contrib/openbsd_libm/src/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 "math.h"
  17 #include "math_private.h"
  18
  19 static float ponef(float), qonef(float);
  20
  21 static const float
  22 huge    = 1e30,
  23 one     = 1.0,
  24 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  25 tpi      =  6.3661974669e-01, /* 0x3f22f983 */
  26         /* R0/S0 on [0,2] */
  27 r00  = -6.2500000000e-02, /* 0xbd800000 */
  28 r01  =  1.4070566976e-03, /* 0x3ab86cfd */
  29 r02  = -1.5995563444e-05, /* 0xb7862e36 */
  30 r03  =  4.9672799207e-08, /* 0x335557d2 */
  31 s01  =  1.9153760746e-02, /* 0x3c9ce859 */
  32 s02  =  1.8594678841e-04, /* 0x3942fab6 */
  33 s03  =  1.1771846857e-06, /* 0x359dffc2 */
  34 s04  =  5.0463624390e-09, /* 0x31ad6446 */
  35 s05  =  1.2354227016e-11; /* 0x2d59567e */
  36
  37 static const float zero    = 0.0;
  38
  39 float
  40 j1f(float x)
  41 {
  42         float z, s,c,ss,cc,r,u,v,y;
  43         int32_t hx,ix;
  44
  45         GET_FLOAT_WORD(hx,x);
  46         ix = hx&0x7fffffff;
  47         if(ix>=0x7f800000) return one/x;
  48         y = fabsf(x);
  49         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  50                 s = sinf(y);
  51                 c = cosf(y);
  52                 ss = -s-c;
  53                 cc = s-c;
  54                 if(ix<0x7f000000) {  /* make sure y+y not overflow */
  55                     z = cosf(y+y);
  56                     if ((s*c)>zero) cc = z/ss;
  57                     else            ss = z/cc;
  58                 }
  59         /*
  60          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
  61          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
  62          */
  63                 if((u_int64_t)ix>0x80000000U) z = (invsqrtpi*cc)/sqrtf(y);
  64                 else {
  65                     u = ponef(y); v = qonef(y);
  66                     z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
  67                 }
  68                 if(hx<0) return -z;
  69                 else     return  z;
  70         }
  71         if(ix<0x32000000) {     /* |x|<2**-27 */
  72             if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
  73         }
  74         z = x*x;
  75         r =  z*(r00+z*(r01+z*(r02+z*r03)));
  76         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
  77         r *= x;
  78         return(x*(float)0.5+r/s);
  79 }
  80
  81 static const float U0[5] = {
  82  -1.9605709612e-01, /* 0xbe48c331 */
  83   5.0443872809e-02, /* 0x3d4e9e3c */
  84  -1.9125689287e-03, /* 0xbafaaf2a */
  85   2.3525259166e-05, /* 0x37c5581c */
  86  -9.1909917899e-08, /* 0xb3c56003 */
  87 };
  88 static const float V0[5] = {
  89   1.9916731864e-02, /* 0x3ca3286a */
  90   2.0255257550e-04, /* 0x3954644b */
  91   1.3560879779e-06, /* 0x35b602d4 */
  92   6.2274145840e-09, /* 0x31d5f8eb */
  93   1.6655924903e-11, /* 0x2d9281cf */
  94 };
  95
  96 float
  97 y1f(float x)
  98 {
  99         float z, s,c,ss,cc,u,v;
 100         int32_t hx,ix;
 101
 102         GET_FLOAT_WORD(hx,x);
 103         ix = 0x7fffffff&hx;
 104     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
 105         if(ix>=0x7f800000) return  one/(x+x*x);
 106         if(ix==0) return -one/zero;
 107         if(hx<0) return zero/zero;
 108         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 109                 s = sinf(x);
 110                 c = cosf(x);
 111                 ss = -s-c;
 112                 cc = s-c;
 113                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
 114                     z = cosf(x+x);
 115                     if ((s*c)>zero) cc = z/ss;
 116                     else            ss = z/cc;
 117                 }
 118         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
 119          * where x0 = x-3pi/4
 120          *      Better formula:
 121          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
 122          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 123          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 124          *                      = -1/sqrt(2) * (cos(x) + sin(x))
 125          * To avoid cancellation, use
 126          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 127          * to compute the worse one.
