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/* mpfr_sub1 -- internal function to perform a "real" subtraction
Copyright 2001-2020 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. */
#include "mpfr-impl.h"
/* compute sign(b) * (|b| - |c|), with |b| > |c|, diff_exp = EXP(b) - EXP(c)
Returns 0 iff result is exact,
a negative value when the result is less than the exact value,
a positive value otherwise.
*/
int
mpfr_sub1 (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode)
{
int sign;
mpfr_exp_t diff_exp, exp_a, exp_b;
mpfr_prec_t cancel, cancel1;
mp_size_t cancel2, an, bn, cn, cn0;
mp_limb_t *ap, *bp, *cp;
mp_limb_t carry, bb, cc;
mpfr_prec_t aq, bq;
int inexact, shift_b, shift_c, add_exp = 0;
int cmp_low = 0; /* used for rounding to nearest: 0 if low(b) = low(c),
negative if low(b) < low(c), positive if low(b) > low(c) */
int sh, k;
MPFR_TMP_DECL(marker);
MPFR_TMP_MARK(marker);
ap = MPFR_MANT(a);
an = MPFR_LIMB_SIZE(a);
(void) MPFR_GET_PREC (a);
(void) MPFR_GET_PREC (b);
(void) MPFR_GET_PREC (c);
sign = mpfr_cmp2 (b, c, &cancel);
if (MPFR_UNLIKELY(sign == 0))
{
MPFR_LOG_MSG (("sign=0\n", 0));
if (rnd_mode == MPFR_RNDD)
MPFR_SET_NEG (a);
else
MPFR_SET_POS (a);
MPFR_SET_ZERO (a);
MPFR_RET (0);
}
/* sign != 0, so that cancel has a valid value. */
MPFR_LOG_MSG (("sign=%d cancel=%Pd\n", sign, cancel));
MPFR_ASSERTD (cancel >= 0 && cancel <= MPFR_PREC_MAX);
/*
* If subtraction: sign(a) = sign * sign(b)
* If addition: sign(a) = sign of the larger argument in absolute value.
*
* Both cases can be simplified in:
* if (sign>0)
* if addition: sign(a) = sign * sign(b) = sign(b)
* if subtraction, b is greater, so sign(a) = sign(b)
* else
* if subtraction, sign(a) = - sign(b)
* if addition, sign(a) = sign(c) (since c is greater)
* But if it is an addition, sign(b) and sign(c) are opposed!
* So sign(a) = - sign(b)
*/
if (sign < 0) /* swap b and c so that |b| > |c| */
{
mpfr_srcptr t;
MPFR_SET_OPPOSITE_SIGN (a,b);
t = b; b = c; c = t;
}
else
MPFR_SET_SAME_SIGN (a,b);
if (MPFR_UNLIKELY (MPFR_IS_UBF (b) || MPFR_IS_UBF (c)))
{
exp_b = MPFR_UBF_GET_EXP (b);
/* Early underflow detection. Rare, but a test is needed anyway
since in the "MAX (aq, bq) + 2 <= diff_exp" branch, the exponent
may decrease and MPFR_EXP_MIN would yield an integer overflow. */
if (MPFR_UNLIKELY (exp_b < __gmpfr_emin - 1))
{
if (rnd_mode == MPFR_RNDN)
rnd_mode = MPFR_RNDZ;
return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a));
}
diff_exp = mpfr_ubf_diff_exp (b, c);
MPFR_LOG_MSG (("UBF: exp_b=%" MPFR_EXP_FSPEC "d%s "
"diff_exp=%" MPFR_EXP_FSPEC "d%s\n",
(mpfr_eexp_t) exp_b,
exp_b == MPFR_EXP_MAX ? "=MPFR_EXP_MAX" : "",
(mpfr_eexp_t) diff_exp,
diff_exp == MPFR_EXP_MAX ? "=MPFR_EXP_MAX" : ""));
/* If diff_exp == MPFR_EXP_MAX, the actual value can be larger,
but anyway, since mpfr_exp_t >= mp_size_t, this will be the
case c small below, and the exact value does not matter. */
/* mpfr_set4 below used with MPFR_RNDF does not support UBF. */
if (rnd_mode == MPFR_RNDF)
rnd_mode = MPFR_RNDN;
}
else
{
exp_b = MPFR_GET_EXP (b);
diff_exp = exp_b - MPFR_GET_EXP (c);
}
MPFR_ASSERTD (diff_exp >= 0);
aq = MPFR_GET_PREC (a);
bq = MPFR_GET_PREC (b);
/* Check if c is too small.
