toolchain/gcc/newlib/libm/mathfp/ef_j1.c

   1 /* ef_j1.c -- float version of e_j1.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 ponef(float), qonef(float);
  20 #else
  21 static float ponef(), qonef();
  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] */
  34 r00  = -6.2500000000e-02, /* 0xbd800000 */
  35 r01  =  1.4070566976e-03, /* 0x3ab86cfd */
  36 r02  = -1.5995563444e-05, /* 0xb7862e36 */
  37 r03  =  4.9672799207e-08, /* 0x335557d2 */
  38 s01  =  1.9153760746e-02, /* 0x3c9ce859 */
  39 s02  =  1.8594678841e-04, /* 0x3942fab6 */
  40 s03  =  1.1771846857e-06, /* 0x359dffc2 */
  41 s04  =  5.0463624390e-09, /* 0x31ad6446 */
  42 s05  =  1.2354227016e-11; /* 0x2d59567e */
  43
  44 #ifdef __STDC__
  45 static const float zero    = 0.0;
  46 #else
  47 static float zero    = 0.0;
  48 #endif
  49
  50 #ifdef __STDC__
  51         float j1f(float x)
  52 #else
  53         float j1f(x)
  54         float x;
  55 #endif
  56 {
  57         float z, s,c,ss,cc,r,u,v,y;
  58         __int32_t hx,ix;
  59
  60         GET_FLOAT_WORD(hx,x);
  61         ix = hx&0x7fffffff;
  62         if(ix>=0x7f800000) return one/x;
  63         y = fabsf(x);
  64         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  65                 s = sinf(y);
  66                 c = cosf(y);
  67                 ss = -s-c;
  68                 cc = s-c;
  69                 if(ix<0x7f000000) {  /* make sure y+y not overflow */
  70                     z = cosf(y+y);
  71                     if ((s*c)>zero) cc = z/ss;
  72                     else            ss = z/cc;
  73                 }
  74         /*
  75          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
  76          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
  77          */
  78                 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y);
  79                 else {
  80                     u = ponef(y); v = qonef(y);
  81                     z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
  82                 }
  83                 if(hx<0) return -z;
  84                 else     return  z;
  85         }
  86         if(ix<0x32000000) {     /* |x|<2**-27 */
  87             if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
  88         }
  89         z = x*x;
  90         r =  z*(r00+z*(r01+z*(r02+z*r03)));
  91         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
  92         r *= x;
  93         return(x*(float)0.5+r/s);
  94 }
  95
  96 #ifdef __STDC__
  97 static const float U0[5] = {
  98 #else
  99 static float U0[5] = {
 100 #endif
 101  -1.9605709612e-01, /* 0xbe48c331 */
 102   5.0443872809e-02, /* 0x3d4e9e3c */
 103  -1.9125689287e-03, /* 0xbafaaf2a */
 104   2.3525259166e-05, /* 0x37c5581c */
 105  -9.1909917899e-08, /* 0xb3c56003 */
 106 };
 107 #ifdef __STDC__
 108 static const float V0[5] = {
 109 #else
 110 static float V0[5] = {
 111 #endif
 112   1.9916731864e-02, /* 0x3ca3286a */
 113   2.0255257550e-04, /* 0x3954644b */
 114   1.3560879779e-06, /* 0x35b602d4 */
 115   6.2274145840e-09, /* 0x31d5f8eb */
 116   1.6655924903e-11, /* 0x2d9281cf */
 117 };
 118
 119 #ifdef __STDC__
 120         float y1f(float x)
 121 #else
 122         float y1f(x)
 123         float x;
 124 #endif
 125 {
 126         float z, s,c,ss,cc,u,v;
 127         __int32_t hx,ix;
 128
 129         GET_FLOAT_WORD(hx,x);
 130         ix = 0x7fffffff&hx;
 131     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
 132         if(ix>=0x7f800000) return  one/(x+x*x);
 133         if(ix==0) return -one/zero;
 134         if(hx<0) return zero/zero;
 135         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 136                 s = sinf(x);
 137                 c = cosf(x);
 138                 ss = -s-c;
 139                 cc = s-c;
 140                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
 141                     z = cosf(x+x);
 142                     if ((s*c)>zero) cc = z/ss;
 143                     else            ss = z/cc;
 144                 }
 145         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
 146          * where x0 = x-3pi/4
 147          *      Better formula:
 148          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
 149          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 150          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 151          *                      = -1/sqrt(2) * (cos(x) + sin(x))
 152          * To avoid cancellation, use
 153          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 154          * to compute the worse one.
