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/* mpn_sqrtrem_new -- integer square root with remainder
(should be directly integrated in a future release of GNU MP)
Copyright 1999, 2000, 2001 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 <stdio.h> /* for NULL */
#include "gmp.h"
#include "gmp-impl.h"
#include "longlong.h"
#define MP_LIMB_T_ONE ((mp_limb_t) 1)
static mp_size_t mpn_sqrtrem1 (mp_ptr, mp_ptr, mp_srcptr);
static mp_limb_t mpn_sqrtrem2 (mp_ptr, mp_ptr, mp_srcptr);
static mp_limb_t mpn_dq_sqrtrem (mp_ptr, mp_ptr, mp_size_t);
mp_size_t mpn_sqrtrem_new (mp_ptr, mp_ptr, mp_srcptr, mp_size_t);
/* table generated by isqrt(2^8*i) $ i=64..255 in MuPAD */
static const unsigned char approx_tab[192] = {
128, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141,
142, 143, 144, 144, 145, 146, 147, 148, 149, 150, 150, 151, 152, 153,
154, 155, 155, 156, 157, 158, 159, 160, 160, 161, 162, 163, 163, 164,
165, 166, 167, 167, 168, 169, 170, 170, 171, 172, 173, 173, 174, 175,
176, 176, 177, 178, 178, 179, 180, 181, 181, 182, 183, 183, 184, 185,
185, 186, 187, 187, 188, 189, 189, 190, 191, 192, 192, 193, 193, 194,
195, 195, 196, 197, 197, 198, 199, 199, 200, 201, 201, 202, 203, 203,
204, 204, 205, 206, 206, 207, 208, 208, 209, 209, 210, 211, 211, 212,
212, 213, 214, 214, 215, 215, 216, 217, 217, 218, 218, 219, 219, 220,
221, 221, 222, 222, 223, 224, 224, 225, 225, 226, 226, 227, 227, 228,
229, 229, 230, 230, 231, 231, 232, 232, 233, 234, 234, 235, 