sysdeps/ieee754/flt-32/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
  16 #include <math.h>
  17 #include <math_private.h>
  18
  19 static float pzerof(float), qzerof(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, 2.00] */
  27 R02  =  1.5625000000e-02, /* 0x3c800000 */
  28 R03  = -1.8997929874e-04, /* 0xb947352e */
  29 R04  =  1.8295404516e-06, /* 0x35f58e88 */
  30 R05  = -4.6183270541e-09, /* 0xb19eaf3c */
  31 S01  =  1.5619102865e-02, /* 0x3c7fe744 */
  32 S02  =  1.1692678527e-04, /* 0x38f53697 */
  33 S03  =  5.1354652442e-07, /* 0x3509daa6 */
  34 S04  =  1.1661400734e-09; /* 0x30a045e8 */
  35
  36 static const float zero = 0.0;
  37
  38 float
  39 __ieee754_j0f(float x)
  40 {
  41         float z, s,c,ss,cc,r,u,v;
  42         int32_t hx,ix;
  43
  44         GET_FLOAT_WORD(hx,x);
  45         ix = hx&0x7fffffff;
  46         if(ix>=0x7f800000) return one/(x*x);
  47         x = fabsf(x);
  48         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  49                 __sincosf (x, &s, &c);
  50                 ss = s-c;
  51                 cc = s+c;
  52                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
  53                     z = -__cosf(x+x);
  54                     if ((s*c)<zero) cc = z/ss;
  55                     else            ss = z/cc;
  56                 }
  57         /*
  58          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
  59          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
  60          */
  61                 if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(x);
  62                 else {
  63                     u = pzerof(x); v = qzerof(x);
  64                     z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(x);
  65                 }
  66                 return z;
  67         }
  68         if(ix<0x39000000) {     /* |x| < 2**-13 */
  69             math_force_eval(huge+x);            /* raise inexact if x != 0 */
  70             if(ix<0x32000000) return one;       /* |x|<2**-27 */
  71             else              return one - (float)0.25*x*x;
  72         }
  73         z = x*x;
  74         r =  z*(R02+z*(R03+z*(R04+z*R05)));
  75         s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
  76         if(ix < 0x3F800000) {   /* |x| < 1.00 */
  77             return one + z*((float)-0.25+(r/s));
  78         } else {
  79             u = (float)0.5*x;
  80             return((one+u)*(one-u)+z*(r/s));
  81         }
  82 }
  83 strong_alias (__ieee754_j0f, __j0f_finite)
  84
  85 static const float
  86 u00  = -7.3804296553e-02, /* 0xbd9726b5 */
  87 u01  =  1.7666645348e-01, /* 0x3e34e80d */
  88 u02  = -1.3818567619e-02, /* 0xbc626746 */
  89 u03  =  3.4745343146e-04, /* 0x39b62a69 */
  90 u04  = -3.8140706238e-06, /* 0xb67ff53c */
  91 u05  =  1.9559013964e-08, /* 0x32a802ba */
  92 u06  = -3.9820518410e-11, /* 0xae2f21eb */
  93 v01  =  1.2730483897e-02, /* 0x3c509385 */
  94 v02  =  7.6006865129e-05, /* 0x389f65e0 */
  95 v03  =  2.5915085189e-07, /* 0x348b216c */
  96 v04  =  4.4111031494e-10; /* 0x2ff280c2 */
  97
  98 float
  99 __ieee754_y0f(float x)
 100 {
 101         float z, s,c,ss,cc,u,v;
 102         int32_t hx,ix;
 103
 104         GET_FLOAT_WORD(hx,x);
 105         ix = 0x7fffffff&hx;
 106     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf.  */
 107         if(ix>=0x7f800000) return  one/(x+x*x);
 108         if(ix==0) return -HUGE_VALF+x;  /* -inf and overflow exception.  */
 109         if(hx<0) return zero/(zero*x);
 110         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 111         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
 112          * where x0 = x-pi/4
 113          *      Better formula:
 114          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 115          *                      =  1/sqrt(2) * (sin(x) + cos(x))
 116          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 117          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 118          * To avoid cancellation, use
 119          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 120          * to compute the worse one.
