sysdeps/ieee754/dbl-64/e_pow.c

   1 /* Double-precision x^y function.
   2    Copyright (C) 2018-2024 Free Software Foundation, Inc.
   3    This file is part of the GNU C Library.
   4
   5    The GNU C Library is free software; you can redistribute it and/or
   6    modify it under the terms of the GNU Lesser General Public
   7    License as published by the Free Software Foundation; either
   8    version 2.1 of the License, or (at your option) any later version.
   9
  10    The GNU C Library is distributed in the hope that it will be useful,
  11    but WITHOUT ANY WARRANTY; without even the implied warranty of
  12    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  13    Lesser General Public License for more details.
  14
  15    You should have received a copy of the GNU Lesser General Public
  16    License along with the GNU C Library; if not, see
  17    <https://www.gnu.org/licenses/>.  */
  18
  19 #include <math.h>
  20 #include <stdint.h>
  21 #include <math-barriers.h>
  22 #include <math-narrow-eval.h>
  23 #include <math-svid-compat.h>
  24 #include <libm-alias-finite.h>
  25 #include <libm-alias-double.h>
  26 #include "math_config.h"
  27
  28 /*
  29 Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53)
  30 relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma)
  31 ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma)
  32 */
  33
  34 #define T __pow_log_data.tab
  35 #define A __pow_log_data.poly
  36 #define Ln2hi __pow_log_data.ln2hi
  37 #define Ln2lo __pow_log_data.ln2lo
  38 #define N (1 << POW_LOG_TABLE_BITS)
  39 #define OFF 0x3fe6955500000000
  40
  41 /* Top 12 bits of a double (sign and exponent bits).  */
  42 static inline uint32_t
  43 top12 (double x)
  44 {
  45   return asuint64 (x) >> 52;
  46 }
  47
  48 /* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about
  49    additional 15 bits precision.  IX is the bit representation of x, but
  50    normalized in the subnormal range using the sign bit for the exponent.  */
  51 static inline double_t
  52 log_inline (uint64_t ix, double_t *tail)
  53 {
  54   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  55   double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p;
  56   uint64_t iz, tmp;
  57   int k, i;
  58
  59   /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
  60      The range is split into N subintervals.
  61      The ith subinterval contains z and c is near its center.  */
  62   tmp = ix - OFF;
  63   i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N;
  64   k = (int64_t) tmp >> 52; /* arithmetic shift */
  65   iz = ix - (tmp & 0xfffULL << 52);
  66   z = asdouble (iz);
  67   kd = (double_t) k;
  68
  69   /* log(x) = k*Ln2 + log(c) + log1p(z/c-1).  */
  70   invc = T[i].invc;
  71   logc = T[i].logc;
  72   logctail = T[i].logctail;
  73
  74   /* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and
  75      |z/c - 1| < 1/N, so r = z/c - 1 is exactly representible.  */
  76 #ifdef __FP_FAST_FMA
  77   r = __builtin_fma (z, invc, -1.0);
  78 #else
  79   /* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|.  */
  80   double_t zhi = asdouble ((iz + (1ULL << 31)) & (-1ULL << 32));
  81   double_t zlo = z - zhi;
  82   double_t rhi = zhi * invc - 1.0;
  83   double_t rlo = zlo * invc;
  84   r = rhi + rlo;
  85 #endif
  86
  87   /* k*Ln2 + log(c) + r.  */
  88   t1 = kd * Ln2hi + logc;
  89   t2 = t1 + r;
  90   lo1 = kd * Ln2lo + logctail;
  91   lo2 = t1 - t2 + r;
  92
  93   /* Evaluation is optimized assuming superscalar pipelined execution.  */
  94   double_t ar, ar2, ar3, lo3, lo4;
  95   ar = A[0] * r; /* A[0] = -0.5.  */
  96   ar2 = r * ar;
  97   ar3 = r * ar2;
