efee5258bc
The MPFR library is a C library for multiple-precision floating-point computations with exact rounding (also called correct rounding). It is based on the GMP multiple-precision library and should replace the MPF class in further releases of GMP. GCC >= 4.2 requires MPFR.
257 lines
8.4 KiB
C
257 lines
8.4 KiB
C
/* mpfr_sqrt -- square root of a floating-point number
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Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
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Contributed by the Arenaire and Cacao projects, INRIA.
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This file is part of the GNU MPFR Library.
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The GNU MPFR Library is free software; you can redistribute it and/or modify
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it under the terms of the GNU Lesser General Public License as published by
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the Free Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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The GNU MPFR Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
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License for more details.
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You should have received a copy of the GNU Lesser General Public License
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along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
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http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
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51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
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#include "mpfr-impl.h"
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int
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mpfr_sqrt (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
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{
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mp_size_t rsize; /* number of limbs of r */
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mp_size_t rrsize;
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mp_size_t usize; /* number of limbs of u */
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mp_size_t tsize; /* number of limbs of the sqrtrem remainder */
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mp_size_t k;
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mp_size_t l;
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mpfr_limb_ptr rp;
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mpfr_limb_ptr up;
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mpfr_limb_ptr sp;
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mpfr_limb_ptr tp;
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mp_limb_t sticky0; /* truncated part of input */
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mp_limb_t sticky1; /* truncated part of rp[0] */
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mp_limb_t sticky;
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int odd_exp;
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int sh; /* number of extra bits in rp[0] */
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int inexact; /* return ternary flag */
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mpfr_exp_t expr;
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MPFR_TMP_DECL(marker);
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MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", u, u, rnd_mode),
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("y[%#R]=%R inexact=%d", r, r, inexact));
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if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(u)))
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{
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if (MPFR_IS_NAN(u))
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{
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MPFR_SET_NAN(r);
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MPFR_RET_NAN;
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}
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else if (MPFR_IS_ZERO(u))
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{
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/* 0+ or 0- */
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MPFR_SET_SAME_SIGN(r, u);
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MPFR_SET_ZERO(r);
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MPFR_RET(0); /* zero is exact */
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}
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else
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{
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MPFR_ASSERTD(MPFR_IS_INF(u));
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/* sqrt(-Inf) = NAN */
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if (MPFR_IS_NEG(u))
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{
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MPFR_SET_NAN(r);
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MPFR_RET_NAN;
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}
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MPFR_SET_POS(r);
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MPFR_SET_INF(r);
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MPFR_RET(0);
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}
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}
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if (MPFR_UNLIKELY(MPFR_IS_NEG(u)))
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{
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MPFR_SET_NAN(r);
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MPFR_RET_NAN;
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}
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MPFR_SET_POS(r);
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rsize = MPFR_LIMB_SIZE(r); /* number of limbs of r */
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rrsize = rsize + rsize;
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usize = MPFR_LIMB_SIZE(u); /* number of limbs of u */
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rp = MPFR_MANT(r);
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up = MPFR_MANT(u);
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sticky0 = MPFR_LIMB_ZERO; /* truncated part of input */
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sticky1 = MPFR_LIMB_ZERO; /* truncated part of rp[0] */
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odd_exp = (unsigned int) MPFR_GET_EXP (u) & 1;
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inexact = -1; /* return ternary flag */
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MPFR_TMP_MARK (marker);
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sp = (mp_limb_t *) MPFR_TMP_ALLOC (rrsize * sizeof (mp_limb_t));
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/* copy the most significant limbs of u to {sp, rrsize} */
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if (MPFR_LIKELY(usize <= rrsize)) /* in case r and u have the same precision,
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we have indeed rrsize = 2 * usize */
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{
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k = rrsize - usize;
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if (MPFR_LIKELY(k))
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MPN_ZERO (sp, k);
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if (odd_exp)
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{
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if (MPFR_LIKELY(k))
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sp[k - 1] = mpn_rshift (sp + k, up, usize, 1);
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else
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sticky0 = mpn_rshift (sp, up, usize, 1);
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}
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else
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MPN_COPY (sp + rrsize - usize, up, usize);
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}
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else /* usize > rrsize: truncate the input */
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{
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k = usize - rrsize;
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if (odd_exp)
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sticky0 = mpn_rshift (sp, up + k, rrsize, 1);
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else
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MPN_COPY (sp, up + k, rrsize);
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l = k;
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while (sticky0 == MPFR_LIMB_ZERO && l != 0)
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sticky0 = up[--l];
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}
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/* sticky0 is non-zero iff the truncated part of the input is non-zero */
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tsize = mpn_sqrtrem (rp, tp = sp, sp, rrsize);
