0x473a0bba */
 211 };
 212 #ifdef __STDC__
 213 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 214 #else
 215 static float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 216 #endif
 217  -1.1412546255e-11, /* 0xad48c58a */
 218  -7.0312492549e-02, /* 0xbd8fffff */
 219  -4.1596107483e+00, /* 0xc0851b88 */
 220  -6.7674766541e+01, /* 0xc287597b */
 221  -3.3123129272e+02, /* 0xc3a59d9b */
 222  -3.4643338013e+02, /* 0xc3ad3779 */
 223 };
 224 #ifdef __STDC__
 225 static const float pS5[5] = {
 226 #else
 227 static float pS5[5] = {
 228 #endif
 229   6.0753936768e+01, /* 0x42730408 */
 230   1.0512523193e+03, /* 0x44836813 */
 231   5.9789707031e+03, /* 0x45bad7c4 */
 232   9.6254453125e+03, /* 0x461665c8 */
 233   2.4060581055e+03, /* 0x451660ee */
 234 };
 235
 236 #ifdef __STDC__
 237 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 238 #else
 239 static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 240 #endif
 241  -2.5470459075e-09, /* 0xb12f081b */
 242  -7.0311963558e-02, /* 0xbd8fffb8 */
 243  -2.4090321064e+00, /* 0xc01a2d95 */
 244  -2.1965976715e+01, /* 0xc1afba52 */
 245  -5.8079170227e+01, /* 0xc2685112 */
 246  -3.1447946548e+01, /* 0xc1fb9565 */
 247 };
 248 #ifdef __STDC__
 249 static const float pS3[5] = {
 250 #else
 251 static float pS3[5] = {
 252 #endif
 253   3.5856033325e+01, /* 0x420f6c94 */
 254   3.6151397705e+02, /* 0x43b4c1ca */
 255   1.1936077881e+03, /* 0x44953373 */
 256   1.1279968262e+03, /* 0x448cffe6 */
 257   1.7358093262e+02, /* 0x432d94b8 */
 258 };
 259
 260 #ifdef __STDC__
 261 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 262 #else
 263 static float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 264 #endif
 265  -8.8753431271e-08, /* 0xb3be98b7 */
 266  -7.0303097367e-02, /* 0xbd8ffb12 */
 267  -1.4507384300e+00, /* 0xbfb9b1cc */
 268  -7.6356959343e+00, /* 0xc0f4579f */
 269  -1.1193166733e+01, /* 0xc1331736 */
 270  -3.2336456776e+00, /* 0xc04ef40d */
 271 };
 272 #ifdef __STDC__
 273 static const float pS2[5] = {
 274 #else
 275 static float pS2[5] = {
 276 #endif
 277   2.2220300674e+01, /* 0x41b1c32d */
 278   1.3620678711e+02, /* 0x430834f0 */
 279   2.7047027588e+02, /* 0x43873c32 */
 280   1.5387539673e+02, /* 0x4319e01a */
 281   1.4657617569e+01, /* 0x416a859a */
 282 };
 283
 284 #ifdef __STDC__
 285         static float pzerof(float x)
 286 #else
 287         static float pzerof(x)
 288         float x;
 289 #endif
 290 {
 291 #ifdef __STDC__
 292         const float *p,*q;
 293 #else
 294         float *p,*q;
 295 #endif
 296         float z,r,s;
 297         int32_t ix;
 298         GET_FLOAT_WORD(ix,x);
 299         ix &= 0x7fffffff;
 300         if(ix>=0x41000000)     {p = pR8; q= pS8;}
 301         else if(ix>=0x40f71c58){p = pR5; q= pS5;}
 302         else if(ix>=0x4036db68){p = pR3; q= pS3;}
 303         else if(ix>=0x40000000){p = pR2; q= pS2;}
 304         z = one/(x*x);
 305         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 306         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 307         return one+ r/s;
 308 }
 309
 310
 311 /* For x >= 8, the asymptotic expansions of qzero is
 312  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 313  * We approximate pzero by
 314  *      qzero(x) = s*(-1.25 + (R/S))
 315  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
 316  *        S = 1 + qS0*s^2 + ... + qS5*s^12
 317  * and
 318  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 319  */
 320 #ifdef __STDC__
 321 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 322 #else
 323 static float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 324 #endif
 325   0.0000000000e+00, /* 0x00000000 */
 326   7.3242187500e-02, /* 0x3d960000 */