  98   /* k*Ln2 + log(c) + r + A[0]*r*r.  */
  99 #ifdef __FP_FAST_FMA
 100   hi = t2 + ar2;
 101   lo3 = __builtin_fma (ar, r, -ar2);
 102   lo4 = t2 - hi + ar2;
 103 #else
 104   double_t arhi = A[0] * rhi;
 105   double_t arhi2 = rhi * arhi;
 106   hi = t2 + arhi2;
 107   lo3 = rlo * (ar + arhi);
 108   lo4 = t2 - hi + arhi2;
 109 #endif
 110   /* p = log1p(r) - r - A[0]*r*r.  */
 111   p = (ar3
 112        * (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6]))));
 113   lo = lo1 + lo2 + lo3 + lo4 + p;
 114   y = hi + lo;
 115   *tail = hi - y + lo;
 116   return y;
 117 }
 118
 119 #undef N
 120 #undef T
 121 #define N (1 << EXP_TABLE_BITS)
 122 #define InvLn2N __exp_data.invln2N
 123 #define NegLn2hiN __exp_data.negln2hiN
 124 #define NegLn2loN __exp_data.negln2loN
 125 #define Shift __exp_data.shift
 126 #define T __exp_data.tab
 127 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
 128 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
 129 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
 130 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
 131 #define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
 132
 133 /* Handle cases that may overflow or underflow when computing the result that
 134    is scale*(1+TMP) without intermediate rounding.  The bit representation of
 135    scale is in SBITS, however it has a computed exponent that may have
 136    overflown into the sign bit so that needs to be adjusted before using it as
 137    a double.  (int32_t)KI is the k used in the argument reduction and exponent
 138    adjustment of scale, positive k here means the result may overflow and
 139    negative k means the result may underflow.  */
 140 static inline double
 141 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
 142 {
 143   double_t scale, y;
 144
 145   if ((ki & 0x80000000) == 0)
 146     {
 147       /* k > 0, the exponent of scale might have overflowed by <= 460.  */
 148       sbits -= 1009ull << 52;
 149       scale = asdouble (sbits);
 150       y = 0x1p1009 * (scale + scale * tmp);
 151       return check_oflow (y);
 152     }
 153   /* k < 0, need special care in the subnormal range.  */
 154   sbits += 1022ull << 52;
 155   /* Note: sbits is signed scale.  */
 156   scale = asdouble (sbits);
 157   y = scale + scale * tmp;
 158   if (fabs (y) < 1.0)
 159     {
 160       /* Round y to the right precision before scaling it into the subnormal
 161          range to avoid double rounding that can cause 0.5+E/2 ulp error where
 162          E is the worst-case ulp error outside the subnormal range.  So this
 163          is only useful if the goal is better than 1 ulp worst-case error.  */
 164       double_t hi, lo, one = 1.0;
 165       if (y < 0.0)
 166         one = -1.0;
 167       lo = scale - y + scale * tmp;
 168       hi = one + y;
 169       lo = one - hi + y + lo;
 170       y = math_narrow_eval (hi + lo) - one;
 171       /* Fix the sign of 0.  */
 172       if (y == 0.0)
 173         y = asdouble (sbits & 0x8000000000000000);
 174       /* The underflow exception needs to be signaled explicitly.  */
 175       math_force_eval (math_opt_barrier (0x1p-1022) * 0x1p-1022);
 176     }
 177   y = 0x1p-1022 * y;
 178   return check_uflow (y);
 179 }
 180
 181 #define SIGN_BIAS (0x800 << EXP_TABLE_BITS)
 182
 183 /* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
 184    The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1.  */
 185 static inline double
 186 exp_inline (double x, double xtail, uint32_t sign_bias)
 187 {
 188   uint32_t abstop;
 189   uint64_t ki, idx, top, sbits;
 190   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