A more precise test is to replace 2 by
(rnd == MPFR_RNDN) + mpfr_power2_raw (b)
but it is more expensive and not very useful */
if (MPFR_UNLIKELY (MAX (aq, bq) + 2 <= diff_exp))
{
MPFR_LOG_MSG (("case c small\n", 0));
/* Remember, we can't have an exact result! */
/* A.AAAAAAAAAAAAAAAAA
= B.BBBBBBBBBBBBBBB
- C.CCCCCCCCCCCCC */
/* A = S*ABS(B) +/- ulp(a) */
/* since we can't have an exact result, for RNDF we can truncate b */
if (rnd_mode == MPFR_RNDF)
return mpfr_set4 (a, b, MPFR_RNDZ, MPFR_SIGN (a));
exp_a = exp_b; /* may be any out-of-range value due to UBF */
MPFR_RNDRAW_EVEN (inexact, a, MPFR_MANT (b), bq,
rnd_mode, MPFR_SIGN (a),
if (exp_a != MPFR_EXP_MAX)
exp_a ++);
MPFR_LOG_MSG (("inexact=%d\n", inexact));
if (inexact == 0 &&
/* a = b, but the exact value of b - c is a bit below. Then,
except for directed rounding similar to toward zero and
before overflow checking: a is the correctly rounded value
and since |b| - |c| < |a|, the ternary value value is given
by the sign of a. */
! MPFR_IS_LIKE_RNDZ (rnd_mode, MPFR_IS_NEG (a)))
{
MPFR_LOG_MSG (("c small, case 1\n", 0));
inexact = MPFR_INT_SIGN (a);
}
else if (inexact != 0 &&
/* A.AAAAAAAAAAAAAA
= B.BBBBBBBBBBBBBBB
- C.CCCCCCCCCCCCC */
/* It isn't exact, so PREC(b) > PREC(a) and the last
PREC(b)-PREC(a) bits of b are not all zeros.
Subtracting c from b will not have an effect on the rounding
except in case of a midpoint in the round-to-nearest mode,
when the even rounding was done away from zero instead of
toward zero.
In case of even rounding:
1.BBBBBBBBBBBBBx10
- 1.CCCCCCCCCCCC
= 1.BBBBBBBBBBBBBx01 Rounded to PREC(b)
= 1.BBBBBBBBBBBBBx Nearest / Rounded to PREC(a)
Set gives:
1.BBBBBBBBBBBBB0 if inexact == EVEN_INEX (x == 0)
1.BBBBBBBBBBBBB1+1 if inexact == -EVEN_INEX (x == 1)
which means we get a wrong rounded result if x == 1,
i.e. inexact == MPFR_EVEN_INEX (for positive numbers). */
MPFR_LIKELY (inexact != MPFR_EVEN_INEX * MPFR_INT_SIGN (a)))
{
MPFR_LOG_MSG (("c small, case 2\n", 0));
/* nothing to do */
}
else
{
/* We need to take the value preceding |a|. We can't use
mpfr_nexttozero due to a possible out-of-range exponent.