 121          */
 122                 __sincosf (x, &s, &c);
 123                 ss = s-c;
 124                 cc = s+c;
 125         /*
 126          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 127          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 128          */
 129                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
 130                     z = -__cosf(x+x);
 131                     if ((s*c)<zero) cc = z/ss;
 132                     else            ss = z/cc;
 133                 }
 134                 if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
 135                 else {
 136                     u = pzerof(x); v = qzerof(x);
 137                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x);
 138                 }
 139                 return z;
 140         }
 141         if(ix<=0x39800000) {    /* x < 2**-13 */
 142             return(u00 + tpi*__ieee754_logf(x));
 143         }
 144         z = x*x;
 145         u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
 146         v = one+z*(v01+z*(v02+z*(v03+z*v04)));
 147         return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
 148 }
 149 strong_alias (__ieee754_y0f, __y0f_finite)
 150
 151 /* The asymptotic expansions of pzero is
 152  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
 153  * For x >= 2, We approximate pzero by
 154  *      pzero(x) = 1 + (R/S)
 155  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
 156  *        S = 1 + pS0*s^2 + ... + pS4*s^10
 157  * and
 158  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
 159  */
 160 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 161   0.0000000000e+00, /* 0x00000000 */
 162  -7.0312500000e-02, /* 0xbd900000 */
 163  -8.0816707611e+00, /* 0xc1014e86 */
 164  -2.5706311035e+02, /* 0xc3808814 */
 165  -2.4852163086e+03, /* 0xc51b5376 */
 166  -5.2530439453e+03, /* 0xc5a4285a */
 167 };
 168 static const float pS8[5] = {
 169   1.1653436279e+02, /* 0x42e91198 */
 170   3.8337448730e+03, /* 0x456f9beb */
 171   4.0597855469e+04, /* 0x471e95db */
 172   1.1675296875e+05, /* 0x47e4087c */
 173   4.7627726562e+04, /* 0x473a0bba */
 174 };
 175 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 176  -1.1412546255e-11, /* 0xad48c58a */
 177  -7.0312492549e-02, /* 0xbd8fffff */
 178  -4.1596107483e+00, /* 0xc0851b88 */
 179  -6.7674766541e+01, /* 0xc287597b */
 180  -3.3123129272e+02, /* 0xc3a59d9b */
 181  -3.4643338013e+02, /* 0xc3ad3779 */
 182 };
 183 static const float pS5[5] = {
 184   6.0753936768e+01, /* 0x42730408 */
 185   1.0512523193e+03, /* 0x44836813 */
 186   5.9789707031e+03, /* 0x45bad7c4 */
 187   9.6254453125e+03, /* 0x461665c8 */
 188   2.4060581055e+03, /* 0x451660ee */
 189 };
 190
 191 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 192  -2.5470459075e-09, /* 0xb12f081b */
 193  -7.0311963558e-02, /* 0xbd8fffb8 */
 194  -2.4090321064e+00, /* 0xc01a2d95 */
 195  -2.1965976715e+01, /* 0xc1afba52 */
 196  -5.8079170227e+01, /* 0xc2685112 */
 197  -3.1447946548e+01, /* 0xc1fb9565 */
 198 };
 199 static const float pS3[5] = {
 200   3.5856033325e+01, /* 0x420f6c94 */
 201   3.6151397705e+02, /* 0x43b4c1ca */
 202   1.1936077881e+03, /* 0x44953373 */
 203   1.1279968262e+03, /* 0x448cffe6 */
 204   1.7358093262e+02, /* 0x432d94b8 */
 205 };
 206
 207 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 208  -8.8753431271e-08, /* 0xb3be98b7 */
 209  -7.0303097367e-02, /* 0xbd8ffb12 */
 210  -1.4507384300e+00, /* 0xbfb9b1cc */
 211  -7.6356959343e+00, /* 0xc0f4579f */
 212  -1.1193166733e+01, /* 0xc1331736 */
 213  -3.2336456776e+00, /* 0xc04ef40d */
 214 };
 215 static const float pS2[5] = {
 216   2.2220300674e+01, /* 0x41b1c32d */
 217   1.3620678711e+02, /* 0x430834f0 */
 218   2.7047027588e+02, /* 0x43873c32 */
 219   1.5387539673e+02, /* 0x4319e01a */
 220   1.4657617569e+01, /* 0x416a859a */
 221 };
 222
 223 static float
 224 pzerof(float x)
 225 {
 226         const float *p,*q;
 227         float z,r,s;
 228         int32_t ix;
