src/math/j1f.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.c */
   2 /*
   3  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
   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 #define _GNU_SOURCE
  17 #include "libm.h"
  18
  19 static float ponef(float), qonef(float);
  20
  21 static const float
  22 invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
  23 tpi       = 6.3661974669e-01; /* 0x3f22f983 */
  24
  25 static float common(uint32_t ix, float x, int y1, int sign)
  26 {
  27         double z,s,c,ss,cc;
  28
  29         s = sinf(x);
  30         if (y1)
  31                 s = -s;
  32         c = cosf(x);
  33         cc = s-c;
  34         if (ix < 0x7f000000) {
  35                 ss = -s-c;
  36                 z = cosf(2*x);
  37                 if (s*c > 0)
  38                         cc = z/ss;
  39                 else
  40                         ss = z/cc;
  41                 if (ix < 0x58800000) {
  42                         if (y1)
  43                                 ss = -ss;
  44                         cc = ponef(x)*cc-qonef(x)*ss;
  45                 }
  46         }
  47         if (sign)
  48                 cc = -cc;
  49         return invsqrtpi*cc/sqrtf(x);
  50 }
  51
  52 /* R0/S0 on [0,2] */
  53 static const float
  54 r00 = -6.2500000000e-02, /* 0xbd800000 */
  55 r01 =  1.4070566976e-03, /* 0x3ab86cfd */
  56 r02 = -1.5995563444e-05, /* 0xb7862e36 */
  57 r03 =  4.9672799207e-08, /* 0x335557d2 */
  58 s01 =  1.9153760746e-02, /* 0x3c9ce859 */
  59 s02 =  1.8594678841e-04, /* 0x3942fab6 */
  60 s03 =  1.1771846857e-06, /* 0x359dffc2 */
  61 s04 =  5.0463624390e-09, /* 0x31ad6446 */
  62 s05 =  1.2354227016e-11; /* 0x2d59567e */
  63
  64 float j1f(float x)
  65 {
  66         float z,r,s;
  67         uint32_t ix;
  68         int sign;
  69
  70         GET_FLOAT_WORD(ix, x);
  71         sign = ix>>31;
  72         ix &= 0x7fffffff;
  73         if (ix >= 0x7f800000)
  74                 return 1/(x*x);
  75         if (ix >= 0x40000000)  /* |x| >= 2 */
  76                 return common(ix, fabsf(x), 0, sign);
  77         if (ix >= 0x39000000) {  /* |x| >= 2**-13 */
  78                 z = x*x;
  79                 r = z*(r00+z*(r01+z*(r02+z*r03)));
  80                 s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
  81                 z = 0.5f + r/s;
  82         } else
  83                 z = 0.5f;
  84         return z*x;
  85 }
  86
  87 static const float U0[5] = {
  88  -1.9605709612e-01, /* 0xbe48c331 */
  89   5.0443872809e-02, /* 0x3d4e9e3c */
  90  -1.9125689287e-03, /* 0xbafaaf2a */
  91   2.3525259166e-05, /* 0x37c5581c */
  92  -9.1909917899e-08, /* 0xb3c56003 */
  93 };
  94 static const float V0[5] = {
  95   1.9916731864e-02, /* 0x3ca3286a */
  96   2.0255257550e-04, /* 0x3954644b */
  97   1.3560879779e-06, /* 0x35b602d4 */
  98   6.2274145840e-09, /* 0x31d5f8eb */
  99   1.6655924903e-11, /* 0x2d9281cf */
 100 };
 101
 102 float y1f(float x)
 103 {
 104         float z,u,v;
 105         uint32_t ix;
 106
 107         GET_FLOAT_WORD(ix, x);
 108         if ((ix & 0x7fffffff) == 0)
 109                 return -1/0.0f;
 110         if (ix>>31)
 111                 return 0/0.0f;
 112         if (ix >= 0x7f800000)
 113                 return 1/x;
 114         if (ix >= 0x40000000)  /* |x| >= 2.0 */
 115                 return common(ix,x,1,0);
 116         if (ix < 0x33000000)  /* x < 2**-25 */
 117                 return -tpi/x;