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l = tsize;
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sticky = sticky0;
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while (sticky == MPFR_LIMB_ZERO && l != 0)
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sticky = tp[--l];
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/* truncated low bits of rp[0] */
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MPFR_UNSIGNED_MINUS_MODULO(sh,MPFR_PREC(r));
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sticky1 = rp[0] & MPFR_LIMB_MASK(sh);
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rp[0] -= sticky1;
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sticky = sticky || sticky1;
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expr = (MPFR_GET_EXP(u) + odd_exp) / 2; /* exact */
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if (rnd_mode == MPFR_RNDZ || rnd_mode == MPFR_RNDD || sticky == MPFR_LIMB_ZERO)
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{
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inexact = (sticky == MPFR_LIMB_ZERO) ? 0 : -1;
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goto truncate;
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}
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else if (rnd_mode == MPFR_RNDN)
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{
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/* if sh>0, the round bit is bit (sh-1) of sticky1
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and the sticky bit is formed by the low sh-1 bits from
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sticky1, together with {tp, tsize} and sticky0. */
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if (sh)
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{
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if (sticky1 & (MPFR_LIMB_ONE << (sh - 1)))
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{ /* round bit is set */
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if (sticky1 == (MPFR_LIMB_ONE << (sh - 1)) && tsize == 0
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&& sticky0 == 0)
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goto even_rule;
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else
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goto add_one_ulp;
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}
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else /* round bit is zero */
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goto truncate; /* with the default inexact=-1 */
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}
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else
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{
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/* if sh=0, we have to compare {tp, tsize} with {rp, rsize}:
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if {tp, tsize} < {rp, rsize}: truncate
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if {tp, tsize} > {rp, rsize}: round up
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if {tp, tsize} = {rp, rsize}: compare the truncated part of the
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input to 1/4
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if < 1/4: truncate
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if > 1/4: round up
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if = 1/4: even rounding rule
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Set inexact = -1 if truncate
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inexact = 1 if add one ulp
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inexact = 0 if even rounding rule
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*/
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if (tsize < rsize)
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inexact = -1;
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else if (tsize > rsize) /* FIXME: may happen? */
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inexact = 1;
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else /* tsize = rsize */
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{
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int cmp;
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cmp = mpn_cmp (tp, rp, rsize);
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if (cmp > 0)
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inexact = 1;
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else if (cmp < 0 || sticky0 == MPFR_LIMB_ZERO)
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inexact = -1;
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/* now tricky case {tp, tsize} = {rp, rsize} */
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/* in case usize <= rrsize, the only case where sticky0 <> 0
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is when the exponent of u is odd and usize = rrsize (k=0),
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but in that case the truncated part is exactly 1/2, thus
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we have to round up.
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If the exponent of u is odd, and up[k] is odd, the truncated
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part is >= 1/2, so we round up too. */
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else if (usize <= rrsize || (odd_exp && (up[k] & MPFR_LIMB_ONE)))
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inexact = 1;
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else
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{
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/* now usize > rrsize:
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(a) if the exponent of u is even, the 1/4 bit is the
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2nd most significant bit of up[k-1];
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(b) if the exponent of u is odd, the 1/4 bit is the
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1st most significant bit of up[k-1]; */
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sticky1 = MPFR_LIMB_ONE << (GMP_NUMB_BITS - 2 + odd_exp);
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if (up[k - 1] < sticky1)
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inexact = -1;
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else if (up[k - 1] > sticky1)
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inexact = 1;
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else
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{
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/* up[k - 1] == sticky1: consider low k-1 limbs */
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while (--k > 0 && up[k - 1] == MPFR_LIMB_ZERO)
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;
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inexact = (k != 0);
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}
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} /* end of case {tp, tsize} = {rp, rsize} */
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} /* end of case tsize = rsize */
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if (inexact == -1)
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goto truncate;
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else if (inexact == 1)
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goto add_one_ulp;
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/* else go through even_rule */
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}
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}
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else /* rnd_mode=GMP_RDNU, necessarily sticky <> 0, thus add 1 ulp */
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goto add_one_ulp;
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even_rule: /* has to set inexact */
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inexact = (rp[0] & (MPFR_LIMB_ONE << sh)) ? 1 : -1;
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if (inexact == -1)
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goto truncate;
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/* else go through add_one_ulp */
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add_one_ulp:
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inexact = 1; /* always here */
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if (mpn_add_1 (rp, rp, rsize, MPFR_LIMB_ONE << sh))
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{
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expr ++;
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rp[rsize - 1] = MPFR_LIMB_HIGHBIT;
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}
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truncate: /* inexact = 0 or -1 */
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MPFR_ASSERTN (expr >= MPFR_EMIN_MIN && expr <= MPFR_EMAX_MAX);
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MPFR_EXP (r) = expr;
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MPFR_TMP_FREE(marker);
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return mpfr_check_range (r, inexact, rnd_mode);
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}
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