sysdeps/ieee754/dbl-64/s_fma.c

   1 /* Compute x * y + z as ternary operation.
   2    Copyright (C) 2010-2013 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    <http://www.gnu.org/licenses/>.  */
  19
  20 #include <float.h>
  21 #include <math.h>
  22 #include <fenv.h>
  23 #include <ieee754.h>
  24 #include <math_private.h>
  25 #include <tininess.h>
  26
  27 /* This implementation uses rounding to odd to avoid problems with
  28    double rounding.  See a paper by Boldo and Melquiond:
  29    http://www.lri.fr/~melquion/doc/08-tc.pdf  */
  30
  31 double
  32 __fma (double x, double y, double z)
  33 {
  34   union ieee754_double u, v, w;
  35   int adjust = 0;
  36   u.d = x;
  37   v.d = y;
  38   w.d = z;
  39   if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
  40                         >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0)
  41       || __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
  42       || __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
  43       || __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
  44       || __builtin_expect (u.ieee.exponent + v.ieee.exponent
  45                            <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG, 0))
  46     {
  47       /* If z is Inf, but x and y are finite, the result should be
  48          z rather than NaN.  */
  49       if (w.ieee.exponent == 0x7ff
  50           && u.ieee.exponent != 0x7ff
  51           && v.ieee.exponent != 0x7ff)
  52         return (z + x) + y;
  53       /* If z is zero and x are y are nonzero, compute the result
  54          as x * y to avoid the wrong sign of a zero result if x * y
  55          underflows to 0.  */
  56       if (z == 0 && x != 0 && y != 0)
  57         return x * y;
  58       /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
  59          x * y + z.  */
  60       if (u.ieee.exponent == 0x7ff
  61           || v.ieee.exponent == 0x7ff
  62           || w.ieee.exponent == 0x7ff
  63           || x == 0
  64           || y == 0)
  65         return x * y + z;
  66       /* If fma will certainly overflow, compute as x * y.  */
  67       if (u.ieee.exponent + v.ieee.exponent > 0x7ff + IEEE754_DOUBLE_BIAS)
  68         return x * y;
  69       /* If x * y is less than 1/4 of DBL_DENORM_MIN, neither the
  70          result nor whether there is underflow depends on its exact
  71          value, only on its sign.  */
  72       if (u.ieee.exponent + v.ieee.exponent
  73           < IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
  74         {
  75           int neg = u.ieee.negative ^ v.ieee.negative;
  76           double tiny = neg ? -0x1p-1074 : 0x1p-1074;
  77           if (w.ieee.exponent >= 3)
  78             return tiny + z;
  79           /* Scaling up, adding TINY and scaling down produces the
  80              correct result, because in round-to-nearest mode adding
  81              TINY has no effect and in other modes double rounding is
  82              harmless.  But it may not produce required underflow
  83              exceptions.  */
  84           v.d = z * 0x1p54 + tiny;
  85           if (TININESS_AFTER_ROUNDING
  86               ? v.ieee.exponent < 55
  87               : (w.ieee.exponent == 0
  88                  || (w.ieee.exponent == 1
  89                      && w.ieee.negative != neg
  90                      && w.ieee.mantissa1 == 0
  91                      && w.ieee.mantissa0 == 0)))
  92             {
  93               volatile double force_underflow = x * y;
  94               (void) force_underflow;
  95             }
  96           return v.d * 0x1p-54;
  97         }
  98       if (u.ieee.exponent + v.ieee.exponent
  99           >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG)
 100         {
 101           /* Compute 1p-53 times smaller result and multiply
 102              at the end.  */
 103           if (u.ieee.exponent > v.ieee.exponent)
 104             u.ieee.exponent -= DBL_MANT_DIG;
 105           else
 106             v.ieee.exponent -= DBL_MANT_DIG;
 107           /* If x + y exponent is very large and z exponent is very small,
 108              it doesn't matter if we don't adjust it.  */
 109           if (w.ieee.exponent > DBL_MANT_DIG)
 110             w.ieee.exponent -= DBL_MANT_DIG;
 111           adjust = 1;
 112         }
 113       else if (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
 114         {
 115           /* Similarly.