 118         z = x*x;
 119         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
 120         v = 1.0f+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
 121         return x*(u/v) + tpi*(j1f(x)*logf(x)-1.0f/x);
 122 }
 123
 124 /* For x >= 8, the asymptotic expansions of pone is
 125  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 126  * We approximate pone by
 127  *      pone(x) = 1 + (R/S)
 128  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 129  *        S = 1 + ps0*s^2 + ... + ps4*s^10
 130  * and
 131  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 132  */
 133
 134 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 135   0.0000000000e+00, /* 0x00000000 */
 136   1.1718750000e-01, /* 0x3df00000 */
 137   1.3239480972e+01, /* 0x4153d4ea */
 138   4.1205184937e+02, /* 0x43ce06a3 */
 139   3.8747453613e+03, /* 0x45722bed */
 140   7.9144794922e+03, /* 0x45f753d6 */
 141 };
 142 static const float ps8[5] = {
 143   1.1420736694e+02, /* 0x42e46a2c */
 144   3.6509309082e+03, /* 0x45642ee5 */
 145   3.6956207031e+04, /* 0x47105c35 */
 146   9.7602796875e+04, /* 0x47bea166 */
 147   3.0804271484e+04, /* 0x46f0a88b */
 148 };
 149
 150 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 151   1.3199052094e-11, /* 0x2d68333f */
 152   1.1718749255e-01, /* 0x3defffff */
 153   6.8027510643e+00, /* 0x40d9b023 */
 154   1.0830818176e+02, /* 0x42d89dca */
 155   5.1763616943e+02, /* 0x440168b7 */
 156   5.2871520996e+02, /* 0x44042dc6 */
 157 };
 158 static const float ps5[5] = {
 159   5.9280597687e+01, /* 0x426d1f55 */
 160   9.9140142822e+02, /* 0x4477d9b1 */
 161   5.3532670898e+03, /* 0x45a74a23 */
 162   7.8446904297e+03, /* 0x45f52586 */
 163   1.5040468750e+03, /* 0x44bc0180 */
 164 };
 165
 166 static const float pr3[6] = {
 167   3.0250391081e-09, /* 0x314fe10d */
 168   1.1718686670e-01, /* 0x3defffab */
 169   3.9329774380e+00, /* 0x407bb5e7 */
 170   3.5119403839e+01, /* 0x420c7a45 */
 171   9.1055007935e+01, /* 0x42b61c2a */
 172   4.8559066772e+01, /* 0x42423c7c */
 173 };
 174 static const float ps3[5] = {
 175   3.4791309357e+01, /* 0x420b2a4d */
 176   3.3676245117e+02, /* 0x43a86198 */
 177   1.0468714600e+03, /* 0x4482dbe3 */
 178   8.9081134033e+02, /* 0x445eb3ed */
 179   1.0378793335e+02, /* 0x42cf936c */
 180 };
 181
 182 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 183   1.0771083225e-07, /* 0x33e74ea8 */
 184   1.1717621982e-01, /* 0x3deffa16 */
 185   2.3685150146e+00, /* 0x401795c0 */
 186   1.2242610931e+01, /* 0x4143e1bc */
 187   1.7693971634e+01, /* 0x418d8d41 */
 188   5.0735230446e+00, /* 0x40a25a4d */
 189 };
 190 static const float ps2[5] = {
 191   2.1436485291e+01, /* 0x41ab7dec */
 192   1.2529022980e+02, /* 0x42fa9499 */
 193   2.3227647400e+02, /* 0x436846c7 */
 194   1.1767937469e+02, /* 0x42eb5bd7 */
 195   8.3646392822e+00, /* 0x4105d590 */
 196 };
 197
 198 static float ponef(float x)
 199 {
 200         const float *p,*q;
 201         float_t z,r,s;
 202         uint32_t ix;
 203
 204         GET_FLOAT_WORD(ix, x);
 205         ix &= 0x7fffffff;
 206         if      (ix >= 0x41000000){p = pr8; q = ps8;}
 207         else if (ix >= 0x409173eb){p = pr5; q = ps5;}
 208         else if (ix >= 0x4036d917){p = pr3; q = ps3;}
 209         else /*ix >= 0x40000000*/ {p = pr2; q = ps2;}
 210         z = 1.0f/(x*x);
 211         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 212         s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 213         return 1.0f + r/s;