But this will allow us to have more specific code. */
MPFR_LOG_MSG (("c small, case 3: correcting the value of a\n", 0));
sh = (mpfr_prec_t) an * GMP_NUMB_BITS - aq;
mpn_sub_1 (ap, ap, an, MPFR_LIMB_ONE << sh);
if (MPFR_UNLIKELY (MPFR_LIMB_MSB (ap[an-1]) == 0))
{
exp_a --;
/* The following is valid whether an = 1 or an > 1. */
ap[an-1] |= MPFR_LIMB_HIGHBIT;
}
inexact = - MPFR_INT_SIGN (a);
}
/* The underflow case is possible only with UBF. The overflow case
is also possible with normal FP due to rounding. */
if (MPFR_UNLIKELY (exp_a > __gmpfr_emax))
return mpfr_overflow (a, rnd_mode, MPFR_SIGN (a));
if (MPFR_UNLIKELY (exp_a < __gmpfr_emin))
{
if (rnd_mode == MPFR_RNDN &&
(exp_a < __gmpfr_emin - 1 ||
(inexact * MPFR_INT_SIGN (a) >= 0 && mpfr_powerof2_raw (a))))
rnd_mode = MPFR_RNDZ;
return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a));
}
MPFR_SET_EXP (a, exp_a);
MPFR_RET (inexact);
}
/* reserve a space to store b aligned with the result, i.e. shifted by
(-cancel) % GMP_NUMB_BITS to the right */
bn = MPFR_LIMB_SIZE (b);
MPFR_UNSIGNED_MINUS_MODULO (shift_b, cancel);
cancel1 = (cancel + shift_b) / GMP_NUMB_BITS;
/* the high cancel1 limbs from b should not be taken into account */
if (MPFR_UNLIKELY (shift_b == 0))
{
bp = MPFR_MANT(b); /* no need of an extra space */
/* Ensure ap != bp */
if (MPFR_UNLIKELY (ap == bp))
{
bp = MPFR_TMP_LIMBS_ALLOC (bn);
MPN_COPY (bp, ap, bn);
}
}
else
{
bp = MPFR_TMP_LIMBS_ALLOC (bn + 1);
bp[0] = mpn_rshift (bp + 1, MPFR_MANT(b), bn++, shift_b);
}
/* reserve a space to store c aligned with the result, i.e. shifted by
(diff_exp-cancel) % GMP_NUMB_BITS to the right */
cn = MPFR_LIMB_SIZE (c);
if (IS_POW2 (GMP_NUMB_BITS))
shift_c = ((mpfr_uexp_t) diff_exp - cancel) % GMP_NUMB_BITS;
else
{
/* The above operation does not work if diff_exp - cancel < 0. */
shift_c = diff_exp - (cancel % GMP_NUMB_BITS);
shift_c = (shift_c + GMP_NUMB_BITS) % GMP_NUMB_BITS;
}
MPFR_ASSERTD (shift_c >= 0 && shift_c < GMP_NUMB_BITS);
if (MPFR_UNLIKELY(shift_c == 0))
{
cp = MPFR_MANT(c);
/* Ensure ap != cp */
if (ap == cp)
{
cp = MPFR_TMP_LIMBS_ALLOC (cn);
MPN_COPY(cp, ap, cn);
}
}
else
{
cp = MPFR_TMP_LIMBS_ALLOC (cn + 1);
cp[0] = mpn_rshift (cp + 1, MPFR_MANT(c), cn++, shift_c);
}
#if 0
MPFR_LOG_MSG (("rnd=%s shift_b=%d shift_c=%d diffexp=%" MPFR_EXP_FSPEC
"d\n", mpfr_print_rnd_mode (rnd_mode), shift_b, shift_c,
(mpfr_eexp_t) diff_exp));
#endif
MPFR_ASSERTD (ap != cp);
MPFR_ASSERTD (bp != cp);
/* here we have shift_c = (diff_exp - cancel) % GMP_NUMB_BITS,
0 <= shift_c < GMP_NUMB_BITS
thus we want cancel2 = ceil((cancel - diff_exp) / GMP_NUMB_BITS) */
/* Possible optimization with a C99 compiler (i.e. well-defined
integer division): if MPFR_PREC_MAX is reduced to
((mpfr_prec_t)((mpfr_uprec_t)(~(mpfr_uprec_t)0)>>1) - GMP_NUMB_BITS + 1)
and diff_exp is of type mpfr_exp_t (no need for mpfr_uexp_t, since
the sum or difference of 2 exponents must be representable, as used
by the multiplication code), then the computation of cancel2 could
be simplified to
cancel2 = (cancel - (diff_exp - shift_c)) / GMP_NUMB_BITS;
because cancel, diff_exp and shift_c are all nonnegative and
these variables are signed. */
MPFR_ASSERTD (cancel >= 0);
if (cancel >= diff_exp)
/* Note that cancel is signed and will be converted to mpfr_uexp_t
(type of diff_exp) in the expression below, so that this will
work even if cancel is very large and diff_exp = 0. */
cancel2 = (cancel - diff_exp + (GMP_NUMB_BITS - 1)) / GMP_NUMB_BITS;
else
cancel2 = - (mp_size_t) ((diff_exp - cancel) / GMP_NUMB_BITS);
/* the high cancel2 limbs from b should not be taken into account */
#if 0
MPFR_LOG_MSG (("cancel=%Pd cancel1=%Pd cancel2=%Pd\n",
cancel, cancel1, cancel2));
#endif
/* ap[an-1] ap[0]
<----------------+-----------|---->
<----------PREC(a)----------><-sh->
cancel1
limbs bp[bn-cancel1-1]
<--...-----><----------------+-----------+----------->
cancel2
limbs cp[cn-cancel2-1] cancel2 >= 0
<--...--><----------------+----------------+---------------->
(-cancel2) cancel2 < 0
limbs <----------------+---------------->
*/
/* first part: put in ap[0..an-1] the value of high(b) - high(c),
where high(b) consists of the high an+cancel1 limbs of b,
and high(c) consists of the high an+cancel2 limbs of c.
