/* mpfr_round_near_x -- Round a floating point number nears another one.
Copyright 2005, 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, and was contributed by Mathieu Dutour.
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"
/* Use MPFR_FAST_COMPUTE_IF_SMALL_INPUT instead (a simple wrapper) */
/* int mpfr_round_near_x (mpfr_ptr y, mpfr_srcptr v, mpfr_uexp_t err, int dir,
mp_rnd_t rnd)
TODO: fix this description.
Assuming y = o(f(x)) = o(x + g(x)) with |g(x)| < 2^(EXP(v)-error)
If x is small enough, y ~= v. This function checks and does this.
It assumes that f(x) is not representable exactly as a FP number.
v must not be a singular value (NAN, INF or ZERO), usual values are
v=1 or v=x.
y is the destination (a mpfr_t), v the value to set (a mpfr_t),
err the error term (a mpfr_uexp_t) such that |g(x)| < 2^(EXP(x)-err),
dir (an int) is the direction of the error (if dir = 0,
it rounds toward 0, if dir=1, it rounds away from 0),
rnd the rounding mode.
It returns 0 if it can't round.
Otherwise it returns the ternary flag (It can't return an exact value).
*/
/* What "small enough" means?
We work with the positive values.
Assuming err > Prec (y)+1
i = [ y = o(x)] // i = inexact flag
If i == 0
Setting x in y is exact. We have:
y = [XXXXXXXXX[...]]0[...] + error where [..] are optional zeros
if dirError = ToInf,
x < f(x) < x + 2^(EXP(x)-err)
since x=y, and ulp (y)/2 > 2^(EXP(x)-err), we have:
y < f(x) < y+ulp(y) and |y-f(x)| < ulp(y)/2
if rnd = RNDN, nothing
if rnd = RNDZ, nothing
if rnd = RNDA, addoneulp
elif dirError = ToZero
x -2^(EXP(x)-err) < f(x) < x
since x=y, and ulp (y)/2 > 2^(EXP(x)-err), we have:
y-ulp(y) < f(x) < y and |y-f(x)| < ulp(y)/2
if rnd = RNDN, nothing
if rnd = RNDZ, nexttozero
if rnd = RNDA, nothing
NOTE: err > prec (y)+1 is needed only for RNDN.
elif i > 0 and i = EVEN_ROUNDING
So rnd = RNDN and we have y = x + ulp(y)/2
if dirError = ToZero,
we have x -2^(EXP(x)-err) < f(x) < x
so y - ulp(y)/2 - 2^(EXP(x)-err) < f(x) < y-ulp(y)/2
so y -ulp(y) < f(x) < y-ulp(y)/2
=> nexttozero(y)
elif dirError = ToInf
we have x < f(x) < x + 2^(EXP(x)-err)
so y - ulp(y)/2 < f(x) < y+ulp(y)/2-ulp(y)/2
so y - ulp(y)/2 < f(x) < y
=> do nothing
elif i < 0 and i = -EVEN_ROUNDING
So rnd = RNDN and we have y = x - ulp(y)/2
if dirError = ToZero,
y < f(x) < y + ulp(y)/2 => do nothing
if dirError = ToInf
y + ulp(y)/2 < f(x) < y + ulp(y) => AddOneUlp
elif i > 0
we can't have rnd = RNDZ, and prec(x) > prec(y), so ulp(x) < ulp(y)
we have y - ulp (y) < x < y
or more exactly y - ulp(y) + ulp(x)/2 <= x <= y - ulp(x)/2
if rnd = RNDA,
if dirError = ToInf,
we have x < f(x) < x + 2^(EXP(x)-err)
if err > prec (x),
we have 2^(EXP(x)-err) < ulp(x), so 2^(EXP(x)-err) <= ulp(x)/2
so f(x) <= y - ulp(x)/2+ulp(x)/2 <= y
and y - ulp(y) < x < f(x)
so we have y - ulp(y) < f(x) < y
so do nothing.
elif we can round, ie y - ulp(y) < x + 2^(EXP(x)-err) < y
we have y - ulp(y) < x < f(x) < x + 2^(EXP(x)-err) < y
so do nothing
otherwise
Wrong. Example X=[0.11101]111111110000
+ 1111111111111111111....
elif dirError = ToZero
we have x - 2^(EXP(x)-err) < f(x) < x
so f(x) < x < y
if err > prec (x)
x-2^(EXP(x)-err) >= x-ulp(x)/2 >= y - ulp(y) + ulp(x)/2-ulp(x)/2
so y - ulp(y) < f(x) < y
so do nothing
elif we can round, ie y - ulp(y) < x - 2^(EXP(x)-err) < y
y - ulp(y) < x - 2^(EXP(x)-err) < f(x) < y
so do nothing
otherwise
Wrong. Example: X=[1.111010]00000010
- 10000001000000000000100....
