vasilito
ff4ff35918
Red Bear OS is a full fork. All sources must be available from git clone with zero network access. Removed gitignore rules that excluded fetched source trees under recipes/*/source/, local/recipes/kde/*/source/, local/recipes/qt/*/source/, and vendor source trees. Build artifacts (target/, build/, source.tar, *.o, *.so) remain excluded. 127291 files added — kernel, relibc, base, bootloader, pkgar, all KDE/Qt frameworks, mesa, wayland, DRM drivers, and every other recipe source.
719 lines
22 KiB
C
719 lines
22 KiB
C
/* mpfr_sqrt -- square root of a floating-point number
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Copyright 1999-2025 Free Software Foundation, Inc.
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Contributed by the Pascaline and Caramba 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.
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If not, see <https://www.gnu.org/licenses/>. */
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#define MPFR_NEED_LONGLONG_H
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#include "mpfr-impl.h"
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#if !defined(MPFR_GENERIC_ABI) && GMP_NUMB_BITS == 64
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#include "invsqrt_limb.h"
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/* Put in rp[1]*2^64+rp[0] an approximation of floor(sqrt(2^128*n)),
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with 2^126 <= n := np[1]*2^64 + np[0] < 2^128. We have:
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{rp, 2} - 4 <= floor(sqrt(2^128*n)) <= {rp, 2} + 26. */
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static void
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mpfr_sqrt2_approx (mpfr_limb_ptr rp, mpfr_limb_srcptr np)
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{
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mp_limb_t x, r1, r0, h, l;
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__gmpfr_sqrt_limb (r1, h, l, x, np[1]);
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/* now r1 = floor(sqrt(2^64*n1)) and h:l = 2^64*n1 - r1^2 with h:l <= 2*r1,
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thus h <= 1, and x is an approximation of 2^96/sqrt(np[1])-2^64 */
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l += np[0];
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h += (l < np[0]);
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/* now 2^64*n1 + n0 - r1^2 = 2^64*h + l with h <= 2 */
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/* divide by 2 */
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l = (h << 63) | (l >> 1);
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h = h >> 1;
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/* now h <= 1 */
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/* now add (2^64+x) * (h*2^64+l) / 2^64 to [r1*2^64, 0] */
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umul_hi (r0, x, l); /* x * l */
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r0 += l;
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r1 += h + (r0 < l); /* now we have added 2^64 * (h*2^64+l) */
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if (h)
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{
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r0 += x;
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r1 += (r0 < x); /* add x */
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}
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MPFR_ASSERTD(r1 & MPFR_LIMB_HIGHBIT);
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rp[0] = r0;
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rp[1] = r1;
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}
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/* Special code for prec(r) = prec(u) < GMP_NUMB_BITS. We cannot have
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prec(u) = GMP_NUMB_BITS here, since when the exponent of u is odd,
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we need to shift u by one bit to the right without losing any bit.
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Assumes GMP_NUMB_BITS = 64. */
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static int
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mpfr_sqrt1 (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
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{
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mpfr_prec_t p = MPFR_GET_PREC(r);
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mpfr_prec_t exp_u = MPFR_EXP(u), exp_r, sh = GMP_NUMB_BITS - p;
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mp_limb_t u0, r0, rb, sb, mask = MPFR_LIMB_MASK(sh);
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mpfr_limb_ptr rp = MPFR_MANT(r);
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MPFR_STAT_STATIC_ASSERT (GMP_NUMB_BITS == 64);
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/* first make the exponent even */
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u0 = MPFR_MANT(u)[0];
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if (((unsigned int) exp_u & 1) != 0)
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{
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u0 >>= 1;
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exp_u ++;
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}
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MPFR_ASSERTD (((unsigned int) exp_u & 1) == 0);
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exp_r = exp_u / 2;
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/* then compute an approximation of the integer square root of
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u0*2^GMP_NUMB_BITS */
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__gmpfr_sqrt_limb_approx (r0, u0);
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sb = 1; /* when we can round correctly with the approximation, the sticky bit
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is non-zero */
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/* the exact square root is in [r0, r0 + 7] */
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if (MPFR_UNLIKELY(((r0 + 7) & (mask >> 1)) <= 7))
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{
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/* We should ensure r0 has its most significant bit set.
