lib/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 <sys/cdefs.h>
  17 #if defined(LIBM_SCCS) && !defined(lint)
  18 __RCSID("$NetBSD: e_j1f.c,v 1.11 2007/08/20 16:01:38 drochner Exp $");
  19 #endif
  20
  21 #include "namespace.h"
  22 #include "math.h"
  23 #include "math_private.h"
  24
  25 static float ponef(float), qonef(float);
  26
  27 static const float
  28 huge    = 1e30,
  29 one     = 1.0,
  30 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  31 tpi      =  6.3661974669e-01, /* 0x3f22f983 */
  32         /* R0/S0 on [0,2] */
  33 r00  = -6.2500000000e-02, /* 0xbd800000 */
  34 r01  =  1.4070566976e-03, /* 0x3ab86cfd */
  35 r02  = -1.5995563444e-05, /* 0xb7862e36 */
  36 r03  =  4.9672799207e-08, /* 0x335557d2 */
  37 s01  =  1.9153760746e-02, /* 0x3c9ce859 */
  38 s02  =  1.8594678841e-04, /* 0x3942fab6 */
  39 s03  =  1.1771846857e-06, /* 0x359dffc2 */
  40 s04  =  5.0463624390e-09, /* 0x31ad6446 */
  41 s05  =  1.2354227016e-11; /* 0x2d59567e */
  42
  43 static const float zero    = 0.0;
  44
  45 float
  46 __ieee754_j1f(float x)
  47 {
  48         float z, s,c,ss,cc,r,u,v,y;
  49         int32_t hx,ix;
  50
  51         GET_FLOAT_WORD(hx,x);
  52         ix = hx&0x7fffffff;
  53         if(ix>=0x7f800000) return one/x;
  54         y = fabsf(x);
  55         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  56                 s = sinf(y);
  57                 c = cosf(y);
  58                 ss = -s-c;
  59                 cc = s-c;
  60                 if(ix<0x7f000000) {  /* make sure y+y not overflow */
  61                     z = cosf(y+y);
  62                     if ((s*c)>zero) cc = z/ss;
  63                     else            ss = z/cc;
  64                 }
  65         /*
  66          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
  67          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
  68          */
  69 #ifdef DEAD_CODE
  70                 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y);
  71                 else
  72 #endif
  73                 {
  74                     u = ponef(y); v = qonef(y);
  75                     z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
  76                 }
  77                 if(hx<0) return -z;
  78                 else     return  z;
  79         }
  80         if(ix<0x32000000) {     /* |x|<2**-27 */
  81             if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
  82         }
  83         z = x*x;
  84         r =  z*(r00+z*(r01+z*(r02+z*r03)));
  85         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
  86         r *= x;
  87         return(x*(float)0.5+r/s);
  88 }
  89
  90 static const float U0[5] = {
  91  -1.9605709612e-01, /* 0xbe48c331 */
  92   5.0443872809e-02, /* 0x3d4e9e3c */
  93  -1.9125689287e-03, /* 0xbafaaf2a */
  94   2.3525259166e-05, /* 0x37c5581c */
  95  -9.1909917899e-08, /* 0xb3c56003 */
  96 };
  97 static const float V0[5] = {
  98   1.9916731864e-02, /* 0x3ca3286a */
  99   2.0255257550e-04, /* 0x3954644b */
 100   1.3560879779e-06, /* 0x35b602d4 */
 101   6.2274145840e-09, /* 0x31d5f8eb */
 102   1.6655924903e-11, /* 0x2d9281cf */
 103 };
 104
 105 float
 106 __ieee754_y1f(float x)
 107 {
 108         float z, s,c,ss,cc,u,v;
 109         int32_t hx,ix;
 110
 111         GET_FLOAT_WORD(hx,x);
 112         ix = 0x7fffffff&hx;
 113     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
 114         if(ix>=0x7f800000) return  one/(x+x*x);
 115         if(ix==0) return -one/zero;
 116         if(hx<0) return zero/zero;
 117         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 118                 s = sinf(x);
 119                 c = cosf(x);
 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)/sqrtf(x);
 139                 else {