 155          */
 156                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
 157                 else {
 158                     u = ponef(x); v = qonef(x);
 159                     z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
 160                 }
 161                 return z;
 162         }
 163         if(ix<=0x24800000) {    /* x < 2**-54 */
 164             return(-tpi/x);
 165         }
 166         z = x*x;
 167         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
 168         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
 169         return(x*(u/v) + tpi*(j1f(x)*logf(x)-one/x));
 170 }
 171
 172 /* For x >= 8, the asymptotic expansions of pone is
 173  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 174  * We approximate pone by
 175  *      pone(x) = 1 + (R/S)
 176  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 177  *        S = 1 + ps0*s^2 + ... + ps4*s^10
 178  * and
 179  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 180  */
 181
 182 #ifdef __STDC__
 183 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 184 #else
 185 static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 186 #endif
 187   0.0000000000e+00, /* 0x00000000 */
 188   1.1718750000e-01, /* 0x3df00000 */
 189   1.3239480972e+01, /* 0x4153d4ea */
 190   4.1205184937e+02, /* 0x43ce06a3 */
 191   3.8747453613e+03, /* 0x45722bed */
 192   7.9144794922e+03, /* 0x45f753d6 */
 193 };
 194 #ifdef __STDC__
 195 static const float ps8[5] = {
 196 #else
 197 static float ps8[5] = {
 198 #endif
 199   1.1420736694e+02, /* 0x42e46a2c */
 200   3.6509309082e+03, /* 0x45642ee5 */
 201   3.6956207031e+04, /* 0x47105c35 */
 202   9.7602796875e+04, /* 0x47bea166 */
 203   3.0804271484e+04, /* 0x46f0a88b */
 204 };
 205
 206 #ifdef __STDC__
 207 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 208 #else
 209 static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 210 #endif
 211   1.3199052094e-11, /* 0x2d68333f */
 212   1.1718749255e-01, /* 0x3defffff */
 213   6.8027510643e+00, /* 0x40d9b023 */
 214   1.0830818176e+02, /* 0x42d89dca */
 215   5.1763616943e+02, /* 0x440168b7 */
 216   5.2871520996e+02, /* 0x44042dc6 */
 217 };
 218 #ifdef __STDC__
 219 static const float ps5[5] = {
 220 #else
 221 static float ps5[5] = {
 222 #endif
 223   5.9280597687e+01, /* 0x426d1f55 */
 224   9.9140142822e+02, /* 0x4477d9b1 */
 225   5.3532670898e+03, /* 0x45a74a23 */
 226   7.8446904297e+03, /* 0x45f52586 */
 227   1.5040468750e+03, /* 0x44bc0180 */
 228 };
 229
 230 #ifdef __STDC__
 231 static const float pr3[6] = {
 232 #else
 233 static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 234 #endif
 235   3.0250391081e-09, /* 0x314fe10d */
 236   1.1718686670e-01, /* 0x3defffab */
 237   3.9329774380e+00, /* 0x407bb5e7 */
 238   3.5119403839e+01, /* 0x420c7a45 */
 239   9.1055007935e+01, /* 0x42b61c2a */
 240   4.8559066772e+01, /* 0x42423c7c */
 241 };
 242 #ifdef __STDC__
 243 static const float ps3[5] = {
 244 #else
 245 static float ps3[5] = {
 246 #endif
 247   3.4791309357e+01, /* 0x420b2a4d */
 248   3.3676245117e+02, /* 0x43a86198 */
 249   1.0468714600e+03, /* 0x4482dbe3 */
 250   8.9081134033e+02, /* 0x445eb3ed */
 251   1.0378793335e+02, /* 0x42cf936c */
 252 };
 253
 254 #ifdef __STDC__
 255 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 256 #else
 257 static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 258 #endif
 259   1.0771083225e-07, /* 0x33e74ea8 */
 260   1.1717621982e-01, /* 0x3deffa16 */
 261   2.3685150146e+00, /* 0x401795c0 */
 262   1.2242610931e+01, /* 0x4143e1bc */
 263   1.7693971634e+01, /* 0x418d8d41 */
 264   5.0735230446e+00, /* 0x40a25a4d */
 265 };
 266 #ifdef __STDC__
 267 static const float ps2[5] = {
 268 #else
 269 static float ps2[5] = {
 270 #endif
 271   2.1436485291e+01, /* 0x41ab7dec */
 272   1.2529022980e+02, /* 0x42fa9499 */
 273   2.3227647400e+02, /* 0x436846c7 */
 274   1.1767937469e+02, /* 0x42eb5bd7 */
 275   8.3646392822e+00, /* 0x4105d590 */
 276 };
 277
 278 #ifdef __STDC__
 279         static float ponef(float x)
 280 #else
