compiler/stdc/math/ld128/e_powl.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /*
  13  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
  14  *
  15  * Permission to use, copy, modify, and distribute this software for any
  16  * purpose with or without fee is hereby granted, provided that the above
  17  * copyright notice and this permission notice appear in all copies.
  18  *
  19  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  20  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  21  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  22  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  23  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  24  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  25  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  26  */
  27
  28 /* powl(x,y) return x**y
  29  *
  30  *                    n
  31  * Method:  Let x =  2   * (1+f)
  32  *      1. Compute and return log2(x) in two pieces:
  33  *              log2(x) = w1 + w2,
  34  *         where w1 has 113-53 = 60 bit trailing zeros.
  35  *      2. Perform y*log2(x) = n+y' by simulating muti-precision
  36  *         arithmetic, where |y'|<=0.5.
  37  *      3. Return x**y = 2**n*exp(y'*log2)
  38  *
  39  * Special cases:
  40  *      1.  (anything) ** 0  is 1
  41  *      2.  (anything) ** 1  is itself
  42  *      3.  (anything) ** NAN is NAN
  43  *      4.  NAN ** (anything except 0) is NAN
  44  *      5.  +-(|x| > 1) **  +INF is +INF
  45  *      6.  +-(|x| > 1) **  -INF is +0
  46  *      7.  +-(|x| < 1) **  +INF is +0
  47  *      8.  +-(|x| < 1) **  -INF is +INF
  48  *      9.  +-1         ** +-INF is NAN
  49  *      10. +0 ** (+anything except 0, NAN)               is +0
  50  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
  51  *      12. +0 ** (-anything except 0, NAN)               is +INF
  52  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
  53  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
  54  *      15. +INF ** (+anything except 0,NAN) is +INF
  55  *      16. +INF ** (-anything except 0,NAN) is +0
  56  *      17. -INF ** (anything)  = -0 ** (-anything)
  57  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  58  *      19. (-anything except 0 and inf) ** (non-integer) is NAN
  59  *
  60  */
  61
  62 #include "math.h"
  63
  64 #include "math_private.h"
  65
  66 static const long double bp[] = {
  67   1.0L,
  68   1.5L,
  69 };
  70
  71 /* log_2(1.5) */
  72 static const long double dp_h[] = {
  73   0.0,
  74   5.8496250072115607565592654282227158546448E-1L
  75 };
  76
  77 /* Low part of log_2(1.5) */
  78 static const long double dp_l[] = {
  79   0.0,
  80   1.0579781240112554492329533686862998106046E-16L
  81 };
  82
  83 static const long double zero = 0.0L,
  84   one = 1.0L,
  85   two = 2.0L,
  86   two113 = 1.0384593717069655257060992658440192E34L,
  87   huge = 1.0e3000L,
  88   tiny = 1.0e-3000L;
  89
  90 /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2))
  91    z = (x-1)/(x+1)
  92    1 <= x <= 1.25
  93    Peak relative error 2.3e-37 */
  94 static const long double LN[] =
  95 {
  96  -3.0779177200290054398792536829702930623200E1L,
  97   6.5135778082209159921251824580292116201640E1L,
  98  -4.6312921812152436921591152809994014413540E1L,
  99   1.2510208195629420304615674658258363295208E1L,
 100  -9.9266909031921425609179910128531667336670E-1L
 101 };
 102 static const long double LD[] =
 103 {
 104  -5.129862866715009066465422805058933131960E1L,
 105   1.452015077564081884387441590064272782044E2L,
 106  -1.524043275549860505277434040464085593165E2L,