 191   double_t kd, z, r, r2, scale, tail, tmp;
 192
 193   abstop = top12 (x) & 0x7ff;
 194   if (__glibc_unlikely (abstop - top12 (0x1p-54)
 195                         >= top12 (512.0) - top12 (0x1p-54)))
 196     {
 197       if (abstop - top12 (0x1p-54) >= 0x80000000)
 198         {
 199           /* Avoid spurious underflow for tiny x.  */
 200           /* Note: 0 is common input.  */
 201           double_t one = WANT_ROUNDING ? 1.0 + x : 1.0;
 202           return sign_bias ? -one : one;
 203         }
 204       if (abstop >= top12 (1024.0))
 205         {
 206           /* Note: inf and nan are already handled.  */
 207           if (asuint64 (x) >> 63)
 208             return __math_uflow (sign_bias);
 209           else
 210             return __math_oflow (sign_bias);
 211         }
 212       /* Large x is special cased below.  */
 213       abstop = 0;
 214     }
 215
 216   /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
 217   /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
 218   z = InvLn2N * x;
 219 #if TOINT_INTRINSICS
 220   /* z - kd is in [-0.5, 0.5] in all rounding modes.  */
 221   kd = roundtoint (z);
 222   ki = converttoint (z);
 223 #else
 224   /* z - kd is in [-1, 1] in non-nearest rounding modes.  */
 225   kd = math_narrow_eval (z + Shift);
 226   ki = asuint64 (kd);
 227   kd -= Shift;
 228 #endif
 229   r = x + kd * NegLn2hiN + kd * NegLn2loN;
 230   /* The code assumes 2^-200 < |xtail| < 2^-8/N.  */
 231   r += xtail;
 232   /* 2^(k/N) ~= scale * (1 + tail).  */
 233   idx = 2 * (ki % N);
 234   top = (ki + sign_bias) << (52 - EXP_TABLE_BITS);
 235   tail = asdouble (T[idx]);
 236   /* This is only a valid scale when -1023*N < k < 1024*N.  */
 237   sbits = T[idx + 1] + top;
 238   /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
 239   /* Evaluation is optimized assuming superscalar pipelined execution.  */
 240   r2 = r * r;
 241   /* Without fma the worst case error is 0.25/N ulp larger.  */
 242   /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
 243   tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
 244   if (__glibc_unlikely (abstop == 0))
 245     return specialcase (tmp, sbits, ki);
 246   scale = asdouble (sbits);
 247   /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
 248      is no spurious underflow here even without fma.  */
 249   return scale + scale * tmp;
 250 }
 251
 252 /* Returns 0 if not int, 1 if odd int, 2 if even int.  The argument is
 253    the bit representation of a non-zero finite floating-point value.  */
 254 static inline int
 255 checkint (uint64_t iy)
 256 {
 257   int e = iy >> 52 & 0x7ff;
 258   if (e < 0x3ff)
 259     return 0;
 260   if (e > 0x3ff + 52)
 261     return 2;
 262   if (iy & ((1ULL << (0x3ff + 52 - e)) - 1))
 263     return 0;
 264   if (iy & (1ULL << (0x3ff + 52 - e)))
 265     return 1;
 266   return 2;
 267 }
 268
 269 /* Returns 1 if input is the bit representation of 0, infinity or nan.  */
 270 static inline int
 271 zeroinfnan (uint64_t i)
 272 {
 273   return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1;
 274 }
 275
 276 #ifndef SECTION
 277 # define SECTION
 278 #endif
 279
 280 double
 281 SECTION
 282 __pow (double x, double y)
 283 {
 284   uint32_t sign_bias = 0;
 285   uint64_t ix, iy;
 286   uint32_t topx, topy;
 287
 288   ix = asuint64 (x);
 289   iy = asuint64 (y);
 290   topx = top12 (x);
 291   topy = top12 (y);
 292   if (__glibc_unlikely (topx - 0x001 >= 0x7ff - 0x001
 293                         || (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be))
 294     {
 295       /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0
