sysdeps/ieee754/flt-32/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 #include <errno.h>
  17 #include <float.h>
  18 #include <math.h>
  19 #include <math_private.h>
  20
  21 static float ponef(float), qonef(float);
  22
  23 static const float
  24 huge    = 1e30,
  25 one     = 1.0,
  26 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  27 tpi      =  6.3661974669e-01, /* 0x3f22f983 */
  28         /* R0/S0 on [0,2] */
  29 r00  = -6.2500000000e-02, /* 0xbd800000 */
  30 r01  =  1.4070566976e-03, /* 0x3ab86cfd */
  31 r02  = -1.5995563444e-05, /* 0xb7862e36 */
  32 r03  =  4.9672799207e-08, /* 0x335557d2 */
  33 s01  =  1.9153760746e-02, /* 0x3c9ce859 */
  34 s02  =  1.8594678841e-04, /* 0x3942fab6 */
  35 s03  =  1.1771846857e-06, /* 0x359dffc2 */
  36 s04  =  5.0463624390e-09, /* 0x31ad6446 */
  37 s05  =  1.2354227016e-11; /* 0x2d59567e */
  38
  39 static const float zero    = 0.0;
  40
  41 float
  42 __ieee754_j1f(float x)
  43 {
  44         float z, s,c,ss,cc,r,u,v,y;
  45         int32_t hx,ix;
  46
  47         GET_FLOAT_WORD(hx,x);
  48         ix = hx&0x7fffffff;
  49         if(__builtin_expect(ix>=0x7f800000, 0)) return one/x;
  50         y = fabsf(x);
  51         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  52                 __sincosf (y, &s, &c);
  53                 ss = -s-c;
  54                 cc = s-c;
  55                 if(ix<0x7f000000) {  /* make sure y+y not overflow */
  56                     z = __cosf(y+y);
  57                     if ((s*c)>zero) cc = z/ss;
  58                     else            ss = z/cc;
  59                 }
  60         /*
  61          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
  62          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
  63          */
  64                 if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(y);
  65                 else {
  66                     u = ponef(y); v = qonef(y);
  67                     z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(y);
  68                 }
  69                 if(hx<0) return -z;
  70                 else     return  z;
  71         }
  72         if(__builtin_expect(ix<0x32000000, 0)) {        /* |x|<2**-27 */
  73             if(huge+x>one) {            /* inexact if x!=0 necessary */
  74                 float ret = math_narrow_eval ((float) 0.5 * x);
  75                 math_check_force_underflow (ret);
  76                 if (ret == 0 && x != 0)
  77                   __set_errno (ERANGE);
  78                 return ret;
  79             }
  80         }
  81         z = x*x;
  82         r =  z*(r00+z*(r01+z*(r02+z*r03)));
  83         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
  84         r *= x;
  85         return(x*(float)0.5+r/s);
  86 }
  87 strong_alias (__ieee754_j1f, __j1f_finite)
  88
  89 static const float U0[5] = {
  90  -1.9605709612e-01, /* 0xbe48c331 */
  91   5.0443872809e-02, /* 0x3d4e9e3c */
  92  -1.9125689287e-03, /* 0xbafaaf2a */
  93   2.3525259166e-05, /* 0x37c5581c */
  94  -9.1909917899e-08, /* 0xb3c56003 */
  95 };
  96 static const float V0[5] = {
  97   1.9916731864e-02, /* 0x3ca3286a */
  98   2.0255257550e-04, /* 0x3954644b */
  99   1.3560879779e-06, /* 0x35b602d4 */
 100   6.2274145840e-09, /* 0x31d5f8eb */
 101   1.6655924903e-11, /* 0x2d9281cf */
 102 };
 103
 104 float
 105 __ieee754_y1f(float x)
 106 {
 107         float z, s,c,ss,cc,u,v;
 108         int32_t hx,ix;
 109
 110         GET_FLOAT_WORD(hx,x);
 111         ix = 0x7fffffff&hx;
 112     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
 113         if(__builtin_expect(ix>=0x7f800000, 0)) return  one/(x+x*x);
 114         if(__builtin_expect(ix==0, 0))
 115                 return -HUGE_VALF+x;  /* -inf and overflow exception.  */
 116         if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
 117         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 118                 SET_RESTORE_ROUNDF (FE_TONEAREST);
 119                 __sincosf (x, &s, &c);
 120                 ss = -s-c;
 121                 cc = s-c;
 122                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
 123                     z = __cosf(x+x);
 124                     if ((s*c)>zero) cc = z/ss;
 125                     else            ss = z/cc;
 126                 }
 127         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
 128          * where x0 = x-3pi/4
 129          *      Better formula:
 130          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
 131          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 132          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 133          *                      = -1/sqrt(2) * (cos(x) + sin(x))
 134          * To avoid cancellation, use
 135          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 136          * to compute the worse one.