 140                     u = ponef(x); v = qonef(x);
 141                     z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
 142                 }
 143                 return z;
 144         }
 145         if(ix<=0x24800000) {    /* x < 2**-54 */
 146             return(-tpi/x);
 147         }
 148         z = x*x;
 149         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
 150         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
 151         return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
 152 }
 153
 154 /* For x >= 8, the asymptotic expansions of pone is
 155  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 156  * We approximate pone by
 157  *      pone(x) = 1 + (R/S)
 158  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 159  *        S = 1 + ps0*s^2 + ... + ps4*s^10
 160  * and
 161  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 162  */
 163
 164 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 165   0.0000000000e+00, /* 0x00000000 */
 166   1.1718750000e-01, /* 0x3df00000 */
 167   1.3239480972e+01, /* 0x4153d4ea */
 168   4.1205184937e+02, /* 0x43ce06a3 */
 169   3.8747453613e+03, /* 0x45722bed */
 170   7.9144794922e+03, /* 0x45f753d6 */
 171 };
 172 static const float ps8[5] = {
 173   1.1420736694e+02, /* 0x42e46a2c */
 174   3.6509309082e+03, /* 0x45642ee5 */
 175   3.6956207031e+04, /* 0x47105c35 */
 176   9.7602796875e+04, /* 0x47bea166 */
 177   3.0804271484e+04, /* 0x46f0a88b */
 178 };
 179
 180 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 181   1.3199052094e-11, /* 0x2d68333f */
 182   1.1718749255e-01, /* 0x3defffff */
 183   6.8027510643e+00, /* 0x40d9b023 */
 184   1.0830818176e+02, /* 0x42d89dca */
 185   5.1763616943e+02, /* 0x440168b7 */
 186   5.2871520996e+02, /* 0x44042dc6 */
 187 };
 188 static const float ps5[5] = {
 189   5.9280597687e+01, /* 0x426d1f55 */
 190   9.9140142822e+02, /* 0x4477d9b1 */
 191   5.3532670898e+03, /* 0x45a74a23 */
 192   7.8446904297e+03, /* 0x45f52586 */
 193   1.5040468750e+03, /* 0x44bc0180 */
 194 };
 195
 196 static const float pr3[6] = {
 197   3.0250391081e-09, /* 0x314fe10d */
 198   1.1718686670e-01, /* 0x3defffab */
 199   3.9329774380e+00, /* 0x407bb5e7 */
 200   3.5119403839e+01, /* 0x420c7a45 */
 201   9.1055007935e+01, /* 0x42b61c2a */
 202   4.8559066772e+01, /* 0x42423c7c */
 203 };
 204 static const float ps3[5] = {
 205   3.4791309357e+01, /* 0x420b2a4d */
 206   3.3676245117e+02, /* 0x43a86198 */
 207   1.0468714600e+03, /* 0x4482dbe3 */
 208   8.9081134033e+02, /* 0x445eb3ed */
 209   1.0378793335e+02, /* 0x42cf936c */
 210 };
 211
 212 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 213   1.0771083225e-07, /* 0x33e74ea8 */
 214   1.1717621982e-01, /* 0x3deffa16 */
 215   2.3685150146e+00, /* 0x401795c0 */
 216   1.2242610931e+01, /* 0x4143e1bc */
 217   1.7693971634e+01, /* 0x418d8d41 */
 218   5.0735230446e+00, /* 0x40a25a4d */
 219 };
 220 static const float ps2[5] = {
 221   2.1436485291e+01, /* 0x41ab7dec */
 222   1.2529022980e+02, /* 0x42fa9499 */
 223   2.3227647400e+02, /* 0x436846c7 */
 224   1.1767937469e+02, /* 0x42eb5bd7 */
 225   8.3646392822e+00, /* 0x4105d590 */
 226 };
 227
 228 static float
 229 ponef(float x)
 230 {
 231         const float *p,*q;
 232         float z,r,s;
 233         int32_t ix;
 234
 235         p = q = 0;
 236         GET_FLOAT_WORD(ix,x);
 237         ix &= 0x7fffffff;
 238         if(ix>=0x41000000)     {p = pr8; q= ps8;}
 239         else if(ix>=0x40f71c58){p = pr5; q= ps5;}
 240         else if(ix>=0x4036db68){p = pr3; q= ps3;}
 241         else if(ix>=0x40000000){p = pr2; q= ps2;}
 242         z = one/(x*x);
 243         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 244         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 245         return one+ r/s;
