lib/libm/src/e_j0f.c

   1 /* e_j0f.c -- float version of e_j0.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  * $NetBSD: e_j0f.c,v 1.7 2002/05/26 22:01:50 wiz Exp $
  16  * $DragonFly: src/lib/libm/src/e_j0f.c,v 1.2 2007/06/17 00:44:00 pavalos Exp $
  17  */
  18
  19 #include <math.h>
  20 #include "math_private.h"
  21
  22 static float pzerof(float), qzerof(float);
  23
  24 static const float
  25 huge    = 1e30,
  26 one     = 1.0,
  27 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  28 tpi      =  6.3661974669e-01, /* 0x3f22f983 */
  29                 /* R0/S0 on [0, 2.00] */
  30 R02  =  1.5625000000e-02, /* 0x3c800000 */
  31 R03  = -1.8997929874e-04, /* 0xb947352e */
  32 R04  =  1.8295404516e-06, /* 0x35f58e88 */
  33 R05  = -4.6183270541e-09, /* 0xb19eaf3c */
  34 S01  =  1.5619102865e-02, /* 0x3c7fe744 */
  35 S02  =  1.1692678527e-04, /* 0x38f53697 */
  36 S03  =  5.1354652442e-07, /* 0x3509daa6 */
  37 S04  =  1.1661400734e-09; /* 0x30a045e8 */
  38
  39 static const float zero = 0.0;
  40
  41 float
  42 j0f(float x)
  43 {
  44         float z, s,c,ss,cc,r,u,v;
  45         int32_t hx,ix;
  46
  47         GET_FLOAT_WORD(hx,x);
  48         ix = hx&0x7fffffff;
  49         if(ix>=0x7f800000) return one/(x*x);
  50         x = fabsf(x);
  51         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  52                 s = sinf(x);
  53                 c = cosf(x);
  54                 ss = s-c;
  55                 cc = s+c;
  56                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
  57                     z = -cosf(x+x);
  58                     if ((s*c)<zero) cc = z/ss;
  59                     else            ss = z/cc;
  60                 }
  61         /*
  62          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
  63          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
  64          */
  65 #ifdef DEAD_CODE
  66                 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x);
  67                 else
  68 #endif
  69                 {
  70                     u = pzerof(x); v = qzerof(x);
  71                     z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
  72                 }
  73                 return z;
  74         }
  75         if(ix<0x39000000) {     /* |x| < 2**-13 */
  76             if(huge+x>one) {    /* raise inexact if x != 0 */
  77                 if(ix<0x32000000) return one;   /* |x|<2**-27 */
  78                 else          return one - (float)0.25*x*x;
  79             }
  80         }
  81         z = x*x;
  82         r =  z*(R02+z*(R03+z*(R04+z*R05)));
  83         s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
  84         if(ix < 0x3F800000) {   /* |x| < 1.00 */
  85             return one + z*((float)-0.25+(r/s));
  86         } else {
  87             u = (float)0.5*x;
  88             return((one+u)*(one-u)+z*(r/s));
  89         }
  90 }
  91
  92 static const float
  93 u00  = -7.3804296553e-02, /* 0xbd9726b5 */
  94 u01  =  1.7666645348e-01, /* 0x3e34e80d */
  95 u02  = -1.3818567619e-02, /* 0xbc626746 */
  96 u03  =  3.4745343146e-04, /* 0x39b62a69 */
  97 u04  = -3.8140706238e-06, /* 0xb67ff53c */
  98 u05  =  1.9559013964e-08, /* 0x32a802ba */
  99 u06  = -3.9820518410e-11, /* 0xae2f21eb */
 100 v01  =  1.2730483897e-02, /* 0x3c509385 */
 101 v02  =  7.6006865129e-05, /* 0x389f65e0 */
 102 v03  =  2.5915085189e-07, /* 0x348b216c */
 103 v04  =  4.4111031494e-10; /* 0x2ff280c2 */
 104
 105 float
 106 y0f(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     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(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         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
 119          * where x0 = x-pi/4
 120          *      Better formula:
 121          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/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) * (sin(x) - cos(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                 s = sinf(x);
