compiler/mlib/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 #ifndef lint
  17 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_j1f.c,v 1.7 2002/05/28 18:15:04 alfred Exp $";
  18 #endif
  19
  20 #include "math.h"
  21 #include "math_private.h"
  22
  23 static float ponef(float), qonef(float);
  24
  25 static const float
  26 huge    = 1e30,
  27 one     = 1.0,
  28 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  29 tpi      =  6.3661974669e-01, /* 0x3f22f983 */
  30         /* R0/S0 on [0,2] */
  31 r00  = -6.2500000000e-02, /* 0xbd800000 */
  32 r01  =  1.4070566976e-03, /* 0x3ab86cfd */
  33 r02  = -1.5995563444e-05, /* 0xb7862e36 */
  34 r03  =  4.9672799207e-08, /* 0x335557d2 */
  35 s01  =  1.9153760746e-02, /* 0x3c9ce859 */
  36 s02  =  1.8594678841e-04, /* 0x3942fab6 */
  37 s03  =  1.1771846857e-06, /* 0x359dffc2 */
  38 s04  =  5.0463624390e-09, /* 0x31ad6446 */
  39 s05  =  1.2354227016e-11; /* 0x2d59567e */
  40
  41 static const float zero    = 0.0;
  42
  43 float
  44 __ieee754_j1f(float x)
  45 {
  46         float z, s,c,ss,cc,r,u,v,y;
  47         int32_t hx,ix;
  48
  49         GET_FLOAT_WORD(hx,x);
  50         ix = hx&0x7fffffff;
  51         if(ix>=0x7f800000) return one/x;
  52         y = fabsf(x);
  53         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  54                 s = sinf(y);
  55                 c = cosf(y);
  56                 ss = -s-c;
  57                 cc = s-c;
  58                 if(ix<0x7f000000) {  /* make sure y+y not overflow */
  59                     z = cosf(y+y);
  60                     if ((s*c)>zero) cc = z/ss;
  61                     else            ss = z/cc;
  62                 }
  63         /*
  64          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
  65          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
  66          */
  67                 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y);
  68                 else {
  69                     u = ponef(y); v = qonef(y);
  70                     z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
  71                 }
  72                 if(hx<0) return -z;
  73                 else     return  z;
  74         }
  75         if(ix<0x32000000) {     /* |x|<2**-27 */
  76             if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
  77         }
  78         z = x*x;
  79         r =  z*(r00+z*(r01+z*(r02+z*r03)));
  80         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
  81         r *= x;
  82         return(x*(float)0.5+r/s);
  83 }
  84
  85 static const float U0[5] = {
  86  -1.9605709612e-01, /* 0xbe48c331 */
  87   5.0443872809e-02, /* 0x3d4e9e3c */
  88  -1.9125689287e-03, /* 0xbafaaf2a */
  89   2.3525259166e-05, /* 0x37c5581c */
  90  -9.1909917899e-08, /* 0xb3c56003 */
  91 };
  92 static const float V0[5] = {
  93   1.9916731864e-02, /* 0x3ca3286a */
  94   2.0255257550e-04, /* 0x3954644b */
  95   1.3560879779e-06, /* 0x35b602d4 */
  96   6.2274145840e-09, /* 0x31d5f8eb */
  97   1.6655924903e-11, /* 0x2d9281cf */
  98 };
  99
 100 float
 101 __ieee754_y1f(float x)
 102 {
 103         float z, s,c,ss,cc,u,v;
 104         int32_t hx,ix;
 105
 106         GET_FLOAT_WORD(hx,x);
 107         ix = 0x7fffffff&hx;
 108     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
 109         if(ix>=0x7f800000) return  one/(x+x*x);
 110         if(ix==0) return -one/zero;
 111         if(hx<0) return zero/zero;
 112         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 113                 s = sinf(x);
 114                 c = cosf(x);
 115                 ss = -s-c;
 116                 cc = s-c;
 117                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
 118                     z = cosf(x+x);
 119                     if ((s*c)>zero) cc = z/ss;
 120                     else            ss = z/cc;
 121                 }
 122         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
 123          * where x0 = x-3pi/4
 124          *      Better formula:
 125          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
 126          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 127          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 128          *                      = -1/sqrt(2) * (cos(x) + sin(x))
 129          * To avoid cancellation, use
 130          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 131          * to compute the worse one.