elif rnd = RNDN,
y - ulp(y)/2 < x < y and we can't have x = y-ulp(y)/2:
so we have:
y - ulp(y)/2 + ulp(x)/2 <= x <= y - ulp(x)/2
if dirError = ToInf
we have x < f(x) < x+2^(EXP(x)-err) and ulp(y) > 2^(EXP(x)-err)
so y - ulp(y)/2 + ulp (x)/2 < f(x) < y + ulp (y)/2 - ulp (x)/2
we can round but we can't compute inexact flag.
if err > prec (x)
y - ulp(y)/2 + ulp (x)/2 < f(x) < y + ulp(x)/2 - ulp(x)/2
so y - ulp(y)/2 + ulp (x)/2 < f(x) < y
we can round and compute inexact flag. do nothing
elif we can round, ie y - ulp(y)/2 < x + 2^(EXP(x)-err) < y
we have y - ulp(y)/2 + ulp (x)/2 < f(x) < y
so do nothing
otherwise
Wrong
elif dirError = ToZero
we have x -2^(EXP(x)-err) < f(x) < x and ulp(y)/2 > 2^(EXP(x)-err)
so y-ulp(y)+ulp(x)/2 < f(x) < y - ulp(x)/2
if err > prec (x)
x- ulp(x)/2 < f(x) < x
so y - ulp(y)/2+ulp(x)/2 - ulp(x)/2 < f(x) < x <= y - ulp(x)/2 < y
do nothing
elif we can round, ie y-ulp(y)/2 < x-2^(EXP(x)-err) < y
we have y-ulp(y)/2 < x-2^(EXP(x)-err) < f(x) < x < y
do nothing
otherwise
Wrong
elif i < 0
same thing?
*/
int
mpfr_round_near_x (mpfr_ptr y, mpfr_srcptr v, mpfr_uexp_t err, int dir,
mp_rnd_t rnd)
{
int inexact, sign;
unsigned int old_flags = __gmpfr_flags;
MPFR_ASSERTD (!MPFR_IS_SINGULAR (v));
MPFR_ASSERTD (dir == 0 || dir == 1);
/* First check if we can round. The test is more restrictive than
necessary. Note that if err is not representable in an mp_exp_t,
then err > MPFR_PREC (v) and the conversion to mp_exp_t will not
occur. */
if (!(err > MPFR_PREC (y) + 1
&& (err > MPFR_PREC (v)
|| mpfr_round_p (MPFR_MANT (v), MPFR_LIMB_SIZE (v),
(mp_exp_t) err,
MPFR_PREC (y) + (rnd == GMP_RNDN)))))
/* If we assume we can not round, return 0, and y is not modified */
return 0;
/* First round v in y */
sign = MPFR_SIGN (v);
MPFR_SET_EXP (y, MPFR_GET_EXP (v));
MPFR_SET_SIGN (y, sign);
MPFR_RNDRAW_GEN (inexact, y, MPFR_MANT (v), MPFR_PREC (v), rnd, sign,
if (dir == 0)
{
inexact = -sign;
goto trunc_doit;
}
else
goto addoneulp;
, if (MPFR_UNLIKELY (++MPFR_EXP (y) > __gmpfr_emax))
mpfr_overflow (y, rnd, sign)
);
/* Fix it in some cases */
MPFR_ASSERTD (!MPFR_IS_NAN (y) && !MPFR_IS_ZERO (y));
/* If inexact == 0, setting y from v is exact but we haven't
take into account yet the error term */
if (inexact == 0)
{
if (dir == 0) /* The error term is negative for v positive */
{
inexact = sign;
if (MPFR_IS_LIKE_RNDZ (rnd, MPFR_IS_NEG_SIGN (sign)))
{
/* case nexttozero */
/* The underflow flag should be set if the result is zero */
__gmpfr_flags = old_flags;
inexact = -sign;
mpfr_nexttozero (y);
if (MPFR_UNLIKELY (MPFR_IS_ZERO (y)))
mpfr_set_underflow ();
}
}
else /* The error term is positive for v positive */
{
inexact = -sign;
/* Round Away */
if (rnd != GMP_RNDN && rnd != GMP_RNDZ
&& MPFR_IS_RNDUTEST_OR_RNDDNOTTEST (rnd, MPFR_IS_POS_SIGN(sign)))
{
/* case nexttoinf */
/* The overflow flag should be set if the result is infinity */
inexact = sign;
mpfr_nexttoinf (y);
if (MPFR_UNLIKELY (MPFR_IS_INF (y)))
mpfr_set_overflow ();
}
}
}
/* the inexact flag cannot be 0, since this would mean an exact value,
and in this case we cannot round correctly */
MPFR_ASSERTD(inexact != 0);
MPFR_RET (inexact);
}