path: root/fms.c

/* mpfr_fms -- Floating multiply-subtract

Copyright 2001, 2002, 2004, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao projects, INRIA.

This file is part of the GNU MPFR Library.

The GNU MPFR Library is free software; you can redistribute
it and/or modify it under the terms of the GNU Lesser
General Public License as published by the Free Software
Foundation; either version 2.1 of the License, or (at your
option) any later version.

The GNU MPFR Library is distributed in the hope that it will
be useful, but WITHOUT ANY WARRANTY; without even the
implied warranty of MERCHANTABILITY or FITNESS FOR A
PARTICULAR PURPOSE.  See the GNU Lesser General Public
License for more details.

You should have received a copy of the GNU Lesser
General Public License along with the GNU MPFR Library; see
the file COPYING.LIB.  If not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
MA 02110-1301, USA. */

#include "mpfr-impl.h"

/* The fused-multiply-subtract (fms) of x, y and z is defined by:
   fms(x,y,z)= x*y - z
   Note: this is neither in IEEE754R, nor in LIA-2, but both the
   PowerPC and the Itanium define fms as x*y - z.
*/

int
mpfr_fms (mpfr_ptr s, mpfr_srcptr x, mpfr_srcptr y, mpfr_srcptr z,
          mp_rnd_t rnd_mode)
{
  int inexact;
  mpfr_t u;
  MPFR_SAVE_EXPO_DECL (expo);

  /* particular cases */
  if (MPFR_UNLIKELY( MPFR_IS_SINGULAR(x) ||
                     MPFR_IS_SINGULAR(y) ||
                     MPFR_IS_SINGULAR(z) ))
    {
      if (MPFR_IS_NAN(x) || MPFR_IS_NAN(y) || MPFR_IS_NAN(z))
        {
          MPFR_SET_NAN(s);
          MPFR_RET_NAN;
        }
      /* now neither x, y or z is NaN */
      else if (MPFR_IS_INF(x) || MPFR_IS_INF(y))
        {
          /* cases Inf*0-z, 0*Inf-z, Inf-Inf */
          if ((MPFR_IS_ZERO(y)) ||
              (MPFR_IS_ZERO(x)) ||
              (MPFR_IS_INF(z) &&
               ((MPFR_MULT_SIGN(MPFR_SIGN(x), MPFR_SIGN(y))) == MPFR_SIGN(z))))
            {
              MPFR_SET_NAN(s);
              MPFR_RET_NAN;
            }
          else if (MPFR_IS_INF(z)) /* case Inf-Inf already checked above */
            {
              MPFR_SET_INF(s);
              MPFR_SET_OPPOSITE_SIGN(s, z);
              MPFR_RET(0);
            }
          else /* z is finite */
            {
              MPFR_SET_INF(s);
              MPFR_SET_SIGN(s, MPFR_MULT_SIGN(MPFR_SIGN(x) , MPFR_SIGN(y)));
              MPFR_RET(0);
            }
        }
      /* now x and y are finite */
      else if (MPFR_IS_INF(z))
        {
          MPFR_SET_INF(s);
          MPFR_SET_OPPOSITE_SIGN(s, z);
          MPFR_RET(0);
        }
      else if (MPFR_IS_ZERO(x) || MPFR_IS_ZERO(y))
        {
          if (MPFR_IS_ZERO(z))
            {
              int sign_p;
              sign_p = MPFR_MULT_SIGN( MPFR_SIGN(x) , MPFR_SIGN(y) );
              MPFR_SET_SIGN(s,(rnd_mode != GMP_RNDD ?
                               ((MPFR_IS_NEG_SIGN(sign_p) && MPFR_IS_POS(z))
                                ? -1 : 1) :
                               ((MPFR_IS_POS_SIGN(sign_p) && MPFR_IS_NEG(z))
                                ? 1 : -1)));
              MPFR_SET_ZERO(s);
              MPFR_RET(0);
            }
          else
            return mpfr_neg (s, z, rnd_mode);
        }
      else /* necessarily z is zero here */
        {
          MPFR_ASSERTD(MPFR_IS_ZERO(z));
          return mpfr_mul (s, x, y, rnd_mode);
        }
    }

