usr/src/lib/libm/common/C/pow.c

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