 132          */
 133                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
 134                 else {
 135                     u = ponef(x); v = qonef(x);
 136                     z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
 137                 }
 138                 return z;
 139         }
 140         if(ix<=0x24800000) {    /* x < 2**-54 */
 141             return(-tpi/x);
 142         }
 143         z = x*x;
 144         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
 145         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
 146         return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
 147 }
 148
 149 /* For x >= 8, the asymptotic expansions of pone is
 150  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 151  * We approximate pone by
 152  *      pone(x) = 1 + (R/S)
 153  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 154  *        S = 1 + ps0*s^2 + ... + ps4*s^10
 155  * and
 156  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 157  */
 158
 159 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 160   0.0000000000e+00, /* 0x00000000 */
 161   1.1718750000e-01, /* 0x3df00000 */
 162   1.3239480972e+01, /* 0x4153d4ea */
 163   4.1205184937e+02, /* 0x43ce06a3 */
 164   3.8747453613e+03, /* 0x45722bed */
 165   7.9144794922e+03, /* 0x45f753d6 */
 166 };
 167 static const float ps8[5] = {
 168   1.1420736694e+02, /* 0x42e46a2c */
 169   3.6509309082e+03, /* 0x45642ee5 */
 170   3.6956207031e+04, /* 0x47105c35 */
 171   9.7602796875e+04, /* 0x47bea166 */
 172   3.0804271484e+04, /* 0x46f0a88b */
 173 };
 174
 175 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 176   1.3199052094e-11, /* 0x2d68333f */
 177   1.1718749255e-01, /* 0x3defffff */
 178   6.8027510643e+00, /* 0x40d9b023 */
 179   1.0830818176e+02, /* 0x42d89dca */
 180   5.1763616943e+02, /* 0x440168b7 */
 181   5.2871520996e+02, /* 0x44042dc6 */
 182 };
 183 static const float ps5[5] = {
 184   5.9280597687e+01, /* 0x426d1f55 */
 185   9.9140142822e+02, /* 0x4477d9b1 */
 186   5.3532670898e+03, /* 0x45a74a23 */
 187   7.8446904297e+03, /* 0x45f52586 */
 188   1.5040468750e+03, /* 0x44bc0180 */
 189 };
 190
 191 static const float pr3[6] = {
 192   3.0250391081e-09, /* 0x314fe10d */
 193   1.1718686670e-01, /* 0x3defffab */
 194   3.9329774380e+00, /* 0x407bb5e7 */
 195   3.5119403839e+01, /* 0x420c7a45 */
 196   9.1055007935e+01, /* 0x42b61c2a */
 197   4.8559066772e+01, /* 0x42423c7c */
 198 };
 199 static const float ps3[5] = {
 200   3.4791309357e+01, /* 0x420b2a4d */
 201   3.3676245117e+02, /* 0x43a86198 */
 202   1.0468714600e+03, /* 0x4482dbe3 */
 203   8.9081134033e+02, /* 0x445eb3ed */
 204   1.0378793335e+02, /* 0x42cf936c */
 205 };
 206
 207 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 208   1.0771083225e-07, /* 0x33e74ea8 */
 209   1.1717621982e-01, /* 0x3deffa16 */
 210   2.3685150146e+00, /* 0x401795c0 */
 211   1.2242610931e+01, /* 0x4143e1bc */
 212   1.7693971634e+01, /* 0x418d8d41 */
 213   5.0735230446e+00, /* 0x40a25a4d */
 214 };
 215 static const float ps2[5] = {
 216   2.1436485291e+01, /* 0x41ab7dec */
 217   1.2529022980e+02, /* 0x42fa9499 */
 218   2.3227647400e+02, /* 0x436846c7 */
 219   1.1767937469e+02, /* 0x42eb5bd7 */
 220   8.3646392822e+00, /* 0x4105d590 */
 221 };
 222
 223         static float ponef(float x)
 224 {
 225         const float *p,*q;
 226         float z,r,s;
 227         int32_t ix;
 228         GET_FLOAT_WORD(ix,x);
 229         ix &= 0x7fffffff;
 230         if(ix>=0x41000000)     {p = pr8; q= ps8;}
 231         else if(ix>=0x40f71c58){p = pr5; q= ps5;}
 232         else if(ix>=0x4036db68){p = pr3; q= ps3;}
