sysdeps/ieee754/ldbl-128/s_fmal.c

   1 /* Compute x * y + z as ternary operation.
   2    Copyright (C) 2010-2016 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 _Float128
  32 __fmal (_Float128 x, _Float128 y, _Float128 z)
  33 {
  34   union ieee854_long_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                         >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
  41                            - LDBL_MANT_DIG, 0)
  42       || __builtin_expect (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
  43       || __builtin_expect (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
  44       || __builtin_expect (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
  45       || __builtin_expect (u.ieee.exponent + v.ieee.exponent
  46                            <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG, 0))
  47     {
  48       /* If z is Inf, but x and y are finite, the result should be
  49          z rather than NaN.  */
  50       if (w.ieee.exponent == 0x7fff
  51           && u.ieee.exponent != 0x7fff
  52           && v.ieee.exponent != 0x7fff)
  53         return (z + x) + y;
  54       /* If z is zero and x are y are nonzero, compute the result
  55          as x * y to avoid the wrong sign of a zero result if x * y
  56          underflows to 0.  */
  57       if (z == 0 && x != 0 && y != 0)
  58         return x * y;
  59       /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
  60          x * y + z.  */
  61       if (u.ieee.exponent == 0x7fff
  62           || v.ieee.exponent == 0x7fff
  63           || w.ieee.exponent == 0x7fff
  64           || x == 0
  65           || y == 0)
  66         return x * y + z;
  67       /* If fma will certainly overflow, compute as x * y.  */
  68       if (u.ieee.exponent + v.ieee.exponent
  69           > 0x7fff + IEEE854_LONG_DOUBLE_BIAS)
  70         return x * y;
  71       /* If x * y is less than 1/4 of LDBL_TRUE_MIN, neither the
  72          result nor whether there is underflow depends on its exact
  73          value, only on its sign.  */
  74       if (u.ieee.exponent + v.ieee.exponent
  75           < IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2)
  76         {
  77           int neg = u.ieee.negative ^ v.ieee.negative;
  78           _Float128 tiny = neg ? L(-0x1p-16494) : L(0x1p-16494);
  79           if (w.ieee.exponent >= 3)
  80             return tiny + z;
  81           /* Scaling up, adding TINY and scaling down produces the
  82              correct result, because in round-to-nearest mode adding
  83              TINY has no effect and in other modes double rounding is
  84              harmless.  But it may not produce required underflow
  85              exceptions.  */
  86           v.d = z * L(0x1p114) + tiny;
  87           if (TININESS_AFTER_ROUNDING
  88               ? v.ieee.exponent < 115
  89               : (w.ieee.exponent == 0
  90                  || (w.ieee.exponent == 1
  91                      && w.ieee.negative != neg
  92                      && w.ieee.mantissa3 == 0
  93                      && w.ieee.mantissa2 == 0
  94                      && w.ieee.mantissa1 == 0
  95                      && w.ieee.mantissa0 == 0)))
  96             {
  97               _Float128 force_underflow = x * y;
  98               math_force_eval (force_underflow);
  99             }
 100           return v.d * L(0x1p-114);
 101         }
 102       if (u.ieee.exponent + v.ieee.exponent
 103           >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)
 104         {
 105           /* Compute 1p-113 times smaller result and multiply
 106              at the end.  */
 107           if (u.ieee.exponent > v.ieee.exponent)
 108             u.ieee.exponent -= LDBL_MANT_DIG;
 109           else
 110             v.ieee.exponent -= LDBL_MANT_DIG;
 111           /* If x + y exponent is very large and z exponent is very small,
 112              it doesn't matter if we don't adjust it.  */
 113           if (w.ieee.exponent > LDBL_MANT_DIG)
 114             w.ieee.exponent -= LDBL_MANT_DIG;
 115           adjust = 1;
 116         }
 117       else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
 118         {
 119           /* Similarly.