*/
/* copy high(b) into a */
if (MPFR_LIKELY(an + (mp_size_t) cancel1 <= bn))
/* a: <----------------+-----------|---->
b: <-----------------------------------------> */
MPN_COPY (ap, bp + bn - (an + cancel1), an);
else
/* a: <----------------+-----------|---->
b: <-------------------------> */
if ((mp_size_t) cancel1 < bn) /* otherwise b does not overlap with a */
{
MPN_ZERO (ap, an + cancel1 - bn);
MPN_COPY (ap + (an + cancel1 - bn), bp, bn - cancel1);
}
else
MPN_ZERO (ap, an);
/* subtract high(c) */
if (MPFR_LIKELY(an + cancel2 > 0)) /* otherwise c does not overlap with a */
{
mp_limb_t *ap2;
if (cancel2 >= 0)
{
if (an + cancel2 <= cn)
/* a: <----------------------------->
c: <-----------------------------------------> */
mpn_sub_n (ap, ap, cp + cn - (an + cancel2), an);
else
/* a: <---------------------------->
c: <-------------------------> */
{
ap2 = ap + an + (cancel2 - cn);
if (cn > cancel2)
mpn_sub_n (ap2, ap2, cp, cn - cancel2);
}
}
else /* cancel2 < 0 */
{
mp_limb_t borrow;
if (an + cancel2 <= cn)
/* a: <----------------------------->
c: <-----------------------------> */
borrow = mpn_sub_n (ap, ap, cp + cn - (an + cancel2),
an + cancel2);
else
/* a: <---------------------------->
c: <----------------> */
{
ap2 = ap + an + cancel2 - cn;
borrow = mpn_sub_n (ap2, ap2, cp, cn);
}
ap2 = ap + an + cancel2;
mpn_sub_1 (ap2, ap2, -cancel2, borrow);
}
}
/* now perform rounding */
sh = (mpfr_prec_t) an * GMP_NUMB_BITS - aq;
/* last unused bits from a */
carry = ap[0] & MPFR_LIMB_MASK (sh);
ap[0] -= carry;
if (rnd_mode == MPFR_RNDF)
{
inexact = 0;
/* truncating is always correct since -1 ulp < low(b) - low(c) < 1 ulp */
goto truncate;
}
else if (rnd_mode == MPFR_RNDN)
{
if (MPFR_LIKELY(sh))
{
/* can decide except when carry = 2^(sh-1) [middle]
or carry = 0 [truncate, but cannot decide inexact flag] */
if (carry > (MPFR_LIMB_ONE << (sh - 1)))
goto add_one_ulp;
else if ((0 < carry) && (carry < (MPFR_LIMB_ONE << (sh - 1))))
{
inexact = -1; /* result if smaller than exact value */
goto truncate;
}
/* now carry = 2^(sh-1), in which case cmp_low=2,
or carry = 0, in which case cmp_low=0 */
cmp_low = (carry == 0) ? 0 : 2;
}
}
else /* directed rounding: set rnd_mode to RNDZ iff toward zero */
{
if (MPFR_IS_RNDUTEST_OR_RNDDNOTTEST(rnd_mode, MPFR_IS_NEG(a)))
rnd_mode = MPFR_RNDZ;
if (carry)
{
if (rnd_mode == MPFR_RNDZ)
{
inexact = -1;
goto truncate;
}
else /* round away */
goto add_one_ulp;
}
}
/* we have to consider the low (bn - (an+cancel1)) limbs from b,
and the (cn - (an+cancel2)) limbs from c. */
bn -= an + cancel1;
cn0 = cn;
cn -= an + cancel2;
#if 0
MPFR_LOG_MSG (("last sh=%d bits from a are %Mu, bn=%Pd, cn=%Pd\n",
sh, carry, (mpfr_prec_t) bn, (mpfr_prec_t) cn));
#endif
/* for rounding to nearest, we couldn't conclude up to here in the following
cases:
1. sh = 0, then cmp_low=0: we can either truncate, subtract one ulp
or add one ulp: -1 ulp < low(b)-low(c) < 1 ulp
2. sh > 0 but the low sh bits from high(b)-high(c) equal 2^(sh-1):
-0.5 ulp <= -1/2^sh < low(b)-low(c)-0.5 < 1/2^sh <= 0.5 ulp
we can't decide the rounding, in that case cmp_low=2:
either we truncate and flag=-1, or we add one ulp and flag=1
3. the low sh>0 bits from high(b)-high(c) equal 0: we know we have to
truncate but we can't decide the ternary value, here cmp_low=0:
-0.5 ulp <= -1/2^sh < low(b)-low(c) < 1/2^sh <= 0.5 ulp
we always truncate and inexact can be any of -1,0,1
*/
/* note: here cn might exceed cn0, in which case we consider a zero limb */
for (k = 0; (bn > 0) || (cn > 0); k = 1)
{
/* if cmp_low < 0, we know low(b) - low(c) < 0
if cmp_low > 0, we know low(b) - low(c) > 0
(more precisely if cmp_low = 2, low(b) - low(c) = 0.5 ulp so far)
if cmp_low = 0, so far low(b) - low(c) = 0 */
/* get next limbs */
bb = (bn > 0) ? bp[--bn] : 0;
if ((cn > 0) && (cn-- <= cn0))
cc = cp[cn];
else
cc = 0;
/* cmp_low compares low(b) and low(c) */
if (cmp_low == 0) /* case 1 or 3 */
cmp_low = (bb < cc) ? -2+k : (bb > cc) ? 1 : 0;
/* Case 1 for k=0 splits into 7 subcases:
1a: bb > cc + half
1b: bb = cc + half
1c: 0 < bb - cc < half
1d: bb = cc
1e: -half < bb - cc < 0
1f: bb - cc = -half
1g: bb - cc < -half
Case 2 splits into 3 subcases:
2a: bb > cc
2b: bb = cc
2c: bb < cc
Case 3 splits into 3 subcases:
3a: bb > cc
3b: bb = cc
3c: bb < cc
*/
/* the case rounding to nearest with sh=0 is special since one couldn't
subtract above 1/2 ulp in the trailing limb of the result */
if (rnd_mode == MPFR_RNDN && sh == 0 && k == 0) /* case 1 for k=0 */
{
mp_limb_t half = MPFR_LIMB_HIGHBIT;
/* add one ulp if bb > cc + half
truncate if cc - half < bb < cc + half
sub one ulp if bb < cc - half
*/
if (cmp_low < 0) /* bb < cc: -1 ulp < low(b) - low(c) < 0,
cases 1e, 1f and 1g */
{
if (cc >= half)
cc -= half;
else /* since bb < cc < half, bb+half < 2*half */
bb += half;
/* now we have bb < cc + half:
we have to subtract one ulp if bb < cc,
and truncate if bb > cc */
}
else if (cmp_low >= 0) /* bb >= cc, cases 1a to 1d */
{
if (cc < half)
cc += half;
else /* since bb >= cc >= half, bb - half >= 0 */
bb -= half;
/* now we have bb > cc - half: we have to add one ulp if bb > cc,
and truncate if bb < cc */
if (cmp_low > 0)
cmp_low = 2;
}
}
#if 0
MPFR_LOG_MSG (("k=%d bb=%Mu cc=%Mu cmp_low=%d\n", k, bb, cc, cmp_low));
#endif
if (cmp_low < 0) /* low(b) - low(c) < 0: either truncate or subtract
one ulp */
{
if (rnd_mode == MPFR_RNDZ)
goto sub_one_ulp; /* set inexact=-1 */
else if (rnd_mode != MPFR_RNDN) /* round away */
{
inexact = 1;
goto truncate;
}
else /* round to nearest */
{
/* If cmp_low < 0 and bb > cc, then -0.5 ulp < low(b)-low(c) < 0,
whatever the value of sh.