 281         static float ponef(x)
 282         float x;
 283 #endif
 284 {
 285 #ifdef __STDC__
 286         const float *p,*q;
 287 #else
 288         float *p,*q;
 289 #endif
 290         float z,r,s;
 291         __int32_t ix;
 292         GET_FLOAT_WORD(ix,x);
 293         ix &= 0x7fffffff;
 294         if(ix>=0x41000000)     {p = pr8; q= ps8;}
 295         else if(ix>=0x40f71c58){p = pr5; q= ps5;}
 296         else if(ix>=0x4036db68){p = pr3; q= ps3;}
 297         else {p = pr2; q= ps2;}
 298         z = one/(x*x);
 299         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 300         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 301         return one+ r/s;
 302 }
 303
 304
 305 /* For x >= 8, the asymptotic expansions of qone is
 306  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 307  * We approximate qone by
 308  *      qone(x) = s*(0.375 + (R/S))
 309  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 310  *        S = 1 + qs1*s^2 + ... + qs6*s^12
 311  * and
 312  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 313  */
 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  -1.0253906250e-01, /* 0xbdd20000 */
 322  -1.6271753311e+01, /* 0xc1822c8d */
 323  -7.5960174561e+02, /* 0xc43de683 */
 324  -1.1849806641e+04, /* 0xc639273a */
 325  -4.8438511719e+04, /* 0xc73d3683 */
 326 };
 327 #ifdef __STDC__
 328 static const float qs8[6] = {
 329 #else
 330 static float qs8[6] = {
 331 #endif
 332   1.6139537048e+02, /* 0x43216537 */
 333   7.8253862305e+03, /* 0x45f48b17 */
 334   1.3387534375e+05, /* 0x4802bcd6 */
 335   7.1965775000e+05, /* 0x492fb29c */
 336   6.6660125000e+05, /* 0x4922be94 */
 337  -2.9449025000e+05, /* 0xc88fcb48 */
 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  -2.0897993405e-11, /* 0xadb7d219 */
 346  -1.0253904760e-01, /* 0xbdd1fffe */
 347  -8.0564479828e+00, /* 0xc100e736 */
 348  -1.8366960144e+02, /* 0xc337ab6b */
 349  -1.3731937256e+03, /* 0xc4aba633 */
 350  -2.6124443359e+03, /* 0xc523471c */
 351 };
 352 #ifdef __STDC__
 353 static const float qs5[6] = {
 354 #else
 355 static float qs5[6] = {
 356 #endif
 357   8.1276550293e+01, /* 0x42a28d98 */
 358   1.9917987061e+03, /* 0x44f8f98f */
 359   1.7468484375e+04, /* 0x468878f8 */
 360   4.9851425781e+04, /* 0x4742bb6d */
 361   2.7948074219e+04, /* 0x46da5826 */
 362  -4.7191835938e+03, /* 0xc5937978 */
 363 };
 364
 365 #ifdef __STDC__
 366 static const float qr3[6] = {
 367 #else
 368 static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 369 #endif
 370  -5.0783124372e-09, /* 0xb1ae7d4f */
 371  -1.0253783315e-01, /* 0xbdd1ff5b */
 372  -4.6101160049e+00, /* 0xc0938612 */
 373  -5.7847221375e+01, /* 0xc267638e */
 374  -2.2824453735e+02, /* 0xc3643e9a */
 375  -2.1921012878e+02, /* 0xc35b35cb */
 376 };
 377 #ifdef __STDC__
 378 static const float qs3[6] = {
 379 #else
 380 static float qs3[6] = {
 381 #endif
 382   4.7665153503e+01, /* 0x423ea91e */
 383   6.7386511230e+02, /* 0x4428775e */
 384   3.3801528320e+03, /* 0x45534272 */
 385   5.5477290039e+03, /* 0x45ad5dd5 */
 386   1.9031191406e+03, /* 0x44ede3d0 */
 387  -1.3520118713e+02, /* 0xc3073381 */
 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  -1.7838172539e-07, /* 0xb43f8932 */
 396  -1.0251704603e-01, /* 0xbdd1f475 */
 397  -2.7522056103e+00, /* 0xc0302423 */
 398  -1.9663616180e+01, /* 0xc19d4f16 */
 399  -4.2325313568e+01, /* 0xc2294d1f */
 400  -2.1371921539e+01, /* 0xc1aaf9b2 */
 401 };
 402 #ifdef __STDC__
 403 static const float qs2[6] = {
 404 #else
 405 static float qs2[6] = {
 406 #endif
 407   2.9533363342e+01, /* 0x41ec4454 */
 408   2.5298155212e+02, /* 0x437cfb47 */
 409   7.5750280762e+02, /* 0x443d602e */
 410   7.3939318848e+02, /* 0x4438d92a */
 411   1.5594900513e+02, /* 0x431bf2f2 */
 412  -4.9594988823e+00, /* 0xc09eb437 */
 413 };
 414
 415 #ifdef __STDC__
 416         static float qonef(float x)
 417 #else
 418         static float qonef(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>=0x40200000)     {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).375 + r/s)/x;
 439 }