 107   7.236063513651544224319663428634139768808E1L,
 108  -1.494198912340228235853027849917095580053E1L
 109   /* 1.0E0 */
 110 };
 111
 112 /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2)))
 113    0 <= x <= 0.5
 114    Peak relative error 5.7e-38  */
 115 static const long double PN[] =
 116 {
 117   5.081801691915377692446852383385968225675E8L,
 118   9.360895299872484512023336636427675327355E6L,
 119   4.213701282274196030811629773097579432957E4L,
 120   5.201006511142748908655720086041570288182E1L,
 121   9.088368420359444263703202925095675982530E-3L,
 122 };
 123 static const long double PD[] =
 124 {
 125   3.049081015149226615468111430031590411682E9L,
 126   1.069833887183886839966085436512368982758E8L,
 127   8.259257717868875207333991924545445705394E5L,
 128   1.872583833284143212651746812884298360922E3L,
 129   /* 1.0E0 */
 130 };
 131
 132 static const long double
 133   /* ln 2 */
 134   lg2 = 6.9314718055994530941723212145817656807550E-1L,
 135   lg2_h = 6.9314718055994528622676398299518041312695E-1L,
 136   lg2_l = 2.3190468138462996154948554638754786504121E-17L,
 137   ovt = 8.0085662595372944372e-0017L,
 138   /* 2/(3*log(2)) */
 139   cp = 9.6179669392597560490661645400126142495110E-1L,
 140   cp_h = 9.6179669392597555432899980587535537779331E-1L,
 141   cp_l = 5.0577616648125906047157785230014751039424E-17L;
 142
 143 long double
 144 powl(long double x, long double y)
 145 {
 146   long double z, ax, z_h, z_l, p_h, p_l;
 147   long double yy1, t1, t2, r, s, t, u, v, w;
 148   long double s2, s_h, s_l, t_h, t_l;
 149   int32_t i, j, k, yisint, n;
 150   uint32_t ix, iy;
 151   int32_t hx, hy;
 152   ieee_quad_shape_type o, p, q;
 153
 154   p.value = x;
 155   hx = p.parts32.mswhi;
 156   ix = hx & 0x7fffffff;
 157
 158   q.value = y;
 159   hy = q.parts32.mswhi;
 160   iy = hy & 0x7fffffff;
 161
 162
 163   /* y==zero: x**0 = 1 */
 164   if ((iy | q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
 165     return one;
 166
 167   /* 1.0**y = 1; -1.0**+-Inf = 1 */
 168   if (x == one)
 169     return one;
 170   if (x == -1.0L && iy == 0x7fff0000
 171       && (q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
 172     return one;
 173
 174   /* +-NaN return x+y */
 175   if ((ix > 0x7fff0000)
 176       || ((ix == 0x7fff0000)
 177           && ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) != 0))
 178       || (iy > 0x7fff0000)
 179       || ((iy == 0x7fff0000)
 180           && ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) != 0)))
 181     return x + y;
 182
 183   /* determine if y is an odd int when x < 0
 184    * yisint = 0       ... y is not an integer
 185    * yisint = 1       ... y is an odd int
 186    * yisint = 2       ... y is an even int
 187    */
 188   yisint = 0;
 189   if (hx < 0)
 190     {
 191       if (iy >= 0x40700000)     /* 2^113 */
 192         yisint = 2;             /* even integer y */
 193       else if (iy >= 0x3fff0000)        /* 1.0 */
 194         {
 195           if (floorl (y) == y)
 196             {
 197               z = 0.5 * y;
 198               if (floorl (z) == z)
 199                 yisint = 2;
 200               else
 201                 yisint = 1;
 202             }
 203         }
 204     }
 205
 206   /* special value of y */
 207   if ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
 208     {
 209       if (iy == 0x7fff0000)     /* y is +-inf */
 210         {
 211           if (((ix - 0x3fff0000) | p.parts32.mswlo | p.parts32.lswhi |
 212             p.parts32.lswlo) == 0)
 213             return y - y;       /* +-1**inf is NaN */