  /* If we take prec(u) >= prec(x) + prec(y), the product u <- x*y
     is exact, except in case of overflow or underflow. */
  MPFR_SAVE_EXPO_MARK (expo);
  mpfr_init2 (u, MPFR_PREC(x) + MPFR_PREC(y));

  if (MPFR_UNLIKELY (mpfr_mul (u, x, y, GMP_RNDN)))
    {
      /* overflow or underflow - this case is regarded as rare, thus
         does not need to be very efficient (even if some tests below
         could have been done earlier).
         It is an overflow iff u is an infinity (since GMP_RNDN was used).
         Alternatively, we could test the overflow flag, but in this case,
         mpfr_clear_flags would have been necessary. */
      if (MPFR_IS_INF (u))  /* overflow */
        {
          /* Let's eliminate the obvious case where x*y and z have the
             same sign. No possible cancellation -> real overflow.
             Also, we know that |z| < 2^emax. If E(x) + E(y) >= emax+3,
             then |x*y| >= 2^(emax+1), and |x*y - z| >= 2^emax. This case
             is also an overflow. */
          if (MPFR_SIGN (u) != MPFR_SIGN (z) ||
              MPFR_GET_EXP (x) + MPFR_GET_EXP (y) >= __gmpfr_emax + 3)
            {
              mpfr_clear (u);
              MPFR_SAVE_EXPO_FREE (expo);
              return mpfr_overflow (s, rnd_mode, - MPFR_SIGN (z));
            }

          /* E(x) + E(y) <= emax+2, therefore |x*y| < 2^(emax+2), and
             (x/4)*y does not overflow (let's recall that the result
             is exact with an unbounded exponent range). It does not
             underflow either, because x*y overflows and the exponent
             range is large enough. */
          inexact = mpfr_div_2ui (u, x, 2, GMP_RNDN);
          MPFR_ASSERTN (inexact == 0);
          inexact = mpfr_mul (u, u, y, GMP_RNDN);
          MPFR_ASSERTN (inexact == 0);

          /* Now, we need to subtract z/4... But it may underflow! */
          {
            mpfr_t zo4;
            mpfr_srcptr zz;
            MPFR_BLOCK_DECL (flags);

            if (MPFR_GET_EXP (u) > MPFR_GET_EXP (z) &&
                MPFR_GET_EXP (u) - MPFR_GET_EXP (z) > MPFR_PREC (u))
              {
                /* |z| < ulp(u)/2, therefore one can use z instead of z/4. */
                zz = z;
              }
            else
              {
                mpfr_init2 (zo4, MPFR_PREC (z));
                if (mpfr_div_2ui (zo4, z, 2, GMP_RNDZ))
                  {
                    /* The division by 4 underflowed! */
                    MPFR_ASSERTN (0); /* TODO... */
                  }
                zz = zo4;
              }

            /* Let's recall that u = x*y/4 and zz = z/4 (or z if the
               following subtraction would give the same result). */
            MPFR_BLOCK (flags, inexact = mpfr_sub (s, u, zz, rnd_mode));
            /* u and zz have the same sign, so that an overflow
               is not possible. But an underflow is theoretically
               possible! */
            if (MPFR_UNDERFLOW (flags))
              {
                MPFR_ASSERTN (zz != z);
                MPFR_ASSERTN (0); /* TODO... */
                mpfr_clears (zo4, u, (mpfr_ptr) 0);
              }
            else
              {
                int inex2;

                if (zz != z)
                  mpfr_clear (zo4);
                mpfr_clear (u);
                MPFR_ASSERTN (! MPFR_OVERFLOW (flags));
                inex2 = mpfr_mul_2ui (s, s, 2, rnd_mode);
                if (inex2)  /* overflow */
                  {
                    inexact = inex2;
                    MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
                  }
                goto end;
              }
          }
        }
      else  /* underflow: one has |xy| < 2^(emin-1). */
        {
          unsigned long scale = 0;
          mpfr_t scaled_z;
          mpfr_srcptr new_z;
          mp_exp_t diffexp;
          mp_prec_t pzs;
          int xy_underflows;

