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/* mpfr_sqrt -- square root of a floating-point number
Copyright 1999, 2000, 2001, 2002, 2003, 2004, 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.
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"
int
mpfr_sqrt (mpfr_ptr r, mpfr_srcptr u, mp_rnd_t rnd_mode)
{
mp_size_t rsize; /* number of limbs of r */
mp_size_t rrsize;
mp_size_t usize; /* number of limbs of u */
mp_size_t tsize; /* number of limbs of the sqrtrem remainder */
mp_size_t k;
mp_size_t l;
mp_ptr rp;
mp_ptr up;
mp_ptr sp;
mp_ptr tp;
mp_limb_t sticky0; /* truncated part of input */
mp_limb_t sticky1; /* truncated part of rp[0] */
mp_limb_t sticky;
int odd_exp;
int sh; /* number of extra bits in rp[0] */
int inexact; /* return ternary flag */
mp_exp_t expr;
MPFR_TMP_DECL(marker);
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", u, u, rnd_mode),
("y[%#R]=%R inexact=%d", r, r, inexact));
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(u)))
{
if (MPFR_IS_NAN(u))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
else if (MPFR_IS_ZERO(u))
{
/* 0+ or 0- */
MPFR_SET_SAME_SIGN(r, u);
MPFR_SET_ZERO(r);
MPFR_RET(0); /* zero is exact */
}
else
{
MPFR_ASSERTD(MPFR_IS_INF(u));
/* sqrt(-Inf) = NAN */
if (MPFR_IS_NEG(u))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
MPFR_SET_POS(r);
MPFR_SET_INF(r);
MPFR_RET(0);
}
}
if (MPFR_UNLIKELY(MPFR_IS_NEG(u)))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
MPFR_CLEAR_FLAGS(r);
MPFR_SET_POS(r);
rsize = MPFR_LIMB_SIZE(r); /* number of limbs of r */
rrsize = rsize + rsize;
usize = MPFR_LIMB_SIZE(u); /* number of limbs of u */
rp = MPFR_MANT(r);
up = MPFR_MANT(u);
sticky0 = MPFR_LIMB_ZERO; /* truncated part of input */
sticky1 = MPFR_LIMB_ZERO; /* truncated part of rp[0] */
odd_exp = (unsigned int) MPFR_GET_EXP (u) & 1;
inexact = -1; /* return ternary flag */
MPFR_TMP_MARK (marker);
sp = (mp_limb_t *) MPFR_TMP_ALLOC (rrsize * sizeof (mp_limb_t));
/* copy the most significant limbs of u to {sp, rrsize} */
if (MPFR_LIKELY(usize <= rrsize)) /* in case r and u have the same precision,
we have indeed rrsize = 2 * usize */
{
k = rrsize - usize;
if (MPFR_LIKELY(k))
MPN_ZERO (sp, k);
if (odd_exp)
{
if (MPFR_LIKELY(k))
sp[k - 1] = mpn_rshift (sp + k, up, usize, 1);
else
sticky0 = mpn_rshift (sp, up, usize, 1);
}
else
MPN_COPY (sp + rrsize - usize, up, usize);
}
else /* usize > rrsize: truncate the input */
{
k = usize - rrsize;
if (odd_exp)
sticky0 = mpn_rshift (sp, up + k, rrsize, 1);
else
MPN_COPY (sp, up + k, rrsize);
l = k;
while (sticky0 == MPFR_LIMB_ZERO && l != 0)
sticky0 = up[--l];
}
/* sticky0 is non-zero iff the truncated part of the input is non-zero */
tsize = mpn_sqrtrem (rp, tp = sp, sp, rrsize);
l = tsize;
sticky = sticky0;
while (sticky == MPFR_LIMB_ZERO && l != 0)
sticky = tp[--l];
/* truncated low bits of rp[0] */
MPFR_UNSIGNED_MINUS_MODULO(sh,MPFR_PREC(r));
sticky1 = rp[0] & MPFR_LIMB_MASK(sh);
rp[0] -= sticky1;
sticky = sticky || sticky1;