 296          and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1.  */
 297       /* Special cases: (x < 0x1p-126 or inf or nan) or
 298          (|y| < 0x1p-65 or |y| >= 0x1p63 or nan).  */
 299       if (__glibc_unlikely (zeroinfnan (iy)))
 300         {
 301           if (2 * iy == 0)
 302             return issignaling_inline (x) ? x + y : 1.0;
 303           if (ix == asuint64 (1.0))
 304             return issignaling_inline (y) ? x + y : 1.0;
 305           if (2 * ix > 2 * asuint64 (INFINITY)
 306               || 2 * iy > 2 * asuint64 (INFINITY))
 307             return x + y;
 308           if (2 * ix == 2 * asuint64 (1.0))
 309             return 1.0;
 310           if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63))
 311             return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf.  */
 312           return y * y;
 313         }
 314       if (__glibc_unlikely (zeroinfnan (ix)))
 315         {
 316           double_t x2 = x * x;
 317           if (ix >> 63 && checkint (iy) == 1)
 318             {
 319               x2 = -x2;
 320               sign_bias = 1;
 321             }
 322           if (WANT_ERRNO && 2 * ix == 0 && iy >> 63)
 323             return __math_divzero (sign_bias);
 324           /* Without the barrier some versions of clang hoist the 1/x2 and
 325              thus division by zero exception can be signaled spuriously.  */
 326           return iy >> 63 ? math_opt_barrier (1 / x2) : x2;
 327         }
 328       /* Here x and y are non-zero finite.  */
 329       if (ix >> 63)
 330         {
 331           /* Finite x < 0.  */
 332           int yint = checkint (iy);
 333           if (yint == 0)
 334             return __math_invalid (x);
 335           if (yint == 1)
 336             sign_bias = SIGN_BIAS;
 337           ix &= 0x7fffffffffffffff;
 338           topx &= 0x7ff;
 339         }
 340       if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)
 341         {
 342           /* Note: sign_bias == 0 here because y is not odd.  */
 343           if (ix == asuint64 (1.0))
 344             return 1.0;
 345           if ((topy & 0x7ff) < 0x3be)
 346             {
 347               /* |y| < 2^-65, x^y ~= 1 + y*log(x).  */
 348               if (WANT_ROUNDING)
 349                 return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y;
 350               else
 351                 return 1.0;
 352             }
 353           return (ix > asuint64 (1.0)) == (topy < 0x800) ? __math_oflow (0)
 354                                                          : __math_uflow (0);
 355         }
 356       if (topx == 0)
 357         {
 358           /* Normalize subnormal x so exponent becomes negative.  */
 359           ix = asuint64 (x * 0x1p52);
 360           ix &= 0x7fffffffffffffff;
 361           ix -= 52ULL << 52;
 362         }
 363     }
 364
 365   double_t lo;
 366   double_t hi = log_inline (ix, &lo);
 367   double_t ehi, elo;
 368 #ifdef __FP_FAST_FMA
 369   ehi = y * hi;
 370   elo = y * lo + __builtin_fma (y, hi, -ehi);
 371 #else
 372   double_t yhi = asdouble (iy & -1ULL << 27);
 373   double_t ylo = y - yhi;
 374   double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27);
 375   double_t llo = hi - lhi + lo;
 376   ehi = yhi * lhi;
 377   elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25.  */
 378 #endif
 379   return exp_inline (ehi, elo, sign_bias);
 380 }
 381 #ifndef __pow
 382 strong_alias (__pow, __ieee754_pow)
 383 libm_alias_finite (__ieee754_pow, __pow)
 384 # if LIBM_SVID_COMPAT
 385 versioned_symbol (libm, __pow, pow, GLIBC_2_29);
 386 libm_alias_double_other (__pow, pow)
 387 # else
 388 libm_alias_double (__pow, pow)
 389 # endif
 390 #endif