sysdeps/ieee754/dbl-64/s_fma.c

   1 /* Compute x * y + z as ternary operation.
   2    Copyright (C) 2010-2020 Free Software Foundation, Inc.
   3    This file is part of the GNU C Library.
   4    Contributed by Jakub Jelinek <jakub@redhat.com>, 2010.
   5
   6    The GNU C Library is free software; you can redistribute it and/or
   7    modify it under the terms of the GNU Lesser General Public
   8    License as published by the Free Software Foundation; either
   9    version 2.1 of the License, or (at your option) any later version.
  10
  11    The GNU C Library is distributed in the hope that it will be useful,
  12    but WITHOUT ANY WARRANTY; without even the implied warranty of
  13    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  14    Lesser General Public License for more details.
  15
  16    You should have received a copy of the GNU Lesser General Public
  17    License along with the GNU C Library; if not, see
  18    <https://www.gnu.org/licenses/>.  */
  19
  20 #include <float.h>
  21 #include <math.h>
  22 #include <fenv.h>
  23 #include <ieee754.h>
  24 #include <math-barriers.h>
  25 #include <fenv_private.h>
  26 #include <libm-alias-double.h>
  27 #include <tininess.h>
  28 #include <math-use-builtins.h>
  29
  30 /* This implementation uses rounding to odd to avoid problems with
  31    double rounding.  See a paper by Boldo and Melquiond:
  32    http://www.lri.fr/~melquion/doc/08-tc.pdf  */
  33
  34 double
  35 __fma (double x, double y, double z)
  36 {
  37 #if USE_FMA_BUILTIN
  38   return __builtin_fma (x, y, z);
  39 #else
  40   /* Use generic implementation.  */
  41   union ieee754_double u, v, w;
  42   int adjust = 0;
  43   u.d = x;
  44   v.d = y;
  45   w.d = z;
  46   if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
  47                         >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0)
  48       || __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
  49       || __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
  50       || __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
  51       || __builtin_expect (u.ieee.exponent + v.ieee.exponent
  52                            <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG, 0))
  53     {
  54       /* If z is Inf, but x and y are finite, the result should be
  55          z rather than NaN.  */
  56       if (w.ieee.exponent == 0x7ff
  57           && u.ieee.exponent != 0x7ff
  58           && v.ieee.exponent != 0x7ff)
  59         return (z + x) + y;
  60       /* If z is zero and x are y are nonzero, compute the result
  61          as x * y to avoid the wrong sign of a zero result if x * y
  62          underflows to 0.  */
  63       if (z == 0 && x != 0 && y != 0)
  64         return x * y;
  65       /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
  66          x * y + z.  */
  67       if (u.ieee.exponent == 0x7ff
  68           || v.ieee.exponent == 0x7ff
  69           || w.ieee.exponent == 0x7ff
  70           || x == 0
  71           || y == 0)
  72         return x * y + z;
  73       /* If fma will certainly overflow, compute as x * y.  */
  74       if (u.ieee.exponent + v.ieee.exponent > 0x7ff + IEEE754_DOUBLE_BIAS)
  75         return x * y;
  76       /* If x * y is less than 1/4 of DBL_TRUE_MIN, neither the
  77          result nor whether there is underflow depends on its exact
  78          value, only on its sign.  */
  79       if (u.ieee.exponent + v.ieee.exponent
  80           < IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
  81         {
  82           int neg = u.ieee.negative ^ v.ieee.negative;
  83           double tiny = neg ? -0x1p-1074 : 0x1p-1074;
  84           if (w.ieee.exponent >= 3)
  85             return tiny + z;
  86           /* Scaling up, adding TINY and scaling down produces the
  87              correct result, because in round-to-nearest mode adding
  88              TINY has no effect and in other modes double rounding is
  89              harmless.  But it may not produce required underflow
  90              exceptions.  */
  91           v.d = z * 0x1p54 + tiny;
  92           if (TININESS_AFTER_ROUNDING
  93               ? v.ieee.exponent < 55
  94               : (w.ieee.exponent == 0
  95                  || (w.ieee.exponent == 1
  96                      && w.ieee.negative != neg
  97                      && w.ieee.mantissa1 == 0
  98                      && w.ieee.mantissa0 == 0)))
  99             {
 100               double force_underflow = x * y;
 101               math_force_eval (force_underflow);
 102             }
 103           return v.d * 0x1p-54;
 104         }
 105       if (u.ieee.exponent + v.ieee.exponent
 106           >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG)
 107         {
 108           /* Compute 1p-53 times smaller result and multiply
 109              at the end.  */
 110           if (u.ieee.exponent > v.ieee.exponent)
 111             u.ieee.exponent -= DBL_MANT_DIG;
 112           else
 113             v.ieee.exponent -= DBL_MANT_DIG;
 114           /* If x + y exponent is very large and z exponent is very small,
 115              it doesn't matter if we don't adjust it.  */
 116           if (w.ieee.exponent > DBL_MANT_DIG)
 117             w.ieee.exponent -= DBL_MANT_DIG;
 118           adjust = 1;
 119         }
 120       else if (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
 121         {
 122           /* Similarly.