 327   1.1768206596e+01, /* 0x413c4a93 */
 328   5.5767340088e+02, /* 0x440b6b19 */
 329   8.8591972656e+03, /* 0x460a6cca */
 330   3.7014625000e+04, /* 0x471096a0 */
 331 };
 332 #ifdef __STDC__
 333 static const float qS8[6] = {
 334 #else
 335 static float qS8[6] = {
 336 #endif
 337   1.6377603149e+02, /* 0x4323c6aa */
 338   8.0983447266e+03, /* 0x45fd12c2 */
 339   1.4253829688e+05, /* 0x480b3293 */
 340   8.0330925000e+05, /* 0x49441ed4 */
 341   8.4050156250e+05, /* 0x494d3359 */
 342  -3.4389928125e+05, /* 0xc8a7eb69 */
 343 };
 344
 345 #ifdef __STDC__
 346 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 347 #else
 348 static float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 349 #endif
 350   1.8408595828e-11, /* 0x2da1ec79 */
 351   7.3242180049e-02, /* 0x3d95ffff */
 352   5.8356351852e+00, /* 0x40babd86 */
 353   1.3511157227e+02, /* 0x43071c90 */
 354   1.0272437744e+03, /* 0x448067cd */
 355   1.9899779053e+03, /* 0x44f8bf4b */
 356 };
 357 #ifdef __STDC__
 358 static const float qS5[6] = {
 359 #else
 360 static float qS5[6] = {
 361 #endif
 362   8.2776611328e+01, /* 0x42a58da0 */
 363   2.0778142090e+03, /* 0x4501dd07 */
 364   1.8847289062e+04, /* 0x46933e94 */
 365   5.6751113281e+04, /* 0x475daf1d */
 366   3.5976753906e+04, /* 0x470c88c1 */
 367  -5.3543427734e+03, /* 0xc5a752be */
 368 };
 369
 370 #ifdef __STDC__
 371 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 372 #else
 373 static float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 374 #endif
 375   4.3774099900e-09, /* 0x3196681b */
 376   7.3241114616e-02, /* 0x3d95ff70 */
 377   3.3442313671e+00, /* 0x405607e3 */
 378   4.2621845245e+01, /* 0x422a7cc5 */
 379   1.7080809021e+02, /* 0x432acedf */
 380   1.6673394775e+02, /* 0x4326bbe4 */
 381 };
 382 #ifdef __STDC__
 383 static const float qS3[6] = {
 384 #else
 385 static float qS3[6] = {
 386 #endif
 387   4.8758872986e+01, /* 0x42430916 */
 388   7.0968920898e+02, /* 0x44316c1c */
 389   3.7041481934e+03, /* 0x4567825f */
 390   6.4604252930e+03, /* 0x45c9e367 */
 391   2.5163337402e+03, /* 0x451d4557 */
 392  -1.4924745178e+02, /* 0xc3153f59 */
 393 };
 394
 395 #ifdef __STDC__
 396 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 397 #else
 398 static float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 399 #endif
 400   1.5044444979e-07, /* 0x342189db */
 401   7.3223426938e-02, /* 0x3d95f62a */
 402   1.9981917143e+00, /* 0x3fffc4bf */
 403   1.4495602608e+01, /* 0x4167edfd */
 404   3.1666231155e+01, /* 0x41fd5471 */
 405   1.6252708435e+01, /* 0x4182058c */
 406 };
 407 #ifdef __STDC__
 408 static const float qS2[6] = {
 409 #else
 410 static float qS2[6] = {
 411 #endif
 412   3.0365585327e+01, /* 0x41f2ecb8 */
 413   2.6934811401e+02, /* 0x4386ac8f */
 414   8.4478375244e+02, /* 0x44533229 */
 415   8.8293585205e+02, /* 0x445cbbe5 */
 416   2.1266638184e+02, /* 0x4354aa98 */
 417  -5.3109550476e+00, /* 0xc0a9f358 */
 418 };
 419
 420 #ifdef __STDC__
 421         static float qzerof(float x)
 422 #else
 423         static float qzerof(x)
 424         float x;
 425 #endif
 426 {
 427 #ifdef __STDC__
 428         const float *p,*q;
 429 #else
 430         float *p,*q;
 431 #endif
 432         float s,r,z;
 433         int32_t ix;
 434         GET_FLOAT_WORD(ix,x);
 435         ix &= 0x7fffffff;
 436         if(ix>=0x41000000)     {p = qR8; q= qS8;}
 437         else if(ix>=0x40f71c58){p = qR5; q= qS5;}
 438         else if(ix>=0x4036db68){p = qR3; q= qS3;}
 439         else if(ix>=0x40000000){p = qR2; q= qS2;}
 440         z = one/(x*x);
 441         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 442         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 443         return (-(float).125 + r/s)/x;
 444 }