          /* Let's scale z so that ulp(z) > 2^emin and ulp(s) > 2^emin
             (the + 1 on MPFR_PREC (s) is necessary because the exponent
             of the result can be EXP(z) - 1). */
          diffexp = MPFR_GET_EXP (z) - __gmpfr_emin;
          pzs = MAX (MPFR_PREC (z), MPFR_PREC (s) + 1);
          if (diffexp <= pzs)
            {
              mp_exp_unsigned_t uscale;
              mpfr_t scaled_v;
              MPFR_BLOCK_DECL (flags);

              uscale = (mp_exp_unsigned_t) pzs - diffexp + 1;
              MPFR_ASSERTN (uscale > 0);
              MPFR_ASSERTN (uscale <= ULONG_MAX);
              scale = uscale;
              mpfr_init2 (scaled_z, MPFR_PREC (z));
              inexact = mpfr_mul_2ui (scaled_z, z, scale, GMP_RNDN);
              MPFR_ASSERTN (inexact == 0);  /* TODO: overflow case */
              new_z = scaled_z;
              /* Now we need to recompute u = xy * 2^scale. */
              MPFR_BLOCK (flags,
                          if (MPFR_GET_EXP (x) < MPFR_GET_EXP (y))
                            {
                              mpfr_init2 (scaled_v, MPFR_PREC (x));
                              mpfr_mul_2ui (scaled_v, x, scale, GMP_RNDN);
                              mpfr_mul (u, scaled_v, y, GMP_RNDN);
                            }
                          else
                            {
                              mpfr_init2 (scaled_v, MPFR_PREC (y));
                              mpfr_mul_2ui (scaled_v, y, scale, GMP_RNDN);
                              mpfr_mul (u, x, scaled_v, GMP_RNDN);
                            });
              mpfr_clear (scaled_v);
              MPFR_ASSERTN (! MPFR_OVERFLOW (flags));
              xy_underflows = MPFR_UNDERFLOW (flags);
            }
          else
            {
              new_z = z;
              xy_underflows = 1;
            }

          if (xy_underflows)
            {
              /* Let's replace xy by sign(xy) * 2^(emin-1). */
              mpfr_set_prec (u, MPFR_PREC_MIN);
              mpfr_setmin (u, __gmpfr_emin);
              MPFR_SET_SIGN (u, MPFR_MULT_SIGN (MPFR_SIGN (x),
                                                MPFR_SIGN (y)));
            }

          {
            MPFR_BLOCK_DECL (flags);

            MPFR_BLOCK (flags, inexact = mpfr_sub (s, u, new_z, rnd_mode));
            mpfr_clear (u);

            if (scale != 0)
              {
                int inex2;

                mpfr_clear (scaled_z);
                /* Here an overflow is theoretically possible, in which case
                   the result may be wrong, hence the assert. An underflow
                   is not possible, but let's check that anyway. */
                MPFR_ASSERTN (! MPFR_OVERFLOW (flags));  /* TODO... */
                MPFR_ASSERTN (! MPFR_UNDERFLOW (flags));  /* not possible */
                inex2 = mpfr_div_2ui (s, s, scale, GMP_RNDN);
                /* FIXME: this seems incorrect. GMP_RNDN -> rnd_mode?
                   Also, handle the double rounding case:
                   s / 2^scale = 2^(emin - 2) in GMP_RNDN. */
                if (inex2)  /* underflow */
                  inexact = inex2;
              }
          }

          /* FIXME/TODO: I'm not sure that the following is correct.
             Check for possible spurious exceptions due to intermediate
             computations. */
          MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
          goto end;
        }
    }

  inexact = mpfr_sub (s, u, z, rnd_mode);
  mpfr_clear (u);
  MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
 end:
  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (s, inexact, rnd_mode);
}