sysdeps/libm-ieee754/e_j1f.c

   1 /* e_j1f.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 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: e_j1f.c,v 1.4 1995/05/10 20:45:31 jtc Exp $";
  18 #endif
  19
  20 #include "math.h"
  21 #include "math_private.h"
  22
  23 #ifdef __STDC__
  24 static float ponef(float), qonef(float);
  25 #else
  26 static float ponef(), qonef();
  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] */
  39 r00  = -6.2500000000e-02, /* 0xbd800000 */
  40 r01  =  1.4070566976e-03, /* 0x3ab86cfd */
  41 r02  = -1.5995563444e-05, /* 0xb7862e36 */
  42 r03  =  4.9672799207e-08, /* 0x335557d2 */
  43 s01  =  1.9153760746e-02, /* 0x3c9ce859 */
  44 s02  =  1.8594678841e-04, /* 0x3942fab6 */
  45 s03  =  1.1771846857e-06, /* 0x359dffc2 */
  46 s04  =  5.0463624390e-09, /* 0x31ad6446 */
  47 s05  =  1.2354227016e-11; /* 0x2d59567e */
  48
  49 #ifdef __STDC__
  50 static const float zero    = 0.0;
  51 #else
  52 static float zero    = 0.0;
  53 #endif
  54
  55 #ifdef __STDC__
  56         float __ieee754_j1f(float x)
  57 #else
  58         float __ieee754_j1f(x)
  59         float x;
  60 #endif
  61 {
  62         float z, s,c,ss,cc,r,u,v,y;
  63         int32_t hx,ix;
  64
  65         GET_FLOAT_WORD(hx,x);
  66         ix = hx&0x7fffffff;
  67         if(ix>=0x7f800000) return one/x;
  68         y = fabsf(x);
  69         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  70                 s = __sinf(y);
  71                 c = __cosf(y);
  72                 ss = -s-c;
  73                 cc = s-c;
  74                 if(ix<0x7f000000) {  /* make sure y+y not overflow */
  75                     z = __cosf(y+y);
  76                     if ((s*c)>zero) cc = z/ss;
  77                     else            ss = z/cc;
  78                 }
  79         /*
  80          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
  81          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
  82          */
  83                 if(ix>0x48000000) z = (invsqrtpi*cc)/__sqrtf(y);
  84                 else {
  85                     u = ponef(y); v = qonef(y);
  86                     z = invsqrtpi*(u*cc-v*ss)/__sqrtf(y);
  87                 }
  88                 if(hx<0) return -z;
  89                 else     return  z;
  90         }
  91         if(ix<0x32000000) {     /* |x|<2**-27 */
  92             if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
  93         }
  94         z = x*x;
  95         r =  z*(r00+z*(r01+z*(r02+z*r03)));
  96         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
  97         r *= x;
  98         return(x*(float)0.5+r/s);
  99 }
 100
 101 #ifdef __STDC__
 102 static const float U0[5] = {
 103 #else
 104 static float U0[5] = {
 105 #endif
 106  -1.9605709612e-01, /* 0xbe48c331 */
 107   5.0443872809e-02, /* 0x3d4e9e3c */
 108  -1.9125689287e-03, /* 0xbafaaf2a */
 109   2.3525259166e-05, /* 0x37c5581c */
 110  -9.1909917899e-08, /* 0xb3c56003 */
 111 };
 112 #ifdef __STDC__
 113 static const float V0[5] = {
 114 #else
 115 static float V0[5] = {
 116 #endif
 117   1.9916731864e-02, /* 0x3ca3286a */
 118   2.0255257550e-04, /* 0x3954644b */
 119   1.3560879779e-06, /* 0x35b602d4 */
 120   6.2274145840e-09, /* 0x31d5f8eb */
 121   1.6655924903e-11, /* 0x2d9281cf */
 122 };
 123
 124 #ifdef __STDC__
 125         float __ieee754_y1f(float x)
 126 #else
 127         float __ieee754_y1f(x)
 128         float x;
 129 #endif
 130 {
 131         float z, s,c,ss,cc,u,v;
 132         int32_t hx,ix;
 133
 134         GET_FLOAT_WORD(hx,x);
 135         ix = 0x7fffffff&hx;
 136     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
 137         if(ix>=0x7f800000) return  one/(x+x*x);
 138         if(ix==0) return -one/zero;
 139         if(hx<0) return zero/zero;
 140         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 141                 s = __sinf(x);
 142                 c = __cosf(x);
 143                 ss = -s-c;
 144                 cc = s-c;
 145                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
 146                     z = __cosf(x+x);
 147                     if ((s*c)>zero) cc = z/ss;
 148                     else            ss = z/cc;
 149                 }
 150         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
 151          * where x0 = x-3pi/4
 152          *      Better formula:
 153          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
 154          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 155          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 156          *                      = -1/sqrt(2) * (cos(x) + sin(x))
 157          * To avoid cancellation, use
 158          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 159          * to compute the worse one.