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Since r0 <= sqrt(2^64*u0) <= r0 + 7, as soon as sqrt(2^64*u0)>=2^63+7,
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which happens for u0>=2^62+8, then r0 >= 2^63.
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It thus remains to check that for 2^62 <= u0 <= 2^62+7,
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__gmpfr_sqrt_limb_approx (r0, u0) gives r0 >= 2^63, which is indeed
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the case:
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u0=4611686018427387904 r0=9223372036854775808
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u0=4611686018427387905 r0=9223372036854775808
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u0=4611686018427387906 r0=9223372036854775809
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u0=4611686018427387907 r0=9223372036854775810
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u0=4611686018427387908 r0=9223372036854775811
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u0=4611686018427387909 r0=9223372036854775812
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u0=4611686018427387910 r0=9223372036854775813
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u0=4611686018427387911 r0=9223372036854775814 */
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MPFR_ASSERTD(r0 >= MPFR_LIMB_HIGHBIT);
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umul_ppmm (rb, sb, r0, r0);
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sub_ddmmss (rb, sb, u0, 0, rb, sb);
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/* for the exact square root, we should have 0 <= rb:sb <= 2*r0 */
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while (!(rb == 0 || (rb == 1 && sb <= 2 * r0)))
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{
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/* subtract 2*r0+1 from rb:sb: subtract r0 before incrementing r0,
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then r0 after (which is r0+1) */
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rb -= (sb < r0);
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sb -= r0;
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r0 ++;
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rb -= (sb < r0);
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sb -= r0;
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}
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/* now we should have rb*2^64 + sb <= 2*r0 */
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MPFR_ASSERTD(rb == 0 || (rb == 1 && sb <= 2 * r0));
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sb = rb | sb;
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}
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rb = r0 & (MPFR_LIMB_ONE << (sh - 1));
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sb |= (r0 & mask) ^ rb;
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rp[0] = r0 & ~mask;
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/* rounding: sb = 0 implies rb = 0, since (rb,sb)=(1,0) is not possible */
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MPFR_ASSERTD (rb == 0 || sb != 0);
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/* Note: if 1 and 2 are in [emin,emax], no overflow nor underflow
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is possible */
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if (MPFR_UNLIKELY (exp_r > __gmpfr_emax))
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return mpfr_overflow (r, rnd_mode, 1);
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/* See comments in mpfr_div_1 */
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if (MPFR_UNLIKELY (exp_r < __gmpfr_emin))
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{
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if (rnd_mode == MPFR_RNDN)
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{
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/* If (1-2^(-p-1))*2^(emin-1) <= sqrt(u) < 2^(emin-1),
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then sqrt(u) would be rounded to 2^(emin-1) with unbounded
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exponent range, and there would be no underflow.
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But this case cannot happen if u has precision p.
