1 /*
   2  * CDDL HEADER START
   3  *
   4  * The contents of this file are subject to the terms of the
   5  * Common Development and Distribution License (the "License").
   6  * You may not use this file except in compliance with the License.
   7  *
   8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
   9  * or http://www.opensolaris.org/os/licensing.
  10  * See the License for the specific language governing permissions
  11  * and limitations under the License.
  12  *
  13  * When distributing Covered Code, include this CDDL HEADER in each
  14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
  15  * If applicable, add the following below this CDDL HEADER, with the
  16  * fields enclosed by brackets "[]" replaced with your own identifying
  17  * information: Portions Copyright [yyyy] [name of copyright owner]
  18  *
  19  * CDDL HEADER END
  20  */
  21 
  22 /*
  23  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
  24  */
  25 /*
  26  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  27  * Use is subject to license terms.
  28  */
  29 
  30 #if defined(ELFOBJ)
  31 #pragma weak pow = __pow
  32 #endif
  33 
  34 /*
  35  * pow(x,y) return x**y
  36  *                    n
  37  * Method:  Let x =  2   * (1+f)
  38  *      1. Compute and return log2(x) in two pieces:
  39  *              log2(x) = w1 + w2,
  40  *         where w1 has 24 bits trailing zero.
  41  *      2. Perform y*log2(x) by simulating muti-precision arithmetic
  42  *      3. Return x**y = exp2(y*log(x))
  43  *
  44  * Special cases:
  45  *      1.  (anything) ** +-0 is 1
  46  *      1'. 1 ** (anything)   is 1      (C99; 1 ** +-INF/NAN used to be NAN)
  47  *      2.  (anything) ** 1   is itself
  48  *      3.  (anything except 1) ** NAN is NAN ("except 1" is C99)
  49  *      4.  NAN ** (anything except 0) is NAN
  50  *      5.  +-(|x| > 1) **  +INF is +INF
  51  *      6.  +-(|x| > 1) **  -INF is +0
  52  *      7.  +-(|x| < 1) **  +INF is +0
  53  *      8.  +-(|x| < 1) **  -INF is +INF
  54  *      9.  -1          ** +-INF is 1   (C99; -1 ** +-INF used to be NAN)
  55  *      10. +0 ** (+anything except 0, NAN)               is +0
  56  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
  57  *      12. +0 ** (-anything except 0, NAN)               is +INF
  58  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
  59  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
  60  *      15. +INF ** (+anything except 0,NAN) is +INF
  61  *      16. +INF ** (-anything except 0,NAN) is +0
  62  *      17. -INF ** (anything)  = -0 ** (-anything)
  63  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  64  *      19. (-anything except 0 and inf) ** (non-integer) is NAN
  65  *
  66  * Accuracy:
  67  *      pow(x,y) returns x**y nearly rounded. In particular
  68  *                      pow(integer,integer)
  69  *      always returns the correct integer provided it is representable.
