/* mpfr_round_near_x -- Round a floating point number nears another one.
Copyright 2005-2017 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
http://www.gnu.org/licenses/ or 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,
mpfr_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,
mpfr_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 mpfr_exp_t,
then err > MPFR_PREC (v) and the conversion to mpfr_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),
(mpfr_exp_t) err,
MPFR_PREC (y) + (rnd == MPFR_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 != MPFR_RNDN && !MPFR_IS_LIKE_RNDZ (rnd, MPFR_IS_NEG_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);
}