 123              If z exponent is very large and x and y exponents are
 124              very small, adjust them up to avoid spurious underflows,
 125              rather than down.  */
 126           if (u.ieee.exponent + v.ieee.exponent
 127               <= IEEE754_DOUBLE_BIAS + 2 * DBL_MANT_DIG)
 128             {
 129               if (u.ieee.exponent > v.ieee.exponent)
 130                 u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
 131               else
 132                 v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
 133             }
 134           else if (u.ieee.exponent > v.ieee.exponent)
 135             {
 136               if (u.ieee.exponent > DBL_MANT_DIG)
 137                 u.ieee.exponent -= DBL_MANT_DIG;
 138             }
 139           else if (v.ieee.exponent > DBL_MANT_DIG)
 140             v.ieee.exponent -= DBL_MANT_DIG;
 141           w.ieee.exponent -= DBL_MANT_DIG;
 142           adjust = 1;
 143         }
 144       else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
 145         {
 146           u.ieee.exponent -= DBL_MANT_DIG;
 147           if (v.ieee.exponent)
 148             v.ieee.exponent += DBL_MANT_DIG;
 149           else
 150             v.d *= 0x1p53;
 151         }
 152       else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
 153         {
 154           v.ieee.exponent -= DBL_MANT_DIG;
 155           if (u.ieee.exponent)
 156             u.ieee.exponent += DBL_MANT_DIG;
 157           else
 158             u.d *= 0x1p53;
 159         }
 160       else /* if (u.ieee.exponent + v.ieee.exponent
 161                   <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
 162         {
 163           if (u.ieee.exponent > v.ieee.exponent)
 164             u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
 165           else
 166             v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
 167           if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 6)
 168             {
 169               if (w.ieee.exponent)
 170                 w.ieee.exponent += 2 * DBL_MANT_DIG + 2;
 171               else
 172                 w.d *= 0x1p108;
 173               adjust = -1;
 174             }
 175           /* Otherwise x * y should just affect inexact
 176              and nothing else.  */
 177         }
 178       x = u.d;
 179       y = v.d;
 180       z = w.d;
 181     }
 182
 183   /* Ensure correct sign of exact 0 + 0.  */
 184   if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
 185     {
 186       x = math_opt_barrier (x);
 187       return x * y + z;
 188     }
 189
 190   fenv_t env;
 191   libc_feholdexcept_setround (&env, FE_TONEAREST);
 192
 193   /* Multiplication m1 + m2 = x * y using Dekker's algorithm.  */
 194 #define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
 195   double x1 = x * C;
 196   double y1 = y * C;
 197   double m1 = x * y;
 198   x1 = (x - x1) + x1;
 199   y1 = (y - y1) + y1;
 200   double x2 = x - x1;
 201   double y2 = y - y1;
 202   double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
 203
 204   /* Addition a1 + a2 = z + m1 using Knuth's algorithm.  */
 205   double a1 = z + m1;
 206   double t1 = a1 - z;
 207   double t2 = a1 - t1;
 208   t1 = m1 - t1;
 209   t2 = z - t2;
 210   double a2 = t1 + t2;
 211   /* Ensure the arithmetic is not scheduled after feclearexcept call.  */
 212   math_force_eval (m2);
 213   math_force_eval (a2);
 214   feclearexcept (FE_INEXACT);