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Indeed, we would have:
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(1-2^(-p-1))^2*2^(2*emin-2) <= u < 2^(2*emin-2), i.e.,
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(1-2^(-p)+2^(-2p-2))*2^(2*emin-2) <= u < 2^(2*emin-2),
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and there is no p-bit number in that interval. */
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/* If the result is <= 0.5^2^(emin-1), we should round to 0. */
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if (exp_r < __gmpfr_emin - 1 ||
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(rp[0] == MPFR_LIMB_HIGHBIT && sb == 0))
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rnd_mode = MPFR_RNDZ;
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}
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else if (MPFR_IS_LIKE_RNDA(rnd_mode, 0))
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{
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if (exp_r == __gmpfr_emin - 1 &&
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rp[0] == ~mask &&
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(rb | sb) != 0)
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goto rounding; /* no underflow */
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}
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return mpfr_underflow (r, rnd_mode, 1);
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}
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rounding:
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MPFR_EXP (r) = exp_r;
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if (sb == 0 /* implies rb = 0 */ || rnd_mode == MPFR_RNDF)
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{
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MPFR_ASSERTD (rb == 0 || rnd_mode == MPFR_RNDF);
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MPFR_ASSERTD(exp_r >= __gmpfr_emin);
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MPFR_ASSERTD(exp_r <= __gmpfr_emax);
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MPFR_RET (0);
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}
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else if (rnd_mode == MPFR_RNDN)
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{
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/* since sb <> 0, only rb is needed to decide how to round, and the exact
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middle is not possible */
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if (rb == 0)
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goto truncate;
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else
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goto add_one_ulp;
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}
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else if (MPFR_IS_LIKE_RNDZ(rnd_mode, 0))
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{
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truncate:
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MPFR_ASSERTD(exp_r >= __gmpfr_emin);
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MPFR_ASSERTD(exp_r <= __gmpfr_emax);
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MPFR_RET(-1);
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}
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else /* round away from zero */
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{
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add_one_ulp:
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rp[0] += MPFR_LIMB_ONE << sh;
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if (rp[0] == 0)
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{
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rp[0] = MPFR_LIMB_HIGHBIT;
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if (MPFR_UNLIKELY(exp_r + 1 > __gmpfr_emax))
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return mpfr_overflow (r, rnd_mode, 1);
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MPFR_ASSERTD(exp_r + 1 <= __gmpfr_emax);
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MPFR_ASSERTD(exp_r + 1 >= __gmpfr_emin);
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MPFR_SET_EXP (r, exp_r + 1);
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}
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MPFR_RET(1);
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}
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}
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/* Special code for prec(r) = prec(u) = GMP_NUMB_BITS. */
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static int
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mpfr_sqrt1n (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
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{
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mpfr_prec_t exp_u = MPFR_EXP(u), exp_r;
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mp_limb_t u0, r0, rb, sb, low;
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mpfr_limb_ptr rp = MPFR_MANT(r);
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MPFR_STAT_STATIC_ASSERT (GMP_NUMB_BITS == 64);
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MPFR_ASSERTD(MPFR_PREC(r) == GMP_NUMB_BITS);
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MPFR_ASSERTD(MPFR_PREC(u) <= GMP_NUMB_BITS);
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/* first make the exponent even */
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u0 = MPFR_MANT(u)[0];
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if (((unsigned int) exp_u & 1) != 0)
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{
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low = u0 << (GMP_NUMB_BITS - 1);
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u0 >>= 1;
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exp_u ++;
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}
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else
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low = 0; /* low part of u0 */
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MPFR_ASSERTD (((unsigned int) exp_u & 1) == 0);
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exp_r = exp_u / 2;
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/* then compute an approximation of the integer square root of
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u0*2^GMP_NUMB_BITS */
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__gmpfr_sqrt_limb_approx (r0, u0);
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/* the exact square root is in [r0, r0 + 7] */
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/* As shown in mpfr_sqrt1 above, r0 has its most significant bit set */
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MPFR_ASSERTD(r0 >= MPFR_LIMB_HIGHBIT);
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umul_ppmm (rb, sb, r0, r0);
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sub_ddmmss (rb, sb, u0, low, rb, sb);
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/* for the exact square root, we should have 0 <= rb:sb <= 2*r0 */
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while (!(rb == 0 || (rb == 1 && sb <= 2 * r0)))
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{
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/* subtract 2*r0+1 from rb:sb: subtract r0 before incrementing r0,
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then r0 after (which is r0+1) */
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rb -= (sb < r0);
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sb -= r0;
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r0 ++;
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rb -= (sb < r0);
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sb -= r0;
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}
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/* now we have u0*2^64+low = r0^2 + rb*2^64+sb, with rb*2^64+sb <= 2*r0 */
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MPFR_ASSERTD(rb == 0 || (rb == 1 && sb <= 2 * r0));
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/* We can't have the middle case u0*2^64 = (r0 + 1/2)^2 since
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(r0 + 1/2)^2 is not an integer.