 246 }
 247
 248
 249 /* For x >= 8, the asymptotic expansions of qone is
 250  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 251  * We approximate pone by
 252  *      qone(x) = s*(0.375 + (R/S))
 253  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 254  *        S = 1 + qs1*s^2 + ... + qs6*s^12
 255  * and
 256  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 257  */
 258
 259 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 260   0.0000000000e+00, /* 0x00000000 */
 261  -1.0253906250e-01, /* 0xbdd20000 */
 262  -1.6271753311e+01, /* 0xc1822c8d */
 263  -7.5960174561e+02, /* 0xc43de683 */
 264  -1.1849806641e+04, /* 0xc639273a */
 265  -4.8438511719e+04, /* 0xc73d3683 */
 266 };
 267 static const float qs8[6] = {
 268   1.6139537048e+02, /* 0x43216537 */
 269   7.8253862305e+03, /* 0x45f48b17 */
 270   1.3387534375e+05, /* 0x4802bcd6 */
 271   7.1965775000e+05, /* 0x492fb29c */
 272   6.6660125000e+05, /* 0x4922be94 */
 273  -2.9449025000e+05, /* 0xc88fcb48 */
 274 };
 275
 276 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 277  -2.0897993405e-11, /* 0xadb7d219 */
 278  -1.0253904760e-01, /* 0xbdd1fffe */
 279  -8.0564479828e+00, /* 0xc100e736 */
 280  -1.8366960144e+02, /* 0xc337ab6b */
 281  -1.3731937256e+03, /* 0xc4aba633 */
 282  -2.6124443359e+03, /* 0xc523471c */
 283 };
 284 static const float qs5[6] = {
 285   8.1276550293e+01, /* 0x42a28d98 */
 286   1.9917987061e+03, /* 0x44f8f98f */
 287   1.7468484375e+04, /* 0x468878f8 */
 288   4.9851425781e+04, /* 0x4742bb6d */
 289   2.7948074219e+04, /* 0x46da5826 */
 290  -4.7191835938e+03, /* 0xc5937978 */
 291 };
 292
 293 static const float qr3[6] = { /* for x in [4.5454,2.8570]=1/[0.22001,0.3499] */
 294  -5.0783124372e-09, /* 0xb1ae7d4f */
 295  -1.0253783315e-01, /* 0xbdd1ff5b */
 296  -4.6101160049e+00, /* 0xc0938612 */
 297  -5.7847221375e+01, /* 0xc267638e */
 298  -2.2824453735e+02, /* 0xc3643e9a */
 299  -2.1921012878e+02, /* 0xc35b35cb */
 300 };
 301 static const float qs3[6] = {
 302   4.7665153503e+01, /* 0x423ea91e */
 303   6.7386511230e+02, /* 0x4428775e */
 304   3.3801528320e+03, /* 0x45534272 */
 305   5.5477290039e+03, /* 0x45ad5dd5 */
 306   1.9031191406e+03, /* 0x44ede3d0 */
 307  -1.3520118713e+02, /* 0xc3073381 */
 308 };
 309
 310 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 311  -1.7838172539e-07, /* 0xb43f8932 */
 312  -1.0251704603e-01, /* 0xbdd1f475 */
 313  -2.7522056103e+00, /* 0xc0302423 */
 314  -1.9663616180e+01, /* 0xc19d4f16 */
 315  -4.2325313568e+01, /* 0xc2294d1f */
 316  -2.1371921539e+01, /* 0xc1aaf9b2 */
 317 };
 318 static const float qs2[6] = {
 319   2.9533363342e+01, /* 0x41ec4454 */
 320   2.5298155212e+02, /* 0x437cfb47 */
 321   7.5750280762e+02, /* 0x443d602e */
 322   7.3939318848e+02, /* 0x4438d92a */
 323   1.5594900513e+02, /* 0x431bf2f2 */
 324  -4.9594988823e+00, /* 0xc09eb437 */
 325 };
 326
 327 static float
 328 qonef(float x)
 329 {
 330         const float *p,*q;
 331         float  s,r,z;
 332         int32_t ix;
 333
 334         p = q = 0;
 335         GET_FLOAT_WORD(ix,x);
 336         ix &= 0x7fffffff;
 337         /* [inf, 8]             (8      41000000) */
 338         if(ix>=0x41000000)     {p = qr8; q= qs8;}
 339         /* [8, 4.5454]          (4.5454 409173eb) */
 340         else if(ix>=0x409173eb){p = qr5; q= qs5;}
 341         /* [4.5454, 2.8570]     (2.8570 4036d917) */
 342         else if(ix>=0x4036d917){p = qr3; q= qs3;}
 343         /* [2.8570, 2]          (2      40000000) */
 344         else if(ix>=0x40000000){p = qr2; q= qs2;}
 345         z = one/(x*x);
 346         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 347         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 348         return ((float).375 + r/s)/x;
 349 }