 160          */
 161                 if(ix>0x48000000) z = (invsqrtpi*ss)/__sqrtf(x);
 162                 else {
 163                     u = ponef(x); v = qonef(x);
 164                     z = invsqrtpi*(u*ss+v*cc)/__sqrtf(x);
 165                 }
 166                 return z;
 167         }
 168         if(ix<=0x24800000) {    /* x < 2**-54 */
 169             return(-tpi/x);
 170         }
 171         z = x*x;
 172         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
 173         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
 174         return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
 175 }
 176
 177 /* For x >= 8, the asymptotic expansions of pone is
 178  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 179  * We approximate pone by
 180  *      pone(x) = 1 + (R/S)
 181  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 182  *        S = 1 + ps0*s^2 + ... + ps4*s^10
 183  * and
 184  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 185  */
 186
 187 #ifdef __STDC__
 188 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 189 #else
 190 static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 191 #endif
 192   0.0000000000e+00, /* 0x00000000 */
 193   1.1718750000e-01, /* 0x3df00000 */
 194   1.3239480972e+01, /* 0x4153d4ea */
 195   4.1205184937e+02, /* 0x43ce06a3 */
 196   3.8747453613e+03, /* 0x45722bed */
 197   7.9144794922e+03, /* 0x45f753d6 */
 198 };
 199 #ifdef __STDC__
 200 static const float ps8[5] = {
 201 #else
 202 static float ps8[5] = {
 203 #endif
 204   1.1420736694e+02, /* 0x42e46a2c */
 205   3.6509309082e+03, /* 0x45642ee5 */
 206   3.6956207031e+04, /* 0x47105c35 */
 207   9.7602796875e+04, /* 0x47bea166 */
 208   3.0804271484e+04, /* 0x46f0a88b */
 209 };
 210
 211 #ifdef __STDC__
 212 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 213 #else
 214 static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 215 #endif
 216   1.3199052094e-11, /* 0x2d68333f */
 217   1.1718749255e-01, /* 0x3defffff */
 218   6.8027510643e+00, /* 0x40d9b023 */
 219   1.0830818176e+02, /* 0x42d89dca */
 220   5.1763616943e+02, /* 0x440168b7 */
 221   5.2871520996e+02, /* 0x44042dc6 */
 222 };
 223 #ifdef __STDC__
 224 static const float ps5[5] = {
 225 #else
 226 static float ps5[5] = {
 227 #endif
 228   5.9280597687e+01, /* 0x426d1f55 */
 229   9.9140142822e+02, /* 0x4477d9b1 */
 230   5.3532670898e+03, /* 0x45a74a23 */
 231   7.8446904297e+03, /* 0x45f52586 */
 232   1.5040468750e+03, /* 0x44bc0180 */
 233 };
 234
 235 #ifdef __STDC__
 236 static const float pr3[6] = {
 237 #else
 238 static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 239 #endif
 240   3.0250391081e-09, /* 0x314fe10d */
 241   1.1718686670e-01, /* 0x3defffab */
 242   3.9329774380e+00, /* 0x407bb5e7 */
 243   3.5119403839e+01, /* 0x420c7a45 */
 244   9.1055007935e+01, /* 0x42b61c2a */
 245   4.8559066772e+01, /* 0x42423c7c */
 246 };
 247 #ifdef __STDC__
 248 static const float ps3[5] = {
 249 #else
 250 static float ps3[5] = {
 251 #endif
 252   3.4791309357e+01, /* 0x420b2a4d */
 253   3.3676245117e+02, /* 0x43a86198 */
 254   1.0468714600e+03, /* 0x4482dbe3 */
 255   8.9081134033e+02, /* 0x445eb3ed */
 256   1.0378793335e+02, /* 0x42cf936c */
 257 };
 258
 259 #ifdef __STDC__
 260 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 261 #else
 262 static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 263 #endif
 264   1.0771083225e-07, /* 0x33e74ea8 */
 265   1.1717621982e-01, /* 0x3deffa16 */
 266   2.3685150146e+00, /* 0x401795c0 */
 267   1.2242610931e+01, /* 0x4143e1bc */
 268   1.7693971634e+01, /* 0x418d8d41 */
 269   5.0735230446e+00, /* 0x40a25a4d */
 270 };
 271 #ifdef __STDC__
 272 static const float ps2[5] = {
 273 #else
 274 static float ps2[5] = {
 275 #endif
 276   2.1436485291e+01, /* 0x41ab7dec */
 277   1.2529022980e+02, /* 0x42fa9499 */
 278   2.3227647400e+02, /* 0x436846c7 */
 279   1.1767937469e+02, /* 0x42eb5bd7 */
 280   8.3646392822e+00, /* 0x4105d590 */
 281 };
 282
 283 #ifdef __STDC__
 284         static float ponef(float x)
 285 #else