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We thus rb = 1 whenever u0*2^64 > (r0 + 1/2)^2, thus rb*2^64 + sb > r0
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and the sticky bit is always 1, unless we had rb = sb = 0. */
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rb = rb || (sb > r0);
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sb = rb | sb;
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rp[0] = r0;
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/* rounding */
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/* Note: if 1 and 2 are in [emin,emax], no overflow nor underflow
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is possible */
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if (MPFR_UNLIKELY (exp_r > __gmpfr_emax))
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return mpfr_overflow (r, rnd_mode, 1);
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/* See comments in mpfr_div_1 */
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if (MPFR_UNLIKELY (exp_r < __gmpfr_emin))
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{
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if (rnd_mode == MPFR_RNDN)
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{
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/* the case rp[0] = 111...111 and rb = 1 cannot happen, since it
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would imply u0 >= (2^64-1/2)^2/2^64 thus u0 >= 2^64 */
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if (exp_r < __gmpfr_emin - 1 ||
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(rp[0] == MPFR_LIMB_HIGHBIT && sb == 0))
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rnd_mode = MPFR_RNDZ;
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}
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else if (MPFR_IS_LIKE_RNDA(rnd_mode, 0))
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{
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if (exp_r == __gmpfr_emin - 1 &&
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rp[0] == MPFR_LIMB_MAX &&
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(rb | sb) != 0)
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goto rounding; /* no underflow */
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}
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return mpfr_underflow (r, rnd_mode, 1);
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}
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/* sb = 0 can only occur when the square root is exact, i.e., rb = 0 */
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rounding:
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MPFR_EXP (r) = exp_r;
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if (sb == 0 /* implies rb = 0 */ || rnd_mode == MPFR_RNDF)
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{
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MPFR_ASSERTD(exp_r >= __gmpfr_emin);
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MPFR_ASSERTD(exp_r <= __gmpfr_emax);
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MPFR_RET (0);
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}
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else if (rnd_mode == MPFR_RNDN)
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{
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/* we can't have sb = 0, thus rb is enough */
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if (rb == 0)
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goto truncate;
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else
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goto add_one_ulp;
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}
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else if (MPFR_IS_LIKE_RNDZ(rnd_mode, 0))
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{
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truncate:
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MPFR_ASSERTD(exp_r >= __gmpfr_emin);
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MPFR_ASSERTD(exp_r <= __gmpfr_emax);
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MPFR_RET(-1);
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}
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else /* round away from zero */
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{
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add_one_ulp:
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rp[0] += MPFR_LIMB_ONE;
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if (rp[0] == 0)
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{
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rp[0] = MPFR_LIMB_HIGHBIT;
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if (MPFR_UNLIKELY(exp_r + 1 > __gmpfr_emax))
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return mpfr_overflow (r, rnd_mode, 1);
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MPFR_ASSERTD(exp_r + 1 <= __gmpfr_emax);
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MPFR_ASSERTD(exp_r + 1 >= __gmpfr_emin);
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MPFR_SET_EXP (r, exp_r + 1);
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}
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MPFR_RET(1);
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}
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}
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/* Special code for GMP_NUMB_BITS < prec(r) = prec(u) < 2*GMP_NUMB_BITS.