 229         GET_FLOAT_WORD(ix,x);
 230         ix &= 0x7fffffff;
 231         /* ix >= 0x40000000 for all calls to this function.  */
 232         if(ix>=0x41000000)     {p = pR8; q= pS8;}
 233         else if(ix>=0x40f71c58){p = pR5; q= pS5;}
 234         else if(ix>=0x4036db68){p = pR3; q= pS3;}
 235         else {p = pR2; q= pS2;}
 236         z = one/(x*x);
 237         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 238         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 239         return one+ r/s;
 240 }
 241
 242
 243 /* For x >= 8, the asymptotic expansions of qzero is
 244  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 245  * We approximate pzero by
 246  *      qzero(x) = s*(-1.25 + (R/S))
 247  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
 248  *        S = 1 + qS0*s^2 + ... + qS5*s^12
 249  * and
 250  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 251  */
 252 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 253   0.0000000000e+00, /* 0x00000000 */
 254   7.3242187500e-02, /* 0x3d960000 */
 255   1.1768206596e+01, /* 0x413c4a93 */
 256   5.5767340088e+02, /* 0x440b6b19 */
 257   8.8591972656e+03, /* 0x460a6cca */
 258   3.7014625000e+04, /* 0x471096a0 */
 259 };
 260 static const float qS8[6] = {
 261   1.6377603149e+02, /* 0x4323c6aa */
 262   8.0983447266e+03, /* 0x45fd12c2 */
 263   1.4253829688e+05, /* 0x480b3293 */
 264   8.0330925000e+05, /* 0x49441ed4 */
 265   8.4050156250e+05, /* 0x494d3359 */
 266  -3.4389928125e+05, /* 0xc8a7eb69 */
 267 };
 268
 269 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 270   1.8408595828e-11, /* 0x2da1ec79 */
 271   7.3242180049e-02, /* 0x3d95ffff */
 272   5.8356351852e+00, /* 0x40babd86 */
 273   1.3511157227e+02, /* 0x43071c90 */
 274   1.0272437744e+03, /* 0x448067cd */
 275   1.9899779053e+03, /* 0x44f8bf4b */
 276 };
 277 static const float qS5[6] = {
 278   8.2776611328e+01, /* 0x42a58da0 */
 279   2.0778142090e+03, /* 0x4501dd07 */
 280   1.8847289062e+04, /* 0x46933e94 */
 281   5.6751113281e+04, /* 0x475daf1d */
 282   3.5976753906e+04, /* 0x470c88c1 */
 283  -5.3543427734e+03, /* 0xc5a752be */
 284 };
 285
 286 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 287   4.3774099900e-09, /* 0x3196681b */
 288   7.3241114616e-02, /* 0x3d95ff70 */
 289   3.3442313671e+00, /* 0x405607e3 */
 290   4.2621845245e+01, /* 0x422a7cc5 */
 291   1.7080809021e+02, /* 0x432acedf */
 292   1.6673394775e+02, /* 0x4326bbe4 */
 293 };
 294 static const float qS3[6] = {
 295   4.8758872986e+01, /* 0x42430916 */
 296   7.0968920898e+02, /* 0x44316c1c */
 297   3.7041481934e+03, /* 0x4567825f */
 298   6.4604252930e+03, /* 0x45c9e367 */
 299   2.5163337402e+03, /* 0x451d4557 */
 300  -1.4924745178e+02, /* 0xc3153f59 */
 301 };
 302
 303 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 304   1.5044444979e-07, /* 0x342189db */
 305   7.3223426938e-02, /* 0x3d95f62a */
 306   1.9981917143e+00, /* 0x3fffc4bf */
 307   1.4495602608e+01, /* 0x4167edfd */
 308   3.1666231155e+01, /* 0x41fd5471 */
 309   1.6252708435e+01, /* 0x4182058c */
 310 };
 311 static const float qS2[6] = {
 312   3.0365585327e+01, /* 0x41f2ecb8 */
 313   2.6934811401e+02, /* 0x4386ac8f */
 314   8.4478375244e+02, /* 0x44533229 */
 315   8.8293585205e+02, /* 0x445cbbe5 */
 316   2.1266638184e+02, /* 0x4354aa98 */
 317  -5.3109550476e+00, /* 0xc0a9f358 */
 318 };
 319
 320 static float
 321 qzerof(float x)
 322 {
 323         const float *p,*q;
 324         float s,r,z;
 325         int32_t ix;
 326         GET_FLOAT_WORD(ix,x);
 327         ix &= 0x7fffffff;
 328         /* ix >= 0x40000000 for all calls to this function.  */
 329         if(ix>=0x41000000)     {p = qR8; q= qS8;}
 330         else if(ix>=0x40f71c58){p = qR5; q= qS5;}
 331         else if(ix>=0x4036db68){p = qR3; q= qS3;}
 332         else {p = qR2; q= qS2;}
 333         z = one/(x*x);
 334         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 335         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 336         return (-(float).125 + r/s)/x;
 337 }