 214           else if (ix >= 0x3fff0000)    /* (|x|>1)**+-inf = inf,0 */
 215             return (hy >= 0) ? y : zero;
 216           else                  /* (|x|<1)**-,+inf = inf,0 */
 217             return (hy < 0) ? -y : zero;
 218         }
 219       if (iy == 0x3fff0000)
 220         {                       /* y is  +-1 */
 221           if (hy < 0)
 222             return one / x;
 223           else
 224             return x;
 225         }
 226       if (hy == 0x40000000)
 227         return x * x;           /* y is  2 */
 228       if (hy == 0x3ffe0000)
 229         {                       /* y is  0.5 */
 230           if (hx >= 0)          /* x >= +0 */
 231             return sqrtl (x);
 232         }
 233     }
 234
 235   ax = fabsl (x);
 236   /* special value of x */
 237   if ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0)
 238     {
 239       if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000)
 240         {
 241           z = ax;               /*x is +-0,+-inf,+-1 */
 242           if (hy < 0)
 243             z = one / z;        /* z = (1/|x|) */
 244           if (hx < 0)
 245             {
 246               if (((ix - 0x3fff0000) | yisint) == 0)
 247                 {
 248                   z = (z - z) / (z - z);        /* (-1)**non-int is NaN */
 249                 }
 250               else if (yisint == 1)
 251                 z = -z;         /* (x<0)**odd = -(|x|**odd) */
 252             }
 253           return z;
 254         }
 255     }
 256
 257   /* (x<0)**(non-int) is NaN */
 258   if (((((uint32_t) hx >> 31) - 1) | yisint) == 0)
 259     return (x - x) / (x - x);
 260
 261   /* |y| is huge.
 262      2^-16495 = 1/2 of smallest representable value.
 263      If (1 - 1/131072)^y underflows, y > 1.4986e9 */
 264   if (iy > 0x401d654b)
 265     {
 266       /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
 267       if (iy > 0x407d654b)
 268         {
 269           if (ix <= 0x3ffeffff)
 270             return (hy < 0) ? huge * huge : tiny * tiny;
 271           if (ix >= 0x3fff0000)
 272             return (hy > 0) ? huge * huge : tiny * tiny;
 273         }
 274       /* over/underflow if x is not close to one */
 275       if (ix < 0x3ffeffff)
 276         return (hy < 0) ? huge * huge : tiny * tiny;
 277       if (ix > 0x3fff0000)
 278         return (hy > 0) ? huge * huge : tiny * tiny;
 279     }
 280
 281   n = 0;
 282   /* take care subnormal number */
 283   if (ix < 0x00010000)
 284     {
 285       ax *= two113;
 286       n -= 113;
 287       o.value = ax;
 288       ix = o.parts32.mswhi;
 289     }
 290   n += ((ix) >> 16) - 0x3fff;
 291   j = ix & 0x0000ffff;
 292   /* determine interval */
 293   ix = j | 0x3fff0000;          /* normalize ix */
 294   if (j <= 0x3988)
 295     k = 0;                      /* |x|<sqrt(3/2) */
 296   else if (j < 0xbb67)
 297     k = 1;                      /* |x|<sqrt(3)   */
 298   else
 299     {
 300       k = 0;
 301       n += 1;
 302       ix -= 0x00010000;
 303     }
 304
 305   o.value = ax;
 306   o.parts32.mswhi = ix;
 307   ax = o.value;
 308
 309   /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
 310   u = ax - bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
 311   v = one / (ax + bp[k]);
 312   s = u * v;
 313   s_h = s;
 314
 315   o.value = s_h;
 316   o.parts32.lswlo = 0;
 317   o.parts32.lswhi &= 0xf8000000;
 318   s_h = o.value;
 319   /* t_h=ax+bp[k] High */
 320   t_h = ax + bp[k];
 321   o.value = t_h;
 322   o.parts32.lswlo = 0;
 323   o.parts32.lswhi &= 0xf8000000;
 324   t_h = o.value;
 325   t_l = ax - (t_h - bp[k]);