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Assumes GMP_NUMB_BITS=64. */
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static int
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mpfr_sqrt2 (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
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{
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mpfr_prec_t p = MPFR_GET_PREC(r);
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mpfr_limb_ptr up = MPFR_MANT(u), rp = MPFR_MANT(r);
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mp_limb_t np[4], rb, sb, mask;
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mpfr_prec_t exp_u = MPFR_EXP(u), exp_r, sh = 2 * GMP_NUMB_BITS - p;
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MPFR_STAT_STATIC_ASSERT (GMP_NUMB_BITS == 64);
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if (((unsigned int) exp_u & 1) != 0)
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{
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np[3] = up[1] >> 1;
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np[2] = (up[1] << (GMP_NUMB_BITS - 1)) | (up[0] >> 1);
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np[1] = up[0] << (GMP_NUMB_BITS - 1);
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exp_u ++;
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}
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else
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{
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np[3] = up[1];
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np[2] = up[0];
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np[1] = 0;
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}
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exp_r = exp_u / 2;
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|
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mask = MPFR_LIMB_MASK(sh);
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mpfr_sqrt2_approx (rp, np + 2);
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/* with n = np[3]*2^64+np[2], we have:
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{rp, 2} - 4 <= floor(sqrt(2^128*n)) <= {rp, 2} + 26, thus we can round
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correctly except when the number formed by the last sh-1 bits
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of rp[0] is in the range [-26, 4]. */
|
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if (MPFR_LIKELY(((rp[0] + 26) & (mask >> 1)) > 30))
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sb = 1;
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else
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{
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mp_limb_t tp[4], h, l;
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|
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np[0] = 0;
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mpn_sqr (tp, rp, 2);
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/* since we know s - 26 <= r <= s + 4 and 0 <= n^2 - s <= 2*s, we have
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-8*s-16 <= n - r^2 <= 54*s - 676, thus it suffices to compute
|
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n - r^2 modulo 2^192 */
|
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mpn_sub_n (tp, np, tp, 3);
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/* invariant: h:l = 2 * {rp, 2}, with upper bit implicit */
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h = (rp[1] << 1) | (rp[0] >> (GMP_NUMB_BITS - 1));
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l = rp[0] << 1;
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while ((mp_limb_signed_t) tp[2] < 0) /* approximation was too large */