If sh>0, then cmp_low < 0 implies that the initial neglected
sh bits were 0 (otherwise cmp_low=2 initially), thus the
weight of the new bits is less than 0.5 ulp too.
If k > 0 (and sh=0) this means that either the first neglected
limbs bb and cc were equal (thus cmp_low was 0 for k=0),
or we had bb - cc = -0.5 ulp or 0.5 ulp.
The last case is not possible here since we would have
cmp_low > 0 which is sticky.
In the first case (where we have cmp_low = -1), we truncate,
whereas in the 2nd case we have cmp_low = -2 and we subtract
one ulp.
*/
if (bb > cc || sh > 0 || cmp_low == -1)
{ /* -0.5 ulp < low(b)-low(c) < 0,
bb > cc corresponds to cases 1e and 1f1
sh > 0 corresponds to cases 3c and 3b3
cmp_low = -1 corresponds to case 1d3 (also 3b3) */
inexact = 1;
goto truncate;
}
else if (bb < cc) /* here sh = 0 and low(b)-low(c) < -0.5 ulp,
this corresponds to cases 1g and 1f3 */
goto sub_one_ulp;
/* the only case where we can't conclude is sh=0 and bb=cc,
i.e., we have low(b) - low(c) = -0.5 ulp (up to now), thus
we don't know if we must truncate or subtract one ulp.
Note: for sh=0 we can't have low(b) - low(c) = -0.5 ulp up to
now, since low(b) - low(c) > 1/2^sh */
}
}
else if (cmp_low > 0) /* 0 < low(b) - low(c): either truncate or
add one ulp */
{
if (rnd_mode == MPFR_RNDZ)
{
inexact = -1;
goto truncate;
}
else if (rnd_mode != MPFR_RNDN) /* round away */
goto add_one_ulp;
else /* round to nearest */
{
if (bb > cc)
{
/* if sh=0, then bb>cc means that low(b)-low(c) > 0.5 ulp,
and similarly when cmp_low=2 */
if (cmp_low == 2) /* cases 1a, 1b1, 2a and 2b1 */
goto add_one_ulp;
/* sh > 0 and cmp_low > 0: this implies that the sh initial
neglected bits were 0, and the remaining low(b)-low(c)>0,
but its weight is less than 0.5 ulp */
else /* 0 < low(b) - low(c) < 0.5 ulp, this corresponds to
cases 3a, 1d1 and 3b1 */
{
inexact = -1;
goto truncate;
}
}
else if (bb < cc) /* 0 < low(b) - low(c) < 0.5 ulp, cases 1c,
1b3, 2b3 and 2c */
{
inexact = -1;
goto truncate;
}
/* the only case where we can't conclude is bb=cc, i.e.,
low(b) - low(c) = 0.5 ulp (up to now), thus we don't know
if we must truncate or add one ulp. */
}
}
/* after k=0, we cannot conclude in the following cases, we split them
according to the values of bb and cc for k=1:
1b. sh=0 and cmp_low = 1 and bb-cc = half [around 0.5 ulp]
1b1. bb > cc: add one ulp, inex = 1
1b2: bb = cc: cannot conclude
1b3: bb < cc: truncate, inex = -1
1d. sh=0 and cmp_low = 0 and bb-cc = 0 [around 0]
1d1: bb > cc: truncate, inex = -1
1d2: bb = cc: cannot conclude
1d3: bb < cc: truncate, inex = +1
1f. sh=0 and cmp_low = -1 and bb-cc = -half [around -0.5 ulp]
1f1: bb > cc: truncate, inex = +1
1f2: bb = cc: cannot conclude
1f3: bb < cc: sub one ulp, inex = -1
2b. sh > 0 and cmp_low = 2 and bb=cc [around 0.5 ulp]
2b1. bb > cc: add one ulp, inex = 1
2b2: bb = cc: cannot conclude
2b3: bb < cc: truncate, inex = -1
3b. sh > 0 and cmp_low = 0 [around 0]
3b1. bb > cc: truncate, inex = -1
3b2: bb = cc: cannot conclude
3b3: bb < cc: truncate, inex = +1
*/
}
if ((rnd_mode == MPFR_RNDN) && cmp_low != 0)
{