 130                 c = cosf(x);
 131                 ss = s-c;
 132                 cc = s+c;
 133         /*
 134          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 135          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 136          */
 137                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
 138                     z = -cosf(x+x);
 139                     if ((s*c)<zero) cc = z/ss;
 140                     else            ss = z/cc;
 141                 }
 142 #ifdef DEAD_CODE
 143                 if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x);
 144                 else
 145 #endif
 146                 {
 147                     u = pzerof(x); v = qzerof(x);
 148                     z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
 149                 }
 150                 return z;
 151         }
 152         if(ix<=0x32000000) {    /* x < 2**-27 */
 153             return(u00 + tpi*logf(x));
 154         }
 155         z = x*x;
 156         u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
 157         v = one+z*(v01+z*(v02+z*(v03+z*v04)));
 158         return(u/v + tpi*(j0f(x)*logf(x)));
 159 }
 160
 161 /* The asymptotic expansions of pzero is
 162  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
 163  * For x >= 2, We approximate pzero by
 164  *      pzero(x) = 1 + (R/S)
 165  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
 166  *        S = 1 + pS0*s^2 + ... + pS4*s^10
 167  * and
 168  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
 169  */
 170 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 171   0.0000000000e+00, /* 0x00000000 */
 172  -7.0312500000e-02, /* 0xbd900000 */
 173  -8.0816707611e+00, /* 0xc1014e86 */
 174  -2.5706311035e+02, /* 0xc3808814 */
 175  -2.4852163086e+03, /* 0xc51b5376 */
 176  -5.2530439453e+03, /* 0xc5a4285a */
 177 };
 178 static const float pS8[5] = {
 179   1.1653436279e+02, /* 0x42e91198 */
 180   3.8337448730e+03, /* 0x456f9beb */
 181   4.0597855469e+04, /* 0x471e95db */
 182   1.1675296875e+05, /* 0x47e4087c */
 183   4.7627726562e+04, /* 0x473a0bba */
 184 };
 185 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 186  -1.1412546255e-11, /* 0xad48c58a */
 187  -7.0312492549e-02, /* 0xbd8fffff */
 188  -4.1596107483e+00, /* 0xc0851b88 */
 189  -6.7674766541e+01, /* 0xc287597b */
 190  -3.3123129272e+02, /* 0xc3a59d9b */
 191  -3.4643338013e+02, /* 0xc3ad3779 */
 192 };
 193 static const float pS5[5] = {
 194   6.0753936768e+01, /* 0x42730408 */
 195   1.0512523193e+03, /* 0x44836813 */
 196   5.9789707031e+03, /* 0x45bad7c4 */
 197   9.6254453125e+03, /* 0x461665c8 */
 198   2.4060581055e+03, /* 0x451660ee */
 199 };
 200
 201 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 202  -2.5470459075e-09, /* 0xb12f081b */
 203  -7.0311963558e-02, /* 0xbd8fffb8 */
 204  -2.4090321064e+00, /* 0xc01a2d95 */
 205  -2.1965976715e+01, /* 0xc1afba52 */
 206  -5.8079170227e+01, /* 0xc2685112 */
 207  -3.1447946548e+01, /* 0xc1fb9565 */
 208 };
 209 static const float pS3[5] = {
 210   3.5856033325e+01, /* 0x420f6c94 */
 211   3.6151397705e+02, /* 0x43b4c1ca */
 212   1.1936077881e+03, /* 0x44953373 */
 213   1.1279968262e+03, /* 0x448cffe6 */
 214   1.7358093262e+02, /* 0x432d94b8 */
 215 };
 216
 217 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 218  -8.8753431271e-08, /* 0xb3be98b7 */
 219  -7.0303097367e-02, /* 0xbd8ffb12 */
 220  -1.4507384300e+00, /* 0xbfb9b1cc */
 221  -7.6356959343e+00, /* 0xc0f4579f */
 222  -1.1193166733e+01, /* 0xc1331736 */
 223  -3.2336456776e+00, /* 0xc04ef40d */
 224 };
 225 static const float pS2[5] = {
 226   2.2220300674e+01, /* 0x41b1c32d */
 227   1.3620678711e+02, /* 0x430834f0 */
 228   2.7047027588e+02, /* 0x43873c32 */
 229   1.5387539673e+02, /* 0x4319e01a */
 230   1.4657617569e+01, /* 0x416a859a */
 231 };
 232
 233 static float