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{
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/* subtract 1 to {rp, 2}, thus 2 to h:l */
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h -= (l <= MPFR_LIMB_ONE);
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l -= 2;
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/* add (1:h:l)+1 to {tp,3} */
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tp[0] += l + 1;
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tp[1] += h + (tp[0] < l);
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/* necessarily rp[1] has its most significant bit set */
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tp[2] += MPFR_LIMB_ONE + (tp[1] < h || (tp[1] == h && tp[0] < l));
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}
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/* now tp[2] >= 0 */
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/* now we want {tp, 4} <= 2 * {rp, 2}, which implies tp[2] <= 1 */
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while (tp[2] > 1 || (tp[2] == 1 && tp[1] > h) ||
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(tp[2] == 1 && tp[1] == h && tp[0] > l))
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{
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/* subtract (1:h:l)+1 from {tp,3} */
|
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tp[2] -= MPFR_LIMB_ONE + (tp[1] < h || (tp[1] == h && tp[0] <= l));
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|
tp[1] -= h + (tp[0] <= l);
|
|
tp[0] -= l + 1;
|
|
/* add 2 to h:l */
|
|
l += 2;
|
|
h += (l <= MPFR_LIMB_ONE);
|
|
}
|
|
/* restore {rp, 2} from h:l */
|
|
rp[1] = MPFR_LIMB_HIGHBIT | (h >> 1);
|
|
rp[0] = (h << (GMP_NUMB_BITS - 1)) | (l >> 1);
|
|
sb = tp[2] | tp[0] | tp[1];
|
|
}
|
|
|
|
rb = rp[0] & (MPFR_LIMB_ONE << (sh - 1));
|
|
sb |= (rp[0] & mask) ^ rb;
|
|
rp[0] = rp[0] & ~mask;
|
|
|
|
/* rounding */
|
|
if (MPFR_UNLIKELY (exp_r > __gmpfr_emax))
|
|
return mpfr_overflow (r, rnd_mode, 1);
|
|
|
|
/* See comments in mpfr_div_1 */
|
|
if (MPFR_UNLIKELY (exp_r < __gmpfr_emin))
|
|
{
|
|
if (rnd_mode == MPFR_RNDN)
|
|
{
|
|
if (exp_r < __gmpfr_emin - 1 || (rp[1] == MPFR_LIMB_HIGHBIT &&
|
|
rp[0] == MPFR_LIMB_ZERO && sb == 0))
|
|
rnd_mode = MPFR_RNDZ;
|
|
}
|
|
else if (MPFR_IS_LIKE_RNDA(rnd_mode, 0))
|
|
{
|
|
if (exp_r == __gmpfr_emin - 1 && (rp[1] == MPFR_LIMB_MAX &&
|
|
rp[0] == ~mask) && (rb | sb))
|
|
goto rounding; /* no underflow */
|
|
}
|
|
return mpfr_underflow (r, rnd_mode, 1);
|
|
}
|
|
|
|
rounding:
|
|
MPFR_EXP (r) = exp_r;
|
|
if (sb == 0 /* implies rb = 0 */ || rnd_mode == MPFR_RNDF)
|
|
{
|
|
MPFR_ASSERTD(exp_r >= __gmpfr_emin);
|
|
MPFR_ASSERTD(exp_r <= __gmpfr_emax);
|
|
MPFR_RET (0);
|
|
}
|
|
else if (rnd_mode == MPFR_RNDN)
|
|
{
|
|
/* since sb <> 0 now, only rb is needed */
|
|
if (rb == 0)
|
|
goto truncate;
|
|
else
|
|
goto add_one_ulp;
|
|
}
|
|
else if (MPFR_IS_LIKE_RNDZ(rnd_mode, 0))
|
|
{
|
|
truncate:
|
|
MPFR_ASSERTD(exp_r >= __gmpfr_emin);
|
|
MPFR_ASSERTD(exp_r <= __gmpfr_emax);
|
|
MPFR_RET(-1);
|
|
}
|
|
else /* round away from zero */
|
|
{
|
|
add_one_ulp:
|
|
rp[0] += MPFR_LIMB_ONE << sh;
|
|
rp[1] += rp[0] == 0;
|
|
if (rp[1] == 0)
|
|
{
|
|
rp[1] = MPFR_LIMB_HIGHBIT;
|
|
if (MPFR_UNLIKELY(exp_r + 1 > __gmpfr_emax))
|
|
return mpfr_overflow (r, rnd_mode, 1);
|
|
MPFR_ASSERTD(exp_r + 1 <= __gmpfr_emax);
|
|
MPFR_ASSERTD(exp_r + 1 >= __gmpfr_emin);
|
|
MPFR_SET_EXP (r, exp_r + 1);
|
|
}
|
|
MPFR_RET(1);
|
|
}
|
|
}
|
|
|
|
#endif /* !defined(MPFR_GENERIC_ABI) && GMP_NUMB_BITS == 64 */
|
|
|
|
int
|
|
mpfr_sqrt (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
|
|
{
|
|