 326   s_l = v * ((u - s_h * t_h) - s_h * t_l);
 327   /* compute log(ax) */
 328   s2 = s * s;
 329   u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4])));
 330   v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2))));
 331   r = s2 * s2 * u / v;
 332   r += s_l * (s_h + s);
 333   s2 = s_h * s_h;
 334   t_h = 3.0 + s2 + r;
 335   o.value = t_h;
 336   o.parts32.lswlo = 0;
 337   o.parts32.lswhi &= 0xf8000000;
 338   t_h = o.value;
 339   t_l = r - ((t_h - 3.0) - s2);
 340   /* u+v = s*(1+...) */
 341   u = s_h * t_h;
 342   v = s_l * t_h + t_l * s;
 343   /* 2/(3log2)*(s+...) */
 344   p_h = u + v;
 345   o.value = p_h;
 346   o.parts32.lswlo = 0;
 347   o.parts32.lswhi &= 0xf8000000;
 348   p_h = o.value;
 349   p_l = v - (p_h - u);
 350   z_h = cp_h * p_h;             /* cp_h+cp_l = 2/(3*log2) */
 351   z_l = cp_l * p_h + p_l * cp + dp_l[k];
 352   /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
 353   t = (long double) n;
 354   t1 = (((z_h + z_l) + dp_h[k]) + t);
 355   o.value = t1;
 356   o.parts32.lswlo = 0;
 357   o.parts32.lswhi &= 0xf8000000;
 358   t1 = o.value;
 359   t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
 360
 361   /* s (sign of result -ve**odd) = -1 else = 1 */
 362   s = one;
 363   if (((((uint32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
 364     s = -one;                   /* (-ve)**(odd int) */
 365
 366   /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
 367   yy1 = y;
 368   o.value = yy1;
 369   o.parts32.lswlo = 0;
 370   o.parts32.lswhi &= 0xf8000000;
 371   yy1 = o.value;
 372   p_l = (y - yy1) * t1 + y * t2;
 373   p_h = yy1 * t1;
 374   z = p_l + p_h;
 375   o.value = z;
 376   j = o.parts32.mswhi;
 377   if (j >= 0x400d0000) /* z >= 16384 */
 378     {
 379       /* if z > 16384 */
 380       if (((j - 0x400d0000) | o.parts32.mswlo | o.parts32.lswhi |
 381         o.parts32.lswlo) != 0)
 382         return s * huge * huge; /* overflow */
 383       else
 384         {
 385           if (p_l + ovt > z - p_h)
 386             return s * huge * huge;     /* overflow */
 387         }
 388     }
 389   else if ((j & 0x7fffffff) >= 0x400d01b9)      /* z <= -16495 */
 390     {
 391       /* z < -16495 */
 392       if (((j - 0xc00d01bc) | o.parts32.mswlo | o.parts32.lswhi |
 393         o.parts32.lswlo)
 394           != 0)
 395         return s * tiny * tiny; /* underflow */
 396       else
 397         {
 398           if (p_l <= z - p_h)
 399             return s * tiny * tiny;     /* underflow */
 400         }
 401     }
 402   /* compute 2**(p_h+p_l) */
 403   i = j & 0x7fffffff;
 404   k = (i >> 16) - 0x3fff;
 405   n = 0;
 406   if (i > 0x3ffe0000)
 407     {                           /* if |z| > 0.5, set n = [z+0.5] */
 408       n = floorl (z + 0.5L);
 409       t = n;
 410       p_h -= t;
 411     }
 412   t = p_l + p_h;
 413   o.value = t;
 414   o.parts32.lswlo = 0;
 415   o.parts32.lswhi &= 0xf8000000;
 416   t = o.value;
 417   u = t * lg2_h;
 418   v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
 419   z = u + v;
 420   w = v - (z - u);
 421   /*  exp(z) */
 422   t = z * z;
 423   u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4])));
 424   v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t)));
 425   t1 = z - t * u / v;
 426   r = (z * t1) / (t1 - two) - (w + z * w);
 427   z = one - (r - z);
 428   o.value = z;
 429   j = o.parts32.mswhi;
 430   j += (n << 16);
 431   if ((j >> 16) <= 0)
 432     z = scalbnl (z, n); /* subnormal output */
 433   else
 434     {
 435       o.parts32.mswhi = j;
 436       z = o.value;
 437     }
 438   return s * z;
 439 }