/* mpfr_fma -- Floating multiply-add 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-add (fma) of x, y and z is defined by: fma(x,y,z)= x*y + z */ int mpfr_fma (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_SAME_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_SAME_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_NEG(z)) ? -1 : 1) : ((MPFR_IS_POS_SIGN(sign_p) && MPFR_IS_POS(z)) ? 1 : -1))); MPFR_SET_ZERO(s); MPFR_RET(0); } else return mpfr_set (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 add 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 addition would give the same result). */ MPFR_BLOCK (flags, inexact = mpfr_add (s, u, zz, rnd_mode)); /* u and zz have different signs, 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_add (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_add (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); }