mp_size_t rsize; /* number of limbs of r (plus 1 if exact limb multiple) */
|
|
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;
|
|
mpfr_limb_ptr rp, rp0;
|
|
mpfr_limb_ptr up;
|
|
mpfr_limb_ptr sp;
|
|
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 */
|
|
mpfr_exp_t expr;
|
|
mpfr_prec_t rq = MPFR_GET_PREC (r);
|
|
MPFR_TMP_DECL(marker);
|
|
|
|
MPFR_LOG_FUNC
|
|
(("x[%Pd]=%.*Rg rnd=%d", mpfr_get_prec (u), mpfr_log_prec, u, rnd_mode),
|
|
("y[%Pd]=%.*Rg inexact=%d",
|
|
mpfr_get_prec (r), mpfr_log_prec, 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_SET_POS(r);
|
|
|
|
#if !defined(MPFR_GENERIC_ABI) && GMP_NUMB_BITS == 64
|
|
{
|
|
mpfr_prec_t uq = MPFR_GET_PREC (u);
|
|
|
|
if (rq == uq)
|
|
{
|
|
if (rq < GMP_NUMB_BITS)
|
|
return mpfr_sqrt1 (r, u, rnd_mode);
|
|
|
|
if (GMP_NUMB_BITS < rq && rq < 2*GMP_NUMB_BITS)
|
|
return mpfr_sqrt2 (r, u, rnd_mode);
|
|
|
|
if (rq == GMP_NUMB_BITS)
|
|
return mpfr_sqrt1n (r, u, rnd_mode);
|
|
}
|
|
}
|
|
#endif
|
|
|
|
MPFR_TMP_MARK (marker);
|
|
MPFR_UNSIGNED_MINUS_MODULO (sh, rq);
|
|
if (sh == 0 && rnd_mode == MPFR_RNDN)
|
|
sh = GMP_NUMB_BITS; /* ugly case */
|
|
rsize = MPFR_LIMB_SIZE(r) + (sh == GMP_NUMB_BITS);
|
|
/* rsize is the number of limbs of r + 1 if exact limb multiple and rounding
|
|
to nearest, this is the number of wanted limbs for the square root */
|
|
rrsize = rsize + rsize;
|
|
usize = MPFR_LIMB_SIZE(u); /* number of limbs of u */
|
|
rp0 = MPFR_MANT(r);
|
|
rp = (sh < GMP_NUMB_BITS) ? rp0 : MPFR_TMP_LIMBS_ALLOC (rsize);
|
|
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 */
|
|
|
|
sp = MPFR_TMP_LIMBS_ALLOC (rrsize);
|
|
|
|
/* 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, NULL, sp, rrsize);
|
|
|
|
/* a return value of zero in mpn_sqrtrem indicates a perfect square */
|
|
sticky = sticky0 || tsize != 0;
|
|
|
|
/* truncate low bits of rp[0] */
|
|
sticky1 = rp[0] & ((sh < GMP_NUMB_BITS) ? MPFR_LIMB_MASK(sh)
|
|
: MPFR_LIMB_MAX);
|
|
rp[0] -= sticky1;
|
|
|
|
sticky = sticky || sticky1;
|
|
|
|
expr = (MPFR_GET_EXP(u) + odd_exp) / 2; /* exact */
|
|
|
|
if (rnd_mode == MPFR_RNDZ || rnd_mode == MPFR_RNDD ||
|
|
sticky == MPFR_LIMB_ZERO)
|
|
{
|
|
inexact = (sticky == MPFR_LIMB_ZERO) ? 0 : -1;
|
|
goto truncate;
|
|
}
|
|
else if (rnd_mode == MPFR_RNDN)
|
|
{
|
|
/* if sh < GMP_NUMB_BITS, 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 the sqrtrem remainder and sticky0. */
|
|
if (sh < GMP_NUMB_BITS)
|
|
{
|
|
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 /* sh = GMP_NUMB_BITS: the round bit is the most significant bit
|
|
of rp[0], and the remaining GMP_NUMB_BITS-1 bits contribute to
|
|
the sticky bit */
|
|
{
|
|
if (sticky1 & MPFR_LIMB_HIGHBIT)
|
|
{ /* round bit is set */
|
|
if (sticky1 == MPFR_LIMB_HIGHBIT && 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 /* rnd_mode=GMP_RDNU, necessarily sticky <> 0, thus add 1 ulp */
|
|
goto add_one_ulp;
|
|
|
|
even_rule: /* has to set inexact */
|
|
if (sh < GMP_NUMB_BITS)
|
|
inexact = (rp[0] & (MPFR_LIMB_ONE << sh)) ? 1 : -1;
|
|
else
|
|
inexact = (rp[1] & MPFR_LIMB_ONE) ? 1 : -1;
|
|
if (inexact == -1)
|
|
goto truncate;
|
|
/* else go through add_one_ulp */
|
|
|
|
add_one_ulp:
|
|
inexact = 1; /* always here */
|
|
if (sh == GMP_NUMB_BITS)
|
|
{
|
|
rp ++;
|
|
rsize --;
|
|
sh = 0;
|
|
}
|
|
/* now rsize = MPFR_LIMB_SIZE(r) */
|
|
if (mpn_add_1 (rp0, rp, rsize, MPFR_LIMB_ONE << sh))
|
|
{
|
|
expr ++;
|
|
rp0[rsize - 1] = MPFR_LIMB_HIGHBIT;
|
|
}
|
|
goto end;
|
|
|
|
truncate: /* inexact = 0 or -1 */
|
|
if (sh == GMP_NUMB_BITS)
|
|
MPN_COPY (rp0, rp + 1, rsize - 1);
|
|
|
|
end:
|
|
/* Do not use MPFR_SET_EXP because the range has not been checked yet. */
|
|
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);
|
|
}
|