expr = (MPFR_GET_EXP(u) + odd_exp) / 2; /* exact */
if (rnd_mode == GMP_RNDZ || rnd_mode == GMP_RNDD || sticky == MPFR_LIMB_ZERO)
{
inexact = (sticky == MPFR_LIMB_ZERO) ? 0 : -1;
goto truncate;
}
else if (rnd_mode == GMP_RNDN)
{
/* if sh>0, the round bit is bit (sh-1) of sticky1
and the sticky bit is formed by the low sh-1 bits from
sticky1, together with {tp, tsize} and sticky0. */
if (sh)
{
if (sticky1 & (MPFR_LIMB_ONE << (sh - 1)))
{ /* round bit is set */
if (sticky1 == (MPFR_LIMB_ONE << (sh - 1)) && tsize == 0
&& sticky0 == 0)
goto even_rule;
else
goto add_one_ulp;
}
else /* round bit is zero */
goto truncate; /* with the default inexact=-1 */
}
else
{
/* if sh=0, we have to compare {tp, tsize} with {rp, rsize}:
if {tp, tsize} < {rp, rsize}: truncate
if {tp, tsize} > {rp, rsize}: round up
if {tp, tsize} = {rp, rsize}: compare the truncated part of the
input to 1/4
if < 1/4: truncate
if > 1/4: round up
if = 1/4: even rounding rule
Set inexact = -1 if truncate
inexact = 1 if add one ulp
inexact = 0 if even rounding rule
*/
if (tsize < rsize)
inexact = -1;
else if (tsize > rsize) /* FIXME: may happen? */
inexact = 1;
else /* tsize = rsize */
{
int cmp;
cmp = mpn_cmp (tp, rp, rsize);
if (cmp > 0)
inexact = 1;
else if (cmp < 0 || sticky0 == MPFR_LIMB_ZERO)
inexact = -1;
/* now tricky case {tp, tsize} = {rp, rsize} */
/* in case usize <= rrsize, the only case where sticky0 <> 0
is when the exponent of u is odd and usize = rrsize (k=0),
but in that case the truncated part is exactly 1/2, thus
we have to round up.
If the exponent of u is odd, and up[k] is odd, the truncated
part is >= 1/2, so we round up too. */
else if (usize <= rrsize || (odd_exp && (up[k] & MPFR_LIMB_ONE)))
inexact = 1;
else
{
/* now usize > rrsize:
(a) if the exponent of u is even, the 1/4 bit is the
2nd most significant bit of up[k-1];
(b) if the exponent of u is odd, the 1/4 bit is the
1st most significant bit of up[k-1]; */
sticky1 = MPFR_LIMB_ONE << (BITS_PER_MP_LIMB - 2 + odd_exp);
if (up[k - 1] < sticky1)
inexact = -1;
else if (up[k - 1] > sticky1)
inexact = 1;
else
{
/* up[k - 1] == sticky1: consider low k-1 limbs */
while (--k > 0 && up[k - 1] == MPFR_LIMB_ZERO)
;
inexact = (k != 0);
}
} /* end of case {tp, tsize} = {rp, rsize} */
} /* end of case tsize = rsize */
if (inexact == -1)
goto truncate;
else if (inexact == 1)
goto add_one_ulp;
/* else go through even_rule */
}
}
else /* rnd_mode=GMP_RDNU, necessarily sticky <> 0, thus add 1 ulp */
goto add_one_ulp;
even_rule: /* has to set inexact */
inexact = (rp[0] & (MPFR_LIMB_ONE << sh)) ? 1 : -1;
if (inexact == -1)
goto truncate;
/* else go through add_one_ulp */
add_one_ulp:
inexact = 1; /* always here */
if (mpn_add_1 (rp, rp, rsize, MPFR_LIMB_ONE << sh))
{
expr ++;
rp[rsize - 1] = MPFR_LIMB_HIGHBIT;
}
truncate: /* inexact = 0 or -1 */
MPFR_ASSERTN (expr >= MPFR_EMIN_MIN && expr <= MPFR_EMAX_MAX);
MPFR_EXP (r) = expr;
MPFR_TMP_FREE(marker);
return mpfr_check_range (r, inexact, rnd_mode);
}
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