 137          */
 138                 if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
 139                 else {
 140                     u = ponef(x); v = qonef(x);
 141                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x);
 142                 }
 143                 return z;
 144         }
 145         if(__builtin_expect(ix<=0x33000000, 0)) {    /* x < 2**-25 */
 146             z = -tpi / x;
 147             if (isinf (z))
 148                 __set_errno (ERANGE);
 149             return z;
 150         }
 151         z = x*x;
 152         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
 153         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
 154         return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
 155 }
 156 strong_alias (__ieee754_y1f, __y1f_finite)
 157
 158 /* For x >= 8, the asymptotic expansions of pone is
 159  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 160  * We approximate pone by
 161  *      pone(x) = 1 + (R/S)
 162  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 163  *        S = 1 + ps0*s^2 + ... + ps4*s^10
 164  * and
 165  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 166  */
 167
 168 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 169   0.0000000000e+00, /* 0x00000000 */
 170   1.1718750000e-01, /* 0x3df00000 */
 171   1.3239480972e+01, /* 0x4153d4ea */
 172   4.1205184937e+02, /* 0x43ce06a3 */
 173   3.8747453613e+03, /* 0x45722bed */
 174   7.9144794922e+03, /* 0x45f753d6 */
 175 };
 176 static const float ps8[5] = {
 177   1.1420736694e+02, /* 0x42e46a2c */
 178   3.6509309082e+03, /* 0x45642ee5 */
 179   3.6956207031e+04, /* 0x47105c35 */
 180   9.7602796875e+04, /* 0x47bea166 */
 181   3.0804271484e+04, /* 0x46f0a88b */
 182 };
 183
 184 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 185   1.3199052094e-11, /* 0x2d68333f */
 186   1.1718749255e-01, /* 0x3defffff */
 187   6.8027510643e+00, /* 0x40d9b023 */
 188   1.0830818176e+02, /* 0x42d89dca */
 189   5.1763616943e+02, /* 0x440168b7 */
 190   5.2871520996e+02, /* 0x44042dc6 */
 191 };
 192 static const float ps5[5] = {
 193   5.9280597687e+01, /* 0x426d1f55 */
 194   9.9140142822e+02, /* 0x4477d9b1 */
 195   5.3532670898e+03, /* 0x45a74a23 */
 196   7.8446904297e+03, /* 0x45f52586 */
 197   1.5040468750e+03, /* 0x44bc0180 */
 198 };
 199
 200 static const float pr3[6] = {
 201   3.0250391081e-09, /* 0x314fe10d */
 202   1.1718686670e-01, /* 0x3defffab */
 203   3.9329774380e+00, /* 0x407bb5e7 */
 204   3.5119403839e+01, /* 0x420c7a45 */
 205   9.1055007935e+01, /* 0x42b61c2a */
 206   4.8559066772e+01, /* 0x42423c7c */
 207 };
 208 static const float ps3[5] = {
 209   3.4791309357e+01, /* 0x420b2a4d */
 210   3.3676245117e+02, /* 0x43a86198 */
 211   1.0468714600e+03, /* 0x4482dbe3 */
 212   8.9081134033e+02, /* 0x445eb3ed */
 213   1.0378793335e+02, /* 0x42cf936c */
 214 };
 215
 216 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 217   1.0771083225e-07, /* 0x33e74ea8 */
 218   1.1717621982e-01, /* 0x3deffa16 */
 219   2.3685150146e+00, /* 0x401795c0 */
 220   1.2242610931e+01, /* 0x4143e1bc */
 221   1.7693971634e+01, /* 0x418d8d41 */
 222   5.0735230446e+00, /* 0x40a25a4d */
 223 };
 224 static const float ps2[5] = {
 225   2.1436485291e+01, /* 0x41ab7dec */
 226   1.2529022980e+02, /* 0x42fa9499 */
 227   2.3227647400e+02, /* 0x436846c7 */
 228   1.1767937469e+02, /* 0x42eb5bd7 */
 229   8.3646392822e+00, /* 0x4105d590 */
 230 };
 231
 232 static float
 233 ponef(float x)
 234 {
 235         const float *p,*q;
 236         float z,r,s;
 237         int32_t ix;
 238         GET_FLOAT_WORD(ix,x);
 239         ix &= 0x7fffffff;
 240         /* ix >= 0x40000000 for all calls to this function.  */
 241         if(ix>=0x41000000)     {p = pr8; q= ps8;}