 128          */
 129                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
 130                 else {
 131                     u = ponef(x); v = qonef(x);
 132                     z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
 133                 }
 134                 return z;
 135         }
 136         if(ix<=0x24800000) {    /* x < 2**-54 */
 137             return(-tpi/x);
 138         }
 139         z = x*x;
 140         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
 141         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
 142         return(x*(u/v) + tpi*(j1f(x)*logf(x)-one/x));
 143 }
 144
 145 /* For x >= 8, the asymptotic expansions of pone is
 146  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 147  * We approximate pone by
 148  *      pone(x) = 1 + (R/S)
 149  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 150  *        S = 1 + ps0*s^2 + ... + ps4*s^10
 151  * and
 152  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 153  */
 154
 155 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 156   0.0000000000e+00, /* 0x00000000 */
 157   1.1718750000e-01, /* 0x3df00000 */
 158   1.3239480972e+01, /* 0x4153d4ea */
 159   4.1205184937e+02, /* 0x43ce06a3 */
 160   3.8747453613e+03, /* 0x45722bed */
 161   7.9144794922e+03, /* 0x45f753d6 */
 162 };
 163 static const float ps8[5] = {
 164   1.1420736694e+02, /* 0x42e46a2c */
 165   3.6509309082e+03, /* 0x45642ee5 */
 166   3.6956207031e+04, /* 0x47105c35 */
 167   9.7602796875e+04, /* 0x47bea166 */
 168   3.0804271484e+04, /* 0x46f0a88b */
 169 };
 170
 171 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 172   1.3199052094e-11, /* 0x2d68333f */
 173   1.1718749255e-01, /* 0x3defffff */
 174   6.8027510643e+00, /* 0x40d9b023 */
 175   1.0830818176e+02, /* 0x42d89dca */
 176   5.1763616943e+02, /* 0x440168b7 */
 177   5.2871520996e+02, /* 0x44042dc6 */
 178 };
 179 static const float ps5[5] = {
 180   5.9280597687e+01, /* 0x426d1f55 */
 181   9.9140142822e+02, /* 0x4477d9b1 */
 182   5.3532670898e+03, /* 0x45a74a23 */
 183   7.8446904297e+03, /* 0x45f52586 */
 184   1.5040468750e+03, /* 0x44bc0180 */
 185 };
 186
 187 static const float pr3[6] = {
 188   3.0250391081e-09, /* 0x314fe10d */
 189   1.1718686670e-01, /* 0x3defffab */
 190   3.9329774380e+00, /* 0x407bb5e7 */
 191   3.5119403839e+01, /* 0x420c7a45 */
 192   9.1055007935e+01, /* 0x42b61c2a */
 193   4.8559066772e+01, /* 0x42423c7c */
 194 };
 195 static const float ps3[5] = {
 196   3.4791309357e+01, /* 0x420b2a4d */
 197   3.3676245117e+02, /* 0x43a86198 */
 198   1.0468714600e+03, /* 0x4482dbe3 */
 199   8.9081134033e+02, /* 0x445eb3ed */
 200   1.0378793335e+02, /* 0x42cf936c */
 201 };
 202
 203 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 204   1.0771083225e-07, /* 0x33e74ea8 */
 205   1.1717621982e-01, /* 0x3deffa16 */
 206   2.3685150146e+00, /* 0x401795c0 */
 207   1.2242610931e+01, /* 0x4143e1bc */
 208   1.7693971634e+01, /* 0x418d8d41 */
 209   5.0735230446e+00, /* 0x40a25a4d */
 210 };
 211 static const float ps2[5] = {
 212   2.1436485291e+01, /* 0x41ab7dec */
 213   1.2529022980e+02, /* 0x42fa9499 */
 214   2.3227647400e+02, /* 0x436846c7 */
 215   1.1767937469e+02, /* 0x42eb5bd7 */
 216   8.3646392822e+00, /* 0x4105d590 */
 217 };
 218
 219 static float
 220 ponef(float x)
 221 {
 222         const float *p,*q;
 223         float z,r,s;
 224         int32_t ix;
 225         GET_FLOAT_WORD(ix,x);
 226         ix &= 0x7fffffff;
 227         if(ix>=0x41000000)     {p = pr8; q= ps8;}
 228         else if(ix>=0x40f71c58){p = pr5; q= ps5;}
 229         else if(ix>=0x4036db68){p = pr3; q= ps3;}
 230         else /*if(ix>=0x40000000)*/ {p = pr2; q= ps2;}