/* even rounding rule */
if ((ap[0] >> sh) & 1)
{
if (cmp_low < 0)
goto sub_one_ulp;
else
goto add_one_ulp;
}
else
inexact = (cmp_low > 0) ? -1 : 1;
}
else
inexact = 0;
goto truncate;
sub_one_ulp: /* sub one unit in last place to a */
mpn_sub_1 (ap, ap, an, MPFR_LIMB_ONE << sh);
inexact = -1;
goto end_of_sub;
add_one_ulp: /* add one unit in last place to a */
if (MPFR_UNLIKELY(mpn_add_1 (ap, ap, an, MPFR_LIMB_ONE << sh)))
/* result is a power of 2: 11111111111111 + 1 = 1000000000000000 */
{
ap[an-1] = MPFR_LIMB_HIGHBIT;
add_exp = 1;
}
inexact = 1; /* result larger than exact value */
truncate:
if (MPFR_UNLIKELY((ap[an-1] >> (GMP_NUMB_BITS - 1)) == 0))
/* case 1 - epsilon */
{
ap[an-1] = MPFR_LIMB_HIGHBIT;
add_exp = 1;
}
end_of_sub:
/* we have to set MPFR_EXP(a) to MPFR_EXP(b) - cancel + add_exp, taking
care of underflows/overflows in that computation, and of the allowed
exponent range */
MPFR_TMP_FREE (marker);
if (MPFR_LIKELY(cancel))
{
cancel -= add_exp; /* OK: add_exp is an int equal to 0 or 1 */
MPFR_ASSERTD (cancel >= 0);
/* Detect an underflow case to avoid a possible integer overflow
with UBF in the computation of exp_a. */
if (MPFR_UNLIKELY (exp_b < __gmpfr_emin - 1))
{
if (rnd_mode == MPFR_RNDN)
rnd_mode = MPFR_RNDZ;
return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a));
}
exp_a = exp_b - cancel;
/* The following assertion corresponds to a limitation of the MPFR
implementation. It may fail with a 32-bit ABI and huge precisions,
but this is practically impossible with a 64-bit ABI. This kind
of issue is not specific to this function. */
MPFR_ASSERTN (exp_b != MPFR_EXP_MAX || exp_a > __gmpfr_emax);
if (MPFR_UNLIKELY (exp_a < __gmpfr_emin))
{
underflow:
if (rnd_mode == MPFR_RNDN &&
(exp_a < __gmpfr_emin - 1 ||
(inexact >= 0 && mpfr_powerof2_raw (a))))
rnd_mode = MPFR_RNDZ;
return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a));
}
/* We cannot have an overflow here, except for UBFs. Indeed:
exp_a = exp_b - cancel + add_exp <= emax - 1 + 1 <= emax.
For UBFs, we can have exp_b > emax. */
if (exp_a > __gmpfr_emax)
{
MPFR_ASSERTD (exp_b > __gmpfr_emax); /* since exp_b >= exp_a */
return mpfr_overflow (a, rnd_mode, MPFR_SIGN (a));
}
}
else /* cancel = 0: MPFR_EXP(a) <- MPFR_EXP(b) + add_exp */
{
/* in case cancel = 0, add_exp can still be 1, in case b is just
below a power of two, c is very small, prec(a) < prec(b),
and rnd=away or nearest */
MPFR_ASSERTD (add_exp == 0 || add_exp == 1);
/* Overflow iff exp_b + add_exp > __gmpfr_emax in Z, but we do
a subtraction below to avoid a potential integer overflow in
the case exp_b == MPFR_EXP_MAX. */
if (MPFR_UNLIKELY (exp_b > __gmpfr_emax - add_exp))
return mpfr_overflow (a, rnd_mode, MPFR_SIGN (a));
exp_a = exp_b + add_exp;
/* Warning: an underflow can happen for UBFs, for example when
mpfr_add is called from mpfr_fmma or mpfr_fmms. */
if (MPFR_UNLIKELY (exp_a < __gmpfr_emin))
goto underflow;
MPFR_ASSERTD (exp_a >= __gmpfr_emin);
}
MPFR_SET_EXP (a, exp_a);
/* check that result is msb-normalized */
MPFR_ASSERTD(ap[an-1] > ~ap[an-1]);
MPFR_RET (inexact * MPFR_INT_SIGN (a));
}
|