 242         else if(ix>=0x40f71c58){p = pr5; q= ps5;}
 243         else if(ix>=0x4036db68){p = pr3; q= ps3;}
 244         else {p = pr2; q= ps2;}
 245         z = one/(x*x);
 246         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 247         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 248         return one+ r/s;
 249 }
 250
 251
 252 /* For x >= 8, the asymptotic expansions of qone is
 253  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 254  * We approximate pone by
 255  *      qone(x) = s*(0.375 + (R/S))
 256  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 257  *        S = 1 + qs1*s^2 + ... + qs6*s^12
 258  * and
 259  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 260  */
 261
 262 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 263   0.0000000000e+00, /* 0x00000000 */
 264  -1.0253906250e-01, /* 0xbdd20000 */
 265  -1.6271753311e+01, /* 0xc1822c8d */
 266  -7.5960174561e+02, /* 0xc43de683 */
 267  -1.1849806641e+04, /* 0xc639273a */
 268  -4.8438511719e+04, /* 0xc73d3683 */
 269 };
 270 static const float qs8[6] = {
 271   1.6139537048e+02, /* 0x43216537 */
 272   7.8253862305e+03, /* 0x45f48b17 */
 273   1.3387534375e+05, /* 0x4802bcd6 */
 274   7.1965775000e+05, /* 0x492fb29c */
 275   6.6660125000e+05, /* 0x4922be94 */
 276  -2.9449025000e+05, /* 0xc88fcb48 */
 277 };
 278
 279 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 280  -2.0897993405e-11, /* 0xadb7d219 */
 281  -1.0253904760e-01, /* 0xbdd1fffe */
 282  -8.0564479828e+00, /* 0xc100e736 */
 283  -1.8366960144e+02, /* 0xc337ab6b */
 284  -1.3731937256e+03, /* 0xc4aba633 */
 285  -2.6124443359e+03, /* 0xc523471c */
 286 };
 287 static const float qs5[6] = {
 288   8.1276550293e+01, /* 0x42a28d98 */
 289   1.9917987061e+03, /* 0x44f8f98f */
 290   1.7468484375e+04, /* 0x468878f8 */
 291   4.9851425781e+04, /* 0x4742bb6d */
 292   2.7948074219e+04, /* 0x46da5826 */
 293  -4.7191835938e+03, /* 0xc5937978 */
 294 };
 295
 296 static const float qr3[6] = {
 297  -5.0783124372e-09, /* 0xb1ae7d4f */
 298  -1.0253783315e-01, /* 0xbdd1ff5b */
 299  -4.6101160049e+00, /* 0xc0938612 */
 300  -5.7847221375e+01, /* 0xc267638e */
 301  -2.2824453735e+02, /* 0xc3643e9a */
 302  -2.1921012878e+02, /* 0xc35b35cb */
 303 };
 304 static const float qs3[6] = {
 305   4.7665153503e+01, /* 0x423ea91e */
 306   6.7386511230e+02, /* 0x4428775e */
 307   3.3801528320e+03, /* 0x45534272 */
 308   5.5477290039e+03, /* 0x45ad5dd5 */
 309   1.9031191406e+03, /* 0x44ede3d0 */
 310  -1.3520118713e+02, /* 0xc3073381 */
 311 };
 312
 313 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 314  -1.7838172539e-07, /* 0xb43f8932 */
 315  -1.0251704603e-01, /* 0xbdd1f475 */
 316  -2.7522056103e+00, /* 0xc0302423 */
 317  -1.9663616180e+01, /* 0xc19d4f16 */
 318  -4.2325313568e+01, /* 0xc2294d1f */
 319  -2.1371921539e+01, /* 0xc1aaf9b2 */
 320 };
 321 static const float qs2[6] = {
 322   2.9533363342e+01, /* 0x41ec4454 */
 323   2.5298155212e+02, /* 0x437cfb47 */
 324   7.5750280762e+02, /* 0x443d602e */
 325   7.3939318848e+02, /* 0x4438d92a */
 326   1.5594900513e+02, /* 0x431bf2f2 */
 327  -4.9594988823e+00, /* 0xc09eb437 */
 328 };
 329
 330 static float
 331 qonef(float x)
 332 {
 333         const float *p,*q;
 334         float  s,r,z;
 335         int32_t ix;
 336         GET_FLOAT_WORD(ix,x);
 337         ix &= 0x7fffffff;
 338         /* ix >= 0x40000000 for all calls to this function.  */
 339         if(ix>=0x40200000)     {p = qr8; q= qs8;}
 340         else if(ix>=0x40f71c58){p = qr5; q= qs5;}
 341         else if(ix>=0x4036db68){p = qr3; q= qs3;}
 342         else {p = qr2; q= qs2;}
 343         z = one/(x*x);
 344         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 345         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 346         return ((float).375 + r/s)/x;
 347 }