  70  */
  71 
  72 #include "libm.h"
  73 #include "xpg6.h"       /* __xpg6 */
  74 #define _C99SUSv3_pow   _C99SUSv3_pow_treats_Inf_as_an_even_int
  75 
  76 static const double zero = 0.0, one = 1.0, two = 2.0;
  77 
  78 extern const double _TBL_log2_hi[], _TBL_log2_lo[];
  79 static const double
  80         two53 = 9007199254740992.0,
  81         A1_hi = 2.8853900432586669921875,
  82         A1_lo = 3.8519259825035041963606002e-8,
  83         A1 = 2.885390081777926817222541963606002026086e+0000,
  84         A2 = 9.617966939207270828380543979852286255862e-0001,
  85         A3 = 5.770807680887875964868853124873696201995e-0001,
  86         B0_hi = 2.8853900432586669921875,
  87         B0_lo = 3.8519259822532793056374320585e-8,
  88         B0 = 2.885390081777926814720293056374320585689e+0000,
  89         B1 = 9.617966939259755138949202350396200257632e-0001,
  90         B2 = 5.770780163585687000782112776448797953382e-0001,
  91         B3 = 4.121985488948771523290174512461778354953e-0001,
  92         B4 = 3.207590534812432970433641789022666850193e-0001;
  93 
  94 static double
  95 log2_x(double x, double *w) {
  96         double f, s, z, qn, h, t;
  97         int *px = (int *) &x;
  98         int *pz = (int *) &z;
  99         int i, j, ix, n;
 100 
 101         n = 0;
 102         ix = px[HIWORD];
 103         if (ix >= 0x3fef03f1 && ix < 0x3ff08208) {        /* 65/63 > x > 63/65 */
 104                 double f1, v;
 105                 f = x - one;
 106                 if (((ix - 0x3ff00000) | px[LOWORD]) == 0) {
 107                         *w = zero;
 108                         return (zero);          /* log2(1)= +0 */
 109                 }
 110                 qn = one / (two + f);
 111                 s = f * qn;                             /* |s|<2**-6 */
 112                 v = s * s;
 113                 h = (double) ((float) s);
 114                 f1 = (double) ((float) f);
 115                 t = qn * (((f - two * h) - h * f1) - h * (f - f1));
 116                                                                 /* s = h+t */
 117                 f1 = h * B0_lo + s * (v * (B1 + v * (B2 + v * (B3 + v * B4))));
 118                 t = f1 + t * B0;
 119                 h *= B0_hi;
 120                 s = (double) ((float) (h + t));
 121                 *w = t - (s - h);
 122                 return (s);
 123         }
 124         if (ix < 0x00100000) {                               /* subnormal x */
 125                 x *= two53;
 126                 n = -53;
 127                 ix = px[HIWORD];
 128         }
 129         /* LARGE N */
 130         n += ((ix + 0x1000) >> 20) - 0x3ff;
 131         ix = (ix & 0x000fffff) | 0x3ff00000;                /* scale x to [1,2] */
 132         px[HIWORD] = ix;
 133         i = ix + 0x1000;
 134         pz[HIWORD] = i & 0xffffe000;
 135         pz[LOWORD] = 0;
 136         qn = one / (x + z);
 137         f = x - z;
 138         s = f * qn;
 139         h = (double) ((float) s);
 140         t = qn * ((f - (h + h) * z) - h * f);
 141         j = (i >> 13) & 0x7f;
 142         f = s * s;
 143         t = t * A1 + h * A1_lo;
 144         t += (s * f) * (A2 + f * A3);
 145         qn = h * A1_hi;
 146         s = n + _TBL_log2_hi[j];
 147         h = qn + s;
 148         t += _TBL_log2_lo[j] - ((h - s) - qn);
 149         f = (double) ((float) (h + t));
 150         *w = t - (f - h);
 151         return (f);
 152 }
 153 
 154 extern const double _TBL_exp2_hi[], _TBL_exp2_lo[];
 155 static const double             /* poly app of 2^x-1 on [-1e-10,2^-7+1e-10] */
 156         E1 = 6.931471805599453100674958533810346197328e-0001,
 157         E2 = 2.402265069587779347846769151717493815979e-0001,
 158         E3 = 5.550410866475410512631124892773937864699e-0002,
 159         E4 = 9.618143209991026824853712740162451423355e-0003,
 160         E5 = 1.333357676549940345096774122231849082991e-0003;
 161 
 162 double
 163 pow(double x, double y) {
 164         double z, ax;
 165         double y1, y2, w1, w2;
 166         int sbx, sby, j, k, yisint;
 167         int hx, hy, ahx, ahy;
 168         unsigned lx, ly;
 169         int *pz = (int *) &z;
 170 
 171         hx = ((int *) &x)[HIWORD];
 172         lx = ((unsigned *) &x)[LOWORD];
 173         hy = ((int *) &y)[HIWORD];
 174         ly = ((unsigned *) &y)[LOWORD];
 175         ahx = hx & ~0x80000000;
