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/* mpfr_sqrt -- square root of a floating-point number
Copyright 1999, 2000, 2001, 2002, 2003, 2004 Free Software Foundation, Inc.
This file is part of the MPFR Library.
The 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 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 MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */
#include "mpfr-impl.h"
int
mpfr_sqrt (mpfr_ptr r, mpfr_srcptr u, mp_rnd_t rnd_mode)
{
mp_size_t rsize = MPFR_LIMB_SIZE(r); /* number of limbs of r */
mp_size_t rrsize = rsize + rsize;
mp_size_t usize = MPFR_LIMB_SIZE(u); /* 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 = MPFR_MANT(r);
mp_ptr up = MPFR_MANT(u);
mp_ptr sp;
mp_ptr tp;
mp_limb_t sticky0 = MPFR_LIMB_ZERO; /* truncated part of input */
mp_limb_t sticky1 = MPFR_LIMB_ZERO; /* truncated part of rp[0] */
mp_limb_t sticky;
int odd_exp = (unsigned int) MPFR_GET_EXP (u) & 1;
int sh; /* number of extra bits in rp[0] */
int inexact = -1; /* return ternary flag */
TMP_DECL(marker);
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);
sp = (mp_limb_t *) 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);
while (sticky0 == MPFR_LIMB_ZERO && k != 0)
sticky0 = up[--k];
}
/* 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;
MPFR_SET_EXP(r, (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 */
{
sh = mpn_cmp (tp, rp, rsize);
if (sh > 0)
inexact = 1;
else if (sh < 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))
{
MPFR_EXP(r) += 1;
rp[rsize - 1] = MPFR_LIMB_HIGHBIT;
}
truncate: /* inexact = 0 or -1 */
TMP_FREE(marker);
return inexact;
}
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