 286         static float ponef(x)
 287         float x;
 288 #endif
 289 {
 290 #ifdef __STDC__
 291         const float *p,*q;
 292 #else
 293         float *p,*q;
 294 #endif
 295         float z,r,s;
 296         int32_t ix;
 297         GET_FLOAT_WORD(ix,x);
 298         ix &= 0x7fffffff;
 299         if(ix>=0x41000000)     {p = pr8; q= ps8;}
 300         else if(ix>=0x40f71c58){p = pr5; q= ps5;}
 301         else if(ix>=0x4036db68){p = pr3; q= ps3;}
 302         else if(ix>=0x40000000){p = pr2; q= ps2;}
 303         z = one/(x*x);
 304         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 305         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 306         return one+ r/s;
 307 }
 308
 309
 310 /* For x >= 8, the asymptotic expansions of qone is
 311  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 312  * We approximate pone by
 313  *      qone(x) = s*(0.375 + (R/S))
 314  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 315  *        S = 1 + qs1*s^2 + ... + qs6*s^12
 316  * and
 317  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 318  */
 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  -1.0253906250e-01, /* 0xbdd20000 */
 327  -1.6271753311e+01, /* 0xc1822c8d */
 328  -7.5960174561e+02, /* 0xc43de683 */
 329  -1.1849806641e+04, /* 0xc639273a */
 330  -4.8438511719e+04, /* 0xc73d3683 */
 331 };
 332 #ifdef __STDC__
 333 static const float qs8[6] = {
 334 #else
 335 static float qs8[6] = {
 336 #endif
 337   1.6139537048e+02, /* 0x43216537 */
 338   7.8253862305e+03, /* 0x45f48b17 */
 339   1.3387534375e+05, /* 0x4802bcd6 */
 340   7.1965775000e+05, /* 0x492fb29c */
 341   6.6660125000e+05, /* 0x4922be94 */
 342  -2.9449025000e+05, /* 0xc88fcb48 */
 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  -2.0897993405e-11, /* 0xadb7d219 */
 351  -1.0253904760e-01, /* 0xbdd1fffe */
 352  -8.0564479828e+00, /* 0xc100e736 */
 353  -1.8366960144e+02, /* 0xc337ab6b */
 354  -1.3731937256e+03, /* 0xc4aba633 */
 355  -2.6124443359e+03, /* 0xc523471c */
 356 };
 357 #ifdef __STDC__
 358 static const float qs5[6] = {
 359 #else
 360 static float qs5[6] = {
 361 #endif
 362   8.1276550293e+01, /* 0x42a28d98 */
 363   1.9917987061e+03, /* 0x44f8f98f */
 364   1.7468484375e+04, /* 0x468878f8 */
 365   4.9851425781e+04, /* 0x4742bb6d */
 366   2.7948074219e+04, /* 0x46da5826 */
 367  -4.7191835938e+03, /* 0xc5937978 */
 368 };
 369
 370 #ifdef __STDC__
 371 static const float qr3[6] = {
 372 #else
 373 static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 374 #endif
 375  -5.0783124372e-09, /* 0xb1ae7d4f */
 376  -1.0253783315e-01, /* 0xbdd1ff5b */
 377  -4.6101160049e+00, /* 0xc0938612 */
 378  -5.7847221375e+01, /* 0xc267638e */
 379  -2.2824453735e+02, /* 0xc3643e9a */
 380  -2.1921012878e+02, /* 0xc35b35cb */
 381 };
 382 #ifdef __STDC__
 383 static const float qs3[6] = {
 384 #else
 385 static float qs3[6] = {
 386 #endif
 387   4.7665153503e+01, /* 0x423ea91e */
 388   6.7386511230e+02, /* 0x4428775e */
 389   3.3801528320e+03, /* 0x45534272 */
 390   5.5477290039e+03, /* 0x45ad5dd5 */
 391   1.9031191406e+03, /* 0x44ede3d0 */
 392  -1.3520118713e+02, /* 0xc3073381 */
 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  -1.7838172539e-07, /* 0xb43f8932 */
 401  -1.0251704603e-01, /* 0xbdd1f475 */
 402  -2.7522056103e+00, /* 0xc0302423 */
 403  -1.9663616180e+01, /* 0xc19d4f16 */
 404  -4.2325313568e+01, /* 0xc2294d1f */
 405  -2.1371921539e+01, /* 0xc1aaf9b2 */
 406 };
 407 #ifdef __STDC__
 408 static const float qs2[6] = {
 409 #else
 410 static float qs2[6] = {
 411 #endif
 412   2.9533363342e+01, /* 0x41ec4454 */
 413   2.5298155212e+02, /* 0x437cfb47 */
 414   7.5750280762e+02, /* 0x443d602e */
 415   7.3939318848e+02, /* 0x4438d92a */
 416   1.5594900513e+02, /* 0x431bf2f2 */
 417  -4.9594988823e+00, /* 0xc09eb437 */
 418 };
 419
 420 #ifdef __STDC__
 421         static float qonef(float x)
 422 #else
 423         static float qonef(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>=0x40200000)     {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).375 + r/s)/x;
 444 }