 215
 216   /* If the result is an exact zero, ensure it has the correct sign.  */
 217   if (a1 == 0 && m2 == 0)
 218     {
 219       libc_feupdateenv (&env);
 220       /* Ensure that round-to-nearest value of z + m1 is not reused.  */
 221       z = math_opt_barrier (z);
 222       return z + m1;
 223     }
 224
 225   libc_fesetround (FE_TOWARDZERO);
 226
 227   /* Perform m2 + a2 addition with round to odd.  */
 228   u.d = a2 + m2;
 229
 230   if (__glibc_unlikely (adjust < 0))
 231     {
 232       if ((u.ieee.mantissa1 & 1) == 0)
 233         u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
 234       v.d = a1 + u.d;
 235       /* Ensure the addition is not scheduled after fetestexcept call.  */
 236       math_force_eval (v.d);
 237     }
 238
 239   /* Reset rounding mode and test for inexact simultaneously.  */
 240   int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;
 241
 242   if (__glibc_likely (adjust == 0))
 243     {
 244       if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
 245         u.ieee.mantissa1 |= j;
 246       /* Result is a1 + u.d.  */
 247       return a1 + u.d;
 248     }
 249   else if (__glibc_likely (adjust > 0))
 250     {
 251       if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
 252         u.ieee.mantissa1 |= j;
 253       /* Result is a1 + u.d, scaled up.  */
 254       return (a1 + u.d) * 0x1p53;
 255     }
 256   else
 257     {
 258       /* If a1 + u.d is exact, the only rounding happens during
 259          scaling down.  */
 260       if (j == 0)
 261         return v.d * 0x1p-108;
 262       /* If result rounded to zero is not subnormal, no double
 263          rounding will occur.  */
 264       if (v.ieee.exponent > 108)
 265         return (a1 + u.d) * 0x1p-108;
 266       /* If v.d * 0x1p-108 with round to zero is a subnormal above
 267          or equal to DBL_MIN / 2, then v.d * 0x1p-108 shifts mantissa
 268          down just by 1 bit, which means v.ieee.mantissa1 |= j would
 269          change the round bit, not sticky or guard bit.
 270          v.d * 0x1p-108 never normalizes by shifting up,
 271          so round bit plus sticky bit should be already enough
 272          for proper rounding.  */
 273       if (v.ieee.exponent == 108)
 274         {
 275           /* If the exponent would be in the normal range when
 276              rounding to normal precision with unbounded exponent
 277              range, the exact result is known and spurious underflows
 278              must be avoided on systems detecting tininess after
 279              rounding.  */
 280           if (TININESS_AFTER_ROUNDING)
 281             {
 282               w.d = a1 + u.d;
 283               if (w.ieee.exponent == 109)
 284                 return w.d * 0x1p-108;
 285             }
 286           /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
 287              v.ieee.mantissa1 & 1 is the round bit and j is our sticky
 288              bit.  */
 289           w.d = 0.0;
 290           w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;
 291           w.ieee.negative = v.ieee.negative;
 292           v.ieee.mantissa1 &= ~3U;
 293           v.d *= 0x1p-108;
 294           w.d *= 0x1p-2;
 295           return v.d + w.d;
 296         }
 297       v.ieee.mantissa1 |= j;
 298       return v.d * 0x1p-108;
 299     }
 300 #endif /* ! USE_FMA_BUILTIN  */
 301 }
 302 #ifndef __fma
 303 libm_alias_double (__fma, fma)
 304 #endif