 176         ahy = hy & ~0x80000000;
 177         if ((ahy | ly) == 0) {  /* y==zero  */
 178                 if ((ahx | lx) == 0)
 179                         z = _SVID_libm_err(x, y, 20);   /* +-0**+-0 */
 180                 else if ((ahx | (((lx | -lx) >> 31) & 1)) > 0x7ff00000)
 181                         z = _SVID_libm_err(x, y, 42);   /* NaN**+-0 */
 182                 else
 183                         z = one;                        /* x**+-0 = 1 */
 184                 return (z);
 185         } else if (hx == 0x3ff00000 && lx == 0 &&
 186                 (__xpg6 & _C99SUSv3_pow) != 0)
 187                 return (one);                   /* C99: 1**anything = 1 */
 188         else if (ahx > 0x7ff00000 || (ahx == 0x7ff00000 && lx != 0) ||
 189                 ahy > 0x7ff00000 || (ahy == 0x7ff00000 && ly != 0))
 190                 return (x * y); /* +-NaN return x*y; + -> * for Cheetah */
 191                                 /* includes Sun: 1**NaN = NaN */
 192         sbx = (unsigned) hx >> 31;
 193         sby = (unsigned) hy >> 31;
 194         ax = fabs(x);
 195 
 196         /*
 197          * determine if y is an odd int when x < 0
 198          * yisint = 0 ... y is not an integer
 199          * yisint = 1 ... y is an odd int
 200          * yisint = 2 ... y is an even int
 201          */
 202         yisint = 0;
 203         if (sbx) {
 204                 if (ahy >= 0x43400000)
 205                         yisint = 2;             /* even integer y */
 206                 else if (ahy >= 0x3ff00000) {
 207                         k = (ahy >> 20) - 0x3ff;  /* exponent */
 208                         if (k > 20) {
 209                                 j = ly >> (52 - k);
 210                                 if ((j << (52 - k)) == ly)
 211                                         yisint = 2 - (j & 1);
 212                         } else if (ly == 0) {
 213                                 j = ahy >> (20 - k);
 214                                 if ((j << (20 - k)) == ahy)
 215                                         yisint = 2 - (j & 1);
 216                         }
 217                 }
 218         }
 219         /* special value of y */
 220         if (ly == 0) {
 221                 if (ahy == 0x7ff00000) {        /* y is +-inf */
 222                         if (((ahx - 0x3ff00000) | lx) == 0) {
 223                                 if ((__xpg6 & _C99SUSv3_pow) != 0)
 224                                         return (one);
 225                                                 /* C99: (-1)**+-inf = 1 */
 226                                 else
 227                                         return (y - y);
 228                                                 /* Sun: (+-1)**+-inf = NaN */
 229                         } else if (ahx >= 0x3ff00000)
 230                                                 /* (|x|>1)**+,-inf = inf,0 */
 231                                 return (sby == 0 ? y : zero);
 232                         else                    /* (|x|<1)**-,+inf = inf,0 */
 233                                 return (sby != 0 ? -y : zero);
 234                 }
 235                 if (ahy == 0x3ff00000) {        /* y is  +-1 */
 236                         if (sby != 0) { /* y is -1 */
 237                                 if (x == zero)  /* divided by zero */
 238                                         return (_SVID_libm_err(x, y, 23));
 239                                 else if (ahx < 0x40000 || ((ahx - 0x40000) |
 240                                         lx) == 0)       /* overflow */
 241                                         return (_SVID_libm_err(x, y, 21));
 242                                 else
 243                                         return (one / x);
 244                         } else
 245                                 return (x);
 246                 }
 247                 if (hy == 0x40000000) {         /* y is  2 */
 248                         if (ahx >= 0x5ff00000 && ahx < 0x7ff00000)
 249                                 return (_SVID_libm_err(x, y, 21));
 250                                                         /* x*x overflow */
 251                         else if (ahx < 0x1e56a09e && (ahx | lx) != 0 ||
 252                                 ahx == 0x1e56a09e && lx < 0x667f3bcd)
 253                                 return (_SVID_libm_err(x, y, 22));