 233         else if(ix>=0x40000000){p = pr2; q= ps2;}
 234         z = one/(x*x);
 235         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 236         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 237         return one+ r/s;
 238 }
 239
 240
 241 /* For x >= 8, the asymptotic expansions of qone is
 242  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 243  * We approximate pone by
 244  *      qone(x) = s*(0.375 + (R/S))
 245  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 246  *        S = 1 + qs1*s^2 + ... + qs6*s^12
 247  * and
 248  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 249  */
 250
 251 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 252   0.0000000000e+00, /* 0x00000000 */
 253  -1.0253906250e-01, /* 0xbdd20000 */
 254  -1.6271753311e+01, /* 0xc1822c8d */
 255  -7.5960174561e+02, /* 0xc43de683 */
 256  -1.1849806641e+04, /* 0xc639273a */
 257  -4.8438511719e+04, /* 0xc73d3683 */
 258 };
 259 static const float qs8[6] = {
 260   1.6139537048e+02, /* 0x43216537 */
 261   7.8253862305e+03, /* 0x45f48b17 */
 262   1.3387534375e+05, /* 0x4802bcd6 */
 263   7.1965775000e+05, /* 0x492fb29c */
 264   6.6660125000e+05, /* 0x4922be94 */
 265  -2.9449025000e+05, /* 0xc88fcb48 */
 266 };
 267
 268 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 269  -2.0897993405e-11, /* 0xadb7d219 */
 270  -1.0253904760e-01, /* 0xbdd1fffe */
 271  -8.0564479828e+00, /* 0xc100e736 */
 272  -1.8366960144e+02, /* 0xc337ab6b */
 273  -1.3731937256e+03, /* 0xc4aba633 */
 274  -2.6124443359e+03, /* 0xc523471c */
 275 };
 276 static const float qs5[6] = {
 277   8.1276550293e+01, /* 0x42a28d98 */
 278   1.9917987061e+03, /* 0x44f8f98f */
 279   1.7468484375e+04, /* 0x468878f8 */
 280   4.9851425781e+04, /* 0x4742bb6d */
 281   2.7948074219e+04, /* 0x46da5826 */
 282  -4.7191835938e+03, /* 0xc5937978 */
 283 };
 284
 285 static const float qr3[6] = {
 286  -5.0783124372e-09, /* 0xb1ae7d4f */
 287  -1.0253783315e-01, /* 0xbdd1ff5b */
 288  -4.6101160049e+00, /* 0xc0938612 */
 289  -5.7847221375e+01, /* 0xc267638e */
 290  -2.2824453735e+02, /* 0xc3643e9a */
 291  -2.1921012878e+02, /* 0xc35b35cb */
 292 };
 293 static const float qs3[6] = {
 294   4.7665153503e+01, /* 0x423ea91e */
 295   6.7386511230e+02, /* 0x4428775e */
 296   3.3801528320e+03, /* 0x45534272 */
 297   5.5477290039e+03, /* 0x45ad5dd5 */
 298   1.9031191406e+03, /* 0x44ede3d0 */
 299  -1.3520118713e+02, /* 0xc3073381 */
 300 };
 301
 302 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 303  -1.7838172539e-07, /* 0xb43f8932 */
 304  -1.0251704603e-01, /* 0xbdd1f475 */
 305  -2.7522056103e+00, /* 0xc0302423 */
 306  -1.9663616180e+01, /* 0xc19d4f16 */
 307  -4.2325313568e+01, /* 0xc2294d1f */
 308  -2.1371921539e+01, /* 0xc1aaf9b2 */
 309 };
 310 static const float qs2[6] = {
 311   2.9533363342e+01, /* 0x41ec4454 */
 312   2.5298155212e+02, /* 0x437cfb47 */
 313   7.5750280762e+02, /* 0x443d602e */
 314   7.3939318848e+02, /* 0x4438d92a */
 315   1.5594900513e+02, /* 0x431bf2f2 */
 316  -4.9594988823e+00, /* 0xc09eb437 */
 317 };
 318
 319         static float qonef(float x)
 320 {
 321         const float *p,*q;
 322         float  s,r,z;
 323         int32_t ix;
 324         GET_FLOAT_WORD(ix,x);
 325         ix &= 0x7fffffff;
 326         if(ix>=0x40200000)     {p = qr8; q= qs8;}
 327         else if(ix>=0x40f71c58){p = qr5; q= qs5;}
 328         else if(ix>=0x4036db68){p = qr3; q= qs3;}
 329         else if(ix>=0x40000000){p = qr2; q= qs2;}
 330         z = one/(x*x);
 331         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 332         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 333         return ((float).375 + r/s)/x;
 334 }