 234 pzerof(float x)
 235 {
 236         const float *p,*q;
 237         float z,r,s;
 238         int32_t ix;
 239
 240         p = q = 0;
 241         GET_FLOAT_WORD(ix,x);
 242         ix &= 0x7fffffff;
 243         if(ix>=0x41000000)     {p = pR8; q= pS8;}
 244         else if(ix>=0x40f71c58){p = pR5; q= pS5;}
 245         else if(ix>=0x4036db68){p = pR3; q= pS3;}
 246         else if(ix>=0x40000000){p = pR2; q= pS2;}
 247         z = one/(x*x);
 248         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 249         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 250         return one+ r/s;
 251 }
 252
 253
 254 /* For x >= 8, the asymptotic expansions of qzero is
 255  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 256  * We approximate pzero by
 257  *      qzero(x) = s*(-1.25 + (R/S))
 258  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
 259  *        S = 1 + qS0*s^2 + ... + qS5*s^12
 260  * and
 261  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 262  */
 263 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 264   0.0000000000e+00, /* 0x00000000 */
 265   7.3242187500e-02, /* 0x3d960000 */
 266   1.1768206596e+01, /* 0x413c4a93 */
 267   5.5767340088e+02, /* 0x440b6b19 */
 268   8.8591972656e+03, /* 0x460a6cca */
 269   3.7014625000e+04, /* 0x471096a0 */
 270 };
 271 static const float qS8[6] = {
 272   1.6377603149e+02, /* 0x4323c6aa */
 273   8.0983447266e+03, /* 0x45fd12c2 */
 274   1.4253829688e+05, /* 0x480b3293 */
 275   8.0330925000e+05, /* 0x49441ed4 */
 276   8.4050156250e+05, /* 0x494d3359 */
 277  -3.4389928125e+05, /* 0xc8a7eb69 */
 278 };
 279
 280 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 281   1.8408595828e-11, /* 0x2da1ec79 */
 282   7.3242180049e-02, /* 0x3d95ffff */
 283   5.8356351852e+00, /* 0x40babd86 */
 284   1.3511157227e+02, /* 0x43071c90 */
 285   1.0272437744e+03, /* 0x448067cd */
 286   1.9899779053e+03, /* 0x44f8bf4b */
 287 };
 288 static const float qS5[6] = {
 289   8.2776611328e+01, /* 0x42a58da0 */
 290   2.0778142090e+03, /* 0x4501dd07 */
 291   1.8847289062e+04, /* 0x46933e94 */
 292   5.6751113281e+04, /* 0x475daf1d */
 293   3.5976753906e+04, /* 0x470c88c1 */
 294  -5.3543427734e+03, /* 0xc5a752be */
 295 };
 296
 297 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 298   4.3774099900e-09, /* 0x3196681b */
 299   7.3241114616e-02, /* 0x3d95ff70 */
 300   3.3442313671e+00, /* 0x405607e3 */
 301   4.2621845245e+01, /* 0x422a7cc5 */
 302   1.7080809021e+02, /* 0x432acedf */
 303   1.6673394775e+02, /* 0x4326bbe4 */
 304 };
 305 static const float qS3[6] = {
 306   4.8758872986e+01, /* 0x42430916 */
 307   7.0968920898e+02, /* 0x44316c1c */
 308   3.7041481934e+03, /* 0x4567825f */
 309   6.4604252930e+03, /* 0x45c9e367 */
 310   2.5163337402e+03, /* 0x451d4557 */
 311  -1.4924745178e+02, /* 0xc3153f59 */
 312 };
 313
 314 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 315   1.5044444979e-07, /* 0x342189db */
 316   7.3223426938e-02, /* 0x3d95f62a */
 317   1.9981917143e+00, /* 0x3fffc4bf */
 318   1.4495602608e+01, /* 0x4167edfd */
 319   3.1666231155e+01, /* 0x41fd5471 */
 320   1.6252708435e+01, /* 0x4182058c */
 321 };
 322 static const float qS2[6] = {
 323   3.0365585327e+01, /* 0x41f2ecb8 */
 324   2.6934811401e+02, /* 0x4386ac8f */
 325   8.4478375244e+02, /* 0x44533229 */
 326   8.8293585205e+02, /* 0x445cbbe5 */
 327   2.1266638184e+02, /* 0x4354aa98 */
 328  -5.3109550476e+00, /* 0xc0a9f358 */
 329 };
 330
 331 static float
 332 qzerof(float x)
 333 {
 334         const float *p,*q;
 335         float s,r,z;
 336         int32_t ix;
 337
 338         p = q = 0;
 339         GET_FLOAT_WORD(ix,x);
 340         ix &= 0x7fffffff;
 341         if(ix>=0x41000000)     {p = qR8; q= qS8;}
 342         else if(ix>=0x40f71c58){p = qR5; q= qS5;}
 343         else if(ix>=0x4036db68){p = qR3; q= qS3;}
 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).125 + r/s)/x;
 349 }