 214 }
 215
 216 /* For x >= 8, the asymptotic expansions of qone is
 217  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 218  * We approximate pone by
 219  *      qone(x) = s*(0.375 + (R/S))
 220  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 221  *        S = 1 + qs1*s^2 + ... + qs6*s^12
 222  * and
 223  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 224  */
 225
 226 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 227   0.0000000000e+00, /* 0x00000000 */
 228  -1.0253906250e-01, /* 0xbdd20000 */
 229  -1.6271753311e+01, /* 0xc1822c8d */
 230  -7.5960174561e+02, /* 0xc43de683 */
 231  -1.1849806641e+04, /* 0xc639273a */
 232  -4.8438511719e+04, /* 0xc73d3683 */
 233 };
 234 static const float qs8[6] = {
 235   1.6139537048e+02, /* 0x43216537 */
 236   7.8253862305e+03, /* 0x45f48b17 */
 237   1.3387534375e+05, /* 0x4802bcd6 */
 238   7.1965775000e+05, /* 0x492fb29c */
 239   6.6660125000e+05, /* 0x4922be94 */
 240  -2.9449025000e+05, /* 0xc88fcb48 */
 241 };
 242
 243 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 244  -2.0897993405e-11, /* 0xadb7d219 */
 245  -1.0253904760e-01, /* 0xbdd1fffe */
 246  -8.0564479828e+00, /* 0xc100e736 */
 247  -1.8366960144e+02, /* 0xc337ab6b */
 248  -1.3731937256e+03, /* 0xc4aba633 */
 249  -2.6124443359e+03, /* 0xc523471c */
 250 };
 251 static const float qs5[6] = {
 252   8.1276550293e+01, /* 0x42a28d98 */
 253   1.9917987061e+03, /* 0x44f8f98f */
 254   1.7468484375e+04, /* 0x468878f8 */
 255   4.9851425781e+04, /* 0x4742bb6d */
 256   2.7948074219e+04, /* 0x46da5826 */
 257  -4.7191835938e+03, /* 0xc5937978 */
 258 };
 259
 260 static const float qr3[6] = {
 261  -5.0783124372e-09, /* 0xb1ae7d4f */
 262  -1.0253783315e-01, /* 0xbdd1ff5b */
 263  -4.6101160049e+00, /* 0xc0938612 */
 264  -5.7847221375e+01, /* 0xc267638e */
 265  -2.2824453735e+02, /* 0xc3643e9a */
 266  -2.1921012878e+02, /* 0xc35b35cb */
 267 };
 268 static const float qs3[6] = {
 269   4.7665153503e+01, /* 0x423ea91e */
 270   6.7386511230e+02, /* 0x4428775e */
 271   3.3801528320e+03, /* 0x45534272 */
 272   5.5477290039e+03, /* 0x45ad5dd5 */
 273   1.9031191406e+03, /* 0x44ede3d0 */
 274  -1.3520118713e+02, /* 0xc3073381 */
 275 };
 276
 277 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 278  -1.7838172539e-07, /* 0xb43f8932 */
 279  -1.0251704603e-01, /* 0xbdd1f475 */
 280  -2.7522056103e+00, /* 0xc0302423 */
 281  -1.9663616180e+01, /* 0xc19d4f16 */
 282  -4.2325313568e+01, /* 0xc2294d1f */
 283  -2.1371921539e+01, /* 0xc1aaf9b2 */
 284 };
 285 static const float qs2[6] = {
 286   2.9533363342e+01, /* 0x41ec4454 */
 287   2.5298155212e+02, /* 0x437cfb47 */
 288   7.5750280762e+02, /* 0x443d602e */
 289   7.3939318848e+02, /* 0x4438d92a */
 290   1.5594900513e+02, /* 0x431bf2f2 */
 291  -4.9594988823e+00, /* 0xc09eb437 */
 292 };
 293
 294 static float qonef(float x)
 295 {
 296         const float *p,*q;
 297         float_t s,r,z;
 298         uint32_t ix;
 299
 300         GET_FLOAT_WORD(ix, x);
 301         ix &= 0x7fffffff;
 302         if      (ix >= 0x41000000){p = qr8; q = qs8;}
 303         else if (ix >= 0x409173eb){p = qr5; q = qs5;}
 304         else if (ix >= 0x4036d917){p = qr3; q = qs3;}
 305         else /*ix >= 0x40000000*/ {p = qr2; q = qs2;}
 306         z = 1.0f/(x*x);
 307         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 308         s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 309         return (.375f + r/s)/x;
 310 }