 116              If z exponent is very large and x and y exponents are
 117              very small, adjust them up to avoid spurious underflows,
 118              rather than down.  */
 119           if (u.ieee.exponent + v.ieee.exponent
 120               <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG)
 121             {
 122               if (u.ieee.exponent > v.ieee.exponent)
 123                 u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
 124               else
 125                 v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
 126             }
 127           else if (u.ieee.exponent > v.ieee.exponent)
 128             {
 129               if (u.ieee.exponent > DBL_MANT_DIG)
 130                 u.ieee.exponent -= DBL_MANT_DIG;
 131             }
 132           else if (v.ieee.exponent > DBL_MANT_DIG)
 133             v.ieee.exponent -= DBL_MANT_DIG;
 134           w.ieee.exponent -= DBL_MANT_DIG;
 135           adjust = 1;
 136         }
 137       else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
 138         {
 139           u.ieee.exponent -= DBL_MANT_DIG;
 140           if (v.ieee.exponent)
 141             v.ieee.exponent += DBL_MANT_DIG;
 142           else
 143             v.d *= 0x1p53;
 144         }
 145       else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
 146         {
 147           v.ieee.exponent -= DBL_MANT_DIG;
 148           if (u.ieee.exponent)
 149             u.ieee.exponent += DBL_MANT_DIG;
 150           else
 151             u.d *= 0x1p53;
 152         }
 153       else /* if (u.ieee.exponent + v.ieee.exponent
 154                   <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
 155         {
 156           if (u.ieee.exponent > v.ieee.exponent)
 157             u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
 158           else
 159             v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
 160           if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 6)
 161             {
 162               if (w.ieee.exponent)
 163                 w.ieee.exponent += 2 * DBL_MANT_DIG + 2;
 164               else
 165                 w.d *= 0x1p108;
 166               adjust = -1;
 167             }
 168           /* Otherwise x * y should just affect inexact
 169              and nothing else.  */
 170         }
 171       x = u.d;
 172       y = v.d;
 173       z = w.d;
 174     }
 175
 176   /* Ensure correct sign of exact 0 + 0.  */
 177   if (__builtin_expect ((x == 0 || y == 0) && z == 0, 0))
 178     return x * y + z;
 179
 180   fenv_t env;
 181   libc_feholdexcept_setround (&env, FE_TONEAREST);
 182
 183   /* Multiplication m1 + m2 = x * y using Dekker's algorithm.  */
 184 #define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
 185   double x1 = x * C;
 186   double y1 = y * C;
 187   double m1 = x * y;
 188   x1 = (x - x1) + x1;
 189   y1 = (y - y1) + y1;
 190   double x2 = x - x1;
 191   double y2 = y - y1;
 192   double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
 193
 194   /* Addition a1 + a2 = z + m1 using Knuth's algorithm.  */
 195   double a1 = z + m1;
 196   double t1 = a1 - z;
 197   double t2 = a1 - t1;
 198   t1 = m1 - t1;
 199   t2 = z - t2;
 200   double a2 = t1 + t2;
 201   feclearexcept (FE_INEXACT);
 202
 203   /* If the result is an exact zero, ensure it has the correct
 204      sign.  */
 205   if (a1 == 0 && m2 == 0)
 206     {
 207       libc_feupdateenv (&env);
 208       /* Ensure that round-to-nearest value of z + m1 is not
 209          reused.  */
 210       asm volatile ("" : "=m" (z) : "m" (z));
 211       return z + m1;
 212     }
 213
 214   libc_fesetround (FE_TOWARDZERO);
 215
 216   /* Perform m2 + a2 addition with round to odd.  */
 217   u.d = a2 + m2;
 218
 219   if (__builtin_expect (adjust < 0, 0))
 220     {
 221       if ((u.ieee.mantissa1 & 1) == 0)
 222         u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
 223       v.d = a1 + u.d;
 224       /* Ensure the addition is not scheduled after fetestexcept call.  */
 225       math_force_eval (v.d);
 226     }
 227
 228   /* Reset rounding mode and test for inexact simultaneously.  */
 229   int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;
 230
 231   if (__builtin_expect (adjust == 0, 1))
 232     {
 233       if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
 234         u.ieee.mantissa1 |= j;
 235       /* Result is a1 + u.d.  */
 236       return a1 + u.d;
 237     }
 238   else if (__builtin_expect (adjust > 0, 1))
 239     {
 240       if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
 241         u.ieee.mantissa1 |= j;
 242       /* Result is a1 + u.d, scaled up.  */
 243       return (a1 + u.d) * 0x1p53;
 244     }
 245   else
 246     {
 247       /* If a1 + u.d is exact, the only rounding happens during
 248          scaling down.  */
 249       if (j == 0)
 250         return v.d * 0x1p-108;
 251       /* If result rounded to zero is not subnormal, no double
 252          rounding will occur.  */
 253       if (v.ieee.exponent > 108)
 254         return (a1 + u.d) * 0x1p-108;
 255       /* If v.d * 0x1p-108 with round to zero is a subnormal above
 256          or equal to DBL_MIN / 2, then v.d * 0x1p-108 shifts mantissa
 257          down just by 1 bit, which means v.ieee.mantissa1 |= j would
 258          change the round bit, not sticky or guard bit.
 259          v.d * 0x1p-108 never normalizes by shifting up,
 260          so round bit plus sticky bit should be already enough
 261          for proper rounding.  */
 262       if (v.ieee.exponent == 108)
 263         {
 264           /* If the exponent would be in the normal range when
 265              rounding to normal precision with unbounded exponent
 266              range, the exact result is known and spurious underflows
 267              must be avoided on systems detecting tininess after
 268              rounding.  */
 269           if (TININESS_AFTER_ROUNDING)
 270             {
 271               w.d = a1 + u.d;
 272               if (w.ieee.exponent == 109)
 273                 return w.d * 0x1p-108;
 274             }
 275           /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
 276              v.ieee.mantissa1 & 1 is the round bit and j is our sticky
 277              bit.  */
 278           w.d = 0.0;
 279           w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;
 280           w.ieee.negative = v.ieee.negative;
 281           v.ieee.mantissa1 &= ~3U;
 282           v.d *= 0x1p-108;
 283           w.d *= 0x1p-2;
 284           return v.d + w.d;
 285         }
 286       v.ieee.mantissa1 |= j;
 287       return v.d * 0x1p-108;
 288     }
 289 }
 290 #ifndef __fma
 291 weak_alias (__fma, fma)
 292 #endif
 293
 294 #ifdef NO_LONG_DOUBLE
 295 strong_alias (__fma, __fmal)
 296 weak_alias (__fmal, fmal)
 297 #endif