 120              If z exponent is very large and x and y exponents are
 121              very small, adjust them up to avoid spurious underflows,
 122              rather than down.  */
 123           if (u.ieee.exponent + v.ieee.exponent
 124               <= IEEE854_LONG_DOUBLE_BIAS + 2 * LDBL_MANT_DIG)
 125             {
 126               if (u.ieee.exponent > v.ieee.exponent)
 127                 u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
 128               else
 129                 v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
 130             }
 131           else if (u.ieee.exponent > v.ieee.exponent)
 132             {
 133               if (u.ieee.exponent > LDBL_MANT_DIG)
 134                 u.ieee.exponent -= LDBL_MANT_DIG;
 135             }
 136           else if (v.ieee.exponent > LDBL_MANT_DIG)
 137             v.ieee.exponent -= LDBL_MANT_DIG;
 138           w.ieee.exponent -= LDBL_MANT_DIG;
 139           adjust = 1;
 140         }
 141       else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
 142         {
 143           u.ieee.exponent -= LDBL_MANT_DIG;
 144           if (v.ieee.exponent)
 145             v.ieee.exponent += LDBL_MANT_DIG;
 146           else
 147             v.d *= L(0x1p113);
 148         }
 149       else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
 150         {
 151           v.ieee.exponent -= LDBL_MANT_DIG;
 152           if (u.ieee.exponent)
 153             u.ieee.exponent += LDBL_MANT_DIG;
 154           else
 155             u.d *= L(0x1p113);
 156         }
 157       else /* if (u.ieee.exponent + v.ieee.exponent
 158                   <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
 159         {
 160           if (u.ieee.exponent > v.ieee.exponent)
 161             u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
 162           else
 163             v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
 164           if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 6)
 165             {
 166               if (w.ieee.exponent)
 167                 w.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
 168               else
 169                 w.d *= L(0x1p228);
 170               adjust = -1;
 171             }
 172           /* Otherwise x * y should just affect inexact
 173              and nothing else.  */
 174         }
 175       x = u.d;
 176       y = v.d;
 177       z = w.d;
 178     }
 179
 180   /* Ensure correct sign of exact 0 + 0.  */
 181   if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
 182     {
 183       x = math_opt_barrier (x);
 184       return x * y + z;
 185     }
 186
 187   fenv_t env;
 188   feholdexcept (&env);
 189   fesetround (FE_TONEAREST);
 190
 191   /* Multiplication m1 + m2 = x * y using Dekker's algorithm.  */
 192 #define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
 193   _Float128 x1 = x * C;
 194   _Float128 y1 = y * C;
 195   _Float128 m1 = x * y;
 196   x1 = (x - x1) + x1;
 197   y1 = (y - y1) + y1;
 198   _Float128 x2 = x - x1;
 199   _Float128 y2 = y - y1;
 200   _Float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
 201
 202   /* Addition a1 + a2 = z + m1 using Knuth's algorithm.  */
 203   _Float128 a1 = z + m1;
 204   _Float128 t1 = a1 - z;
 205   _Float128 t2 = a1 - t1;
 206   t1 = m1 - t1;
 207   t2 = z - t2;
 208   _Float128 a2 = t1 + t2;
 209   /* Ensure the arithmetic is not scheduled after feclearexcept call.  */
 210   math_force_eval (m2);
 211   math_force_eval (a2);
 212   feclearexcept (FE_INEXACT);
 213
 214   /* If the result is an exact zero, ensure it has the correct sign.  */
 215   if (a1 == 0 && m2 == 0)
 216     {
 217       feupdateenv (&env);
 218       /* Ensure that round-to-nearest value of z + m1 is not reused.  */
 219       z = math_opt_barrier (z);
 220       return z + m1;
 221     }
 222
 223   fesetround (FE_TOWARDZERO);
 224   /* Perform m2 + a2 addition with round to odd.  */
 225   u.d = a2 + m2;
 226
 227   if (__glibc_likely (adjust == 0))
 228     {
 229       if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
 230         u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
 231       feupdateenv (&env);
 232       /* Result is a1 + u.d.  */
 233       return a1 + u.d;
 234     }
 235   else if (__glibc_likely (adjust > 0))
 236     {
 237       if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
 238         u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
 239       feupdateenv (&env);
 240       /* Result is a1 + u.d, scaled up.  */
 241       return (a1 + u.d) * L(0x1p113);
 242     }
 243   else
 244     {
 245       if ((u.ieee.mantissa3 & 1) == 0)
 246         u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
 247       v.d = a1 + u.d;
 248       /* Ensure the addition is not scheduled after fetestexcept call.  */
 249       math_force_eval (v.d);
 250       int j = fetestexcept (FE_INEXACT) != 0;
 251       feupdateenv (&env);
 252       /* Ensure the following computations are performed in default rounding
 253          mode instead of just reusing the round to zero computation.  */
 254       asm volatile ("" : "=m" (u) : "m" (u));
 255       /* If a1 + u.d is exact, the only rounding happens during
 256          scaling down.  */
 257       if (j == 0)
 258         return v.d * L(0x1p-228);
 259       /* If result rounded to zero is not subnormal, no double
 260          rounding will occur.  */
 261       if (v.ieee.exponent > 228)
 262         return (a1 + u.d) * L(0x1p-228);
 263       /* If v.d * 0x1p-228L with round to zero is a subnormal above
 264          or equal to LDBL_MIN / 2, then v.d * 0x1p-228L shifts mantissa
 265          down just by 1 bit, which means v.ieee.mantissa3 |= j would
 266          change the round bit, not sticky or guard bit.
 267          v.d * 0x1p-228L never normalizes by shifting up,
 268          so round bit plus sticky bit should be already enough
 269          for proper rounding.  */
 270       if (v.ieee.exponent == 228)
 271         {
 272           /* If the exponent would be in the normal range when
 273              rounding to normal precision with unbounded exponent
 274              range, the exact result is known and spurious underflows
 275              must be avoided on systems detecting tininess after
 276              rounding.  */
 277           if (TININESS_AFTER_ROUNDING)
 278             {
 279               w.d = a1 + u.d;
 280               if (w.ieee.exponent == 229)
 281                 return w.d * L(0x1p-228);
 282             }
 283           /* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding,
 284              v.ieee.mantissa3 & 1 is the round bit and j is our sticky
 285              bit.  */
 286           w.d = 0;
 287           w.ieee.mantissa3 = ((v.ieee.mantissa3 & 3) << 1) | j;
 288           w.ieee.negative = v.ieee.negative;
 289           v.ieee.mantissa3 &= ~3U;
 290           v.d *= L(0x1p-228);
 291           w.d *= L(0x1p-2);
 292           return v.d + w.d;
 293         }
 294       v.ieee.mantissa3 |= j;
 295       return v.d * L(0x1p-228);
 296     }
 297 }
 298 weak_alias (__fmal, fmal)