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/* mpfr_hypot -- Euclidean distance
Copyright 2001, 2002, 2003, 2004, 2005 Free Software Foundation, Inc.
This file is part of the MPFR Library.
The 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 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 MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin Place, Fifth Floor, Boston,
MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* The computation of hypot of x and y is done by *
* hypot(x,y)= sqrt(x^2+y^2) = z */
int
mpfr_hypot (mpfr_ptr z, mpfr_srcptr x, mpfr_srcptr y, mp_rnd_t rnd_mode)
{
int inexact, exact;
mpfr_t t, te, ti; /* auxiliary variables */
mp_prec_t Nx, Ny, Nz; /* size variables */
mp_prec_t Nt; /* precision of the intermediary variable */
mp_exp_t Ex, Ey, sh;
mp_exp_unsigned_t diff_exp;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
/* particular cases */
if (MPFR_ARE_SINGULAR (x, y))
{
if (MPFR_IS_INF (x) || MPFR_IS_INF (y))
{
/* Return +inf, even when the other number is NaN. */
MPFR_SET_INF (z);
MPFR_SET_POS (z);
MPFR_RET (0);
}
else if (MPFR_IS_NAN (x) || MPFR_IS_NAN (y))
{
MPFR_SET_NAN (z);
MPFR_RET_NAN;
}
else if (MPFR_IS_ZERO (x))
return mpfr_abs (z, y, rnd_mode);
else /* y is necessarily 0 */
return mpfr_abs (z, x, rnd_mode);
}
MPFR_CLEAR_FLAGS(z);
if (mpfr_cmpabs (x, y) < 0)
{
mpfr_srcptr t;
t = x;
x = y;
y = t;
}
/* now |x| >= |y| */
Ex = MPFR_GET_EXP (x);
Ey = MPFR_GET_EXP (y);
diff_exp = (mp_exp_unsigned_t) Ex - Ey;
Nx = MPFR_PREC (x); /* Precision of input variable */
Nz = MPFR_PREC (z); /* Precision of output variable */
/* we have x < 2^Ex thus x^2 < 2^(2*Ex),
and ulp(x) = 2^(Ex-Nx) thus ulp(x^2) >= 2^(2*Ex-2*Nx).
y does not overlap with the result when
x^2+y^2 < (|x| + 1/2*ulp(x,Nz))^2 = x^2 + |x|*ulp(x,Nz) + 1/4*ulp(x,Nz)^2,
i.e. a sufficient condition is y^2 < |x|*ulp(x,Nz),
or 2^(2*Ey) <= 2^(2*Ex-1-Nz), i.e. 2*diff_exp > Nz.
Warning: this is true only for Nx <= Nz, otherwise the trailing bits
of x may be already very close to 1/2*ulp(x,Nz)!
If Nx > Nz, then we can notice that it is possible to round on Nx bits
if 2*diff_exp > Nx (see above as if Nz = Nx), therefore on Nz bits.
Hence the condition: 2*diff_exp > MAX(Nz,Nx).
*/
if (diff_exp > MAX (Nz, Nx) / 2)
/* result is |x| or |x|+ulp(|x|,Nz) */
{
if (MPFR_UNLIKELY (rnd_mode == GMP_RNDU))
{
/* If z > abs(x), then it was already rounded up; otherwise
z = abs(x), and we need to add one ulp due to y. */
if (mpfr_abs (z, x, rnd_mode) == 0)
mpfr_nexttoinf (z);
return 1;
}
else /* GMP_RNDZ, GMP_RNDD, GMP_RNDN */
{
if (MPFR_LIKELY (Nz >= Nx))
{
mpfr_abs (z, x, rnd_mode); /* exact */
return -1;
}
else
{
MPFR_SET_EXP (z, Ex);
MPFR_SET_SIGN (z, 1);
MPFR_RNDRAW_GEN (inexact, z, MPFR_MANT (x), Nx, rnd_mode, 1,
goto addoneulp,
if (MPFR_UNLIKELY (++MPFR_EXP (z) > __gmpfr_emax))
return mpfr_overflow (z, rnd_mode, 1);
);
return inexact ? inexact : -1;
}
}
}
/* General case */
Ny = MPFR_PREC(y); /* Precision of input variable */
/* compute the working precision -- see algorithms.ps */
Nt = MAX (MAX (MAX (Nx, Ny), Nz), 8);
/* FIXME: if Nx or Ny are very large with respect to the target precision
Nz, this may be overkill! */
Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 2;
/* initialise the intermediary variables */
mpfr_init2 (t, Nt);
mpfr_init2 (te, Nt);
mpfr_init2 (ti, Nt);
MPFR_SAVE_EXPO_MARK (expo);
sh = MAX (0, MIN (Ex, Ey));
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* computations of hypot */
mpfr_div_2ui (te, x, sh, GMP_RNDZ); /* exact since Nt >= Nx */
mpfr_div_2ui (ti, y, sh, GMP_RNDZ); /* exact since Nt >= Ny */
exact = mpfr_mul (te, te, te, GMP_RNDZ); /* x^2 */
exact |= mpfr_mul (ti, ti, ti, GMP_RNDZ); /* y^2 */
exact |= mpfr_add (t, te, ti, GMP_RNDZ); /* x^2+y^2 */
exact |= mpfr_sqrt (t, t, GMP_RNDZ); /* sqrt(x^2+y^2)*/
if (MPFR_LIKELY (exact == 0
|| MPFR_CAN_ROUND (t, Nt-2, Nz, rnd_mode)))
break;
/* reactualization of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
mpfr_set_prec (te, Nt);
mpfr_set_prec (ti, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_mul_2ui (z, t, sh, rnd_mode);
MPFR_ASSERTD (exact == 0 || inexact != 0);
mpfr_clear (t);
mpfr_clear (ti);
mpfr_clear (te);
/*
exact inexact
0 0 result is exact, ternary flag is 0
0 non zero t is exact, ternary flag given by inexact
1 0 impossible (see above)
1 non zero ternary flag given by inexact
*/
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (z, inexact, rnd_mode);
}
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