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/* mpfr_hypot -- Euclidean distance
Copyright 2001-2019 Free Software Foundation, Inc.
Contributed by the AriC and Caramba 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 3 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.LESSER. If not, see
https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, 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, mpfr_rnd_t rnd_mode)
{
int inexact, exact;
mpfr_t t, te, ti; /* auxiliary variables */
mpfr_prec_t N, Nz; /* size variables */
mpfr_prec_t Nt; /* precision of the intermediary variable */
mpfr_prec_t threshold;
mpfr_exp_t Ex, sh;
mpfr_uexp_t diff_exp;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_BLOCK_DECL (flags);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg y[%Pu]=%.*Rg rnd=%d",
mpfr_get_prec (x), mpfr_log_prec, x,
mpfr_get_prec (y), mpfr_log_prec, y, rnd_mode),
("z[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (z), mpfr_log_prec, z, inexact));
/* 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);
}
if (mpfr_cmpabs (x, y) < 0)
{
mpfr_srcptr u;
u = x;
x = y;
y = u;
}
/* now |x| >= |y| */
Ex = MPFR_GET_EXP (x);
diff_exp = (mpfr_uexp_t) Ex - MPFR_GET_EXP (y);
N = MPFR_PREC (x); /* Precision of input variable */
Nz = MPFR_PREC (z); /* Precision of output variable */
threshold = (MAX (N, Nz) + (rnd_mode == MPFR_RNDN ? 1 : 0)) << 1;
if (rnd_mode == MPFR_RNDA)
rnd_mode = MPFR_RNDU; /* since the result is positive, RNDA = RNDU */
/* Is |x| a suitable approximation to the precision Nz ?
(see algorithms.tex for explanations) */
if (diff_exp > threshold)
/* result is |x| or |x|+ulp(|x|,Nz) */
{
if (MPFR_UNLIKELY (rnd_mode == MPFR_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);
/* since mpfr_nexttoinf does not set the overflow flag,
we have to check manually for overflow */
if (MPFR_UNLIKELY (MPFR_IS_INF (z)))
MPFR_SET_OVERFLOW ();
}
MPFR_RET (1);
}
else /* MPFR_RNDZ, MPFR_RNDD, MPFR_RNDN */
{
if (MPFR_LIKELY (Nz >= N))
{
mpfr_abs (z, x, rnd_mode); /* exact */
MPFR_RET (-1);
}
else
{
MPFR_SET_EXP (z, Ex);
MPFR_SET_SIGN (z, MPFR_SIGN_POS);
MPFR_RNDRAW_GEN (inexact, z, MPFR_MANT (x), N, rnd_mode, 1,
goto addoneulp,
if (MPFR_UNLIKELY (++ MPFR_EXP (z) >
__gmpfr_emax))
return mpfr_overflow (z, rnd_mode, 1);
);
if (MPFR_UNLIKELY (inexact == 0))
inexact = -1;
MPFR_RET (inexact);
}
}
}
/* General case */
N = MAX (MPFR_PREC (x), MPFR_PREC (y));
/* working precision */
Nt = Nz + MPFR_INT_CEIL_LOG2 (Nz) + 4;
mpfr_init2 (t, Nt);
mpfr_init2 (te, Nt);
mpfr_init2 (ti, Nt);
MPFR_SAVE_EXPO_MARK (expo);
/* Scale x and y to avoid any overflow and underflow in x^2
(as |x| >= |y|), and to limit underflow cases for y or y^2.
We have: x = Mx * 2^Ex with 1/2 <= |Mx| < 1, and we choose:
sh = floor((Emax - 1) / 2) - Ex.
Thus (x * 2^sh)^2 = Mx^2 * 2^(2*floor((Emax-1)/2)), which has an
exponent of at most Emax - 1. Therefore (x * 2^sh)^2 + (y * 2^sh)^2
will have an exponent of at most Emax, even after rounding as we
round toward zero. Using a larger sh wouldn't guarantee an absence
of overflow. Note that the scaling of y can underflow only when the
target precision is huge, otherwise the case would already have been
handled by the diff_exp > threshold code.
FIXME: Friedland in "Algorithm 312: Absolute Value and Square Root of a
Complex Number" (Communications of the ACM, 1967) avoids overflow by
computing |x|*sqrt(1+(y/x)^2) if |x| >= |y|, and |y|*sqrt(1+(x/y)^2)
otherwise.
[VL] This trick (which is a scaling by a non-power of 2, thus doesn't
really bring new behavior w.r.t. overflow/underflow exceptions) may be
useful for hardware floating-point formats because a whole power-of-2
scaling code is likely to take more time than the additional division,
but in the context of multiple-precision, I doubt that it is a good
idea. Ideally scaling by a power of 2 could be done in a constant time,
e.g. with MPFR_ALIAS; but one needs to be very careful... */
sh = (mpfr_get_emax () - 1) / 2 - Ex;
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
mpfr_prec_t err;
exact = mpfr_mul_2si (te, x, sh, MPFR_RNDZ);
exact |= mpfr_mul_2si (ti, y, sh, MPFR_RNDZ);
exact |= mpfr_sqr (te, te, MPFR_RNDZ);
/* Use fma in order to avoid underflow when diff_exp<=MPFR_EMAX_MAX-2 */
exact |= mpfr_fma (t, ti, ti, te, MPFR_RNDZ);
exact |= mpfr_sqrt (t, t, MPFR_RNDZ);
err = Nt < N ? 4 : 2;
if (MPFR_LIKELY (exact == 0
|| MPFR_CAN_ROUND (t, Nt-err, Nz, rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
mpfr_set_prec (te, Nt);
mpfr_set_prec (ti, Nt);
}
MPFR_ZIV_FREE (loop);
MPFR_BLOCK (flags, inexact = mpfr_div_2si (z, t, sh, rnd_mode));
MPFR_ASSERTD (exact == 0 || inexact != 0 || rnd_mode == MPFR_RNDF);
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);
if (MPFR_OVERFLOW (flags))
MPFR_SET_OVERFLOW ();
/* hypot(x,y) >= |x|, thus underflow is not possible. */
return mpfr_check_range (z, inexact, rnd_mode);
}
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