 231         z = one/(x*x);
 232         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 233         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 234         return one+ r/s;
 235 }
 236
 237
 238 /* For x >= 8, the asymptotic expansions of qone is
 239  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 240  * We approximate pone by
 241  *      qone(x) = s*(0.375 + (R/S))
 242  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 243  *        S = 1 + qs1*s^2 + ... + qs6*s^12
 244  * and
 245  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 246  */
 247
 248 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 249   0.0000000000e+00, /* 0x00000000 */
 250  -1.0253906250e-01, /* 0xbdd20000 */
 251  -1.6271753311e+01, /* 0xc1822c8d */
 252  -7.5960174561e+02, /* 0xc43de683 */
 253  -1.1849806641e+04, /* 0xc639273a */
 254  -4.8438511719e+04, /* 0xc73d3683 */
 255 };
 256 static const float qs8[6] = {
 257   1.6139537048e+02, /* 0x43216537 */
 258   7.8253862305e+03, /* 0x45f48b17 */
 259   1.3387534375e+05, /* 0x4802bcd6 */
 260   7.1965775000e+05, /* 0x492fb29c */
 261   6.6660125000e+05, /* 0x4922be94 */
 262  -2.9449025000e+05, /* 0xc88fcb48 */
 263 };
 264
 265 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 266  -2.0897993405e-11, /* 0xadb7d219 */
 267  -1.0253904760e-01, /* 0xbdd1fffe */
 268  -8.0564479828e+00, /* 0xc100e736 */
 269  -1.8366960144e+02, /* 0xc337ab6b */
 270  -1.3731937256e+03, /* 0xc4aba633 */
 271  -2.6124443359e+03, /* 0xc523471c */
 272 };
 273 static const float qs5[6] = {
 274   8.1276550293e+01, /* 0x42a28d98 */
 275   1.9917987061e+03, /* 0x44f8f98f */
 276   1.7468484375e+04, /* 0x468878f8 */
 277   4.9851425781e+04, /* 0x4742bb6d */
 278   2.7948074219e+04, /* 0x46da5826 */
 279  -4.7191835938e+03, /* 0xc5937978 */
 280 };
 281
 282 static const float qr3[6] = {
 283  -5.0783124372e-09, /* 0xb1ae7d4f */
 284  -1.0253783315e-01, /* 0xbdd1ff5b */
 285  -4.6101160049e+00, /* 0xc0938612 */
 286  -5.7847221375e+01, /* 0xc267638e */
 287  -2.2824453735e+02, /* 0xc3643e9a */
 288  -2.1921012878e+02, /* 0xc35b35cb */
 289 };
 290 static const float qs3[6] = {
 291   4.7665153503e+01, /* 0x423ea91e */
 292   6.7386511230e+02, /* 0x4428775e */
 293   3.3801528320e+03, /* 0x45534272 */
 294   5.5477290039e+03, /* 0x45ad5dd5 */
 295   1.9031191406e+03, /* 0x44ede3d0 */
 296  -1.3520118713e+02, /* 0xc3073381 */
 297 };
 298
 299 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 300  -1.7838172539e-07, /* 0xb43f8932 */
 301  -1.0251704603e-01, /* 0xbdd1f475 */
 302  -2.7522056103e+00, /* 0xc0302423 */
 303  -1.9663616180e+01, /* 0xc19d4f16 */
 304  -4.2325313568e+01, /* 0xc2294d1f */
 305  -2.1371921539e+01, /* 0xc1aaf9b2 */
 306 };
 307 static const float qs2[6] = {
 308   2.9533363342e+01, /* 0x41ec4454 */
 309   2.5298155212e+02, /* 0x437cfb47 */
 310   7.5750280762e+02, /* 0x443d602e */
 311   7.3939318848e+02, /* 0x4438d92a */
 312   1.5594900513e+02, /* 0x431bf2f2 */
 313  -4.9594988823e+00, /* 0xc09eb437 */
 314 };
 315
 316 static float
 317 qonef(float x)
 318 {
 319         const float *p,*q;
 320         float  s,r,z;
 321         int32_t ix;
 322         GET_FLOAT_WORD(ix,x);
 323         ix &= 0x7fffffff;
 324         if(ix>=0x40200000)     {p = qr8; q= qs8;}
 325         else if(ix>=0x40f71c58){p = qr5; q= qs5;}
 326         else if(ix>=0x4036db68){p = qr3; q= qs3;}
 327         else /*if(ix>=0x40000000)*/ {p = qr2; q= qs2;}
 328         z = one/(x*x);
 329         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 330         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 331         return ((float).375 + r/s)/x;
 332 }