 254                                                         /* x*x underflow */
 255                         else
 256                                 return (x * x);
 257                 }
 258                 if (hy == 0x3fe00000) {
 259                         if (!((ahx | lx) == 0 || ((ahx - 0x7ff00000) | lx) ==
 260                                 0 || sbx == 1))
 261                                 return (sqrt(x));       /* y is 0.5 and x > 0 */
 262                 }
 263         }
 264         /* special value of x */
 265         if (lx == 0) {
 266                 if (ahx == 0x7ff00000 || ahx == 0 || ahx == 0x3ff00000) {
 267                         /* x is +-0,+-inf,-1 */
 268                         z = ax;
 269                         if (sby == 1) {
 270                                 z = one / z;    /* z = |x|**y */
 271                                 if (ahx == 0)
 272                                         return (_SVID_libm_err(x, y, 23));
 273                         }
 274                         if (sbx == 1) {
 275                                 if (ahx == 0x3ff00000 && yisint == 0)
 276                                         z = _SVID_libm_err(x, y, 24);
 277                                         /* neg**non-integral is NaN + invalid */
 278                                 else if (yisint == 1)
 279                                         z = -z; /* (x<0)**odd = -(|x|**odd) */
 280                         }
 281                         return (z);
 282                 }
 283         }
 284         /* (x<0)**(non-int) is NaN */
 285         if (sbx == 1 && yisint == 0)
 286                 return (_SVID_libm_err(x, y, 24));
 287         /* Now ax is finite, y is finite */
 288         /* first compute log2(ax) = w1+w2, with 24 bits w1 */
 289         w1 = log2_x(ax, &w2);
 290 
 291         /* split up y into y1+y2 and compute (y1+y2)*(w1+w2) */
 292         if (((ly & 0x07ffffff) == 0) || ahy >= 0x47e00000 ||
 293                 ahy <= 0x38100000) {
 294                 /* no need to split if y is short or too large or too small */
 295                 y1 = y * w1;
 296                 y2 = y * w2;
 297         } else {
 298                 y1 = (double) ((float) y);
 299                 y2 = (y - y1) * w1 + y * w2;
 300                 y1 *= w1;
 301         }
 302         z = y1 + y2;
 303         j = pz[HIWORD];
 304         if (j >= 0x40900000) {                               /* z >= 1024 */
 305                 if (!(j == 0x40900000 && pz[LOWORD] == 0))      /* z > 1024 */
 306                         return (_SVID_libm_err(x, y, 21));      /* overflow */
 307                 else {
 308                         w2 = y1 - z;
 309                         w2 += y2;
 310                                                         /* rounded to inf */
 311                         if (w2 >= -8.008566259537296567160e-17)
 312                                 return (_SVID_libm_err(x, y, 21));
 313                                                                 /* overflow */
 314                 }
 315         } else if ((j & ~0x80000000) >= 0x4090cc00) {    /* z <= -1075 */
 316                 if (!(j == 0xc090cc00 && pz[LOWORD] == 0))      /* z < -1075 */
 317                         return (_SVID_libm_err(x, y, 22));      /* underflow */
 318                 else {
 319                         w2 = y1 - z;
 320                         w2 += y2;
 321                         if (w2 <= zero)                      /* underflow */
 322                                 return (_SVID_libm_err(x, y, 22));
 323                 }
 324         }
 325         /*
 326          * compute 2**(k+f[j]+g)
 327          */
 328         k = (int) (z * 64.0 + (((hy ^ (ahx - 0x3ff00000)) > 0) ? 0.5 : -0.5));
 329         j = k & 63;
 330         w1 = y2 - ((double) k * 0.015625 - y1);
 331         w2 = _TBL_exp2_hi[j];
 332         z = _TBL_exp2_lo[j] + (w2 * w1) * (E1 + w1 * (E2 + w1 * (E3 + w1 *
 333                 (E4 + w1 * E5))));
 334         z += w2;
 335         k >>= 6;
 336         if (k < -1021)
 337                 z = scalbn(z, k);
 338         else                    /* subnormal output */
 339                 pz[HIWORD] += k << 20;
 340         if (sbx == 1 && yisint == 1)
 341                 z = -z;         /* (-ve)**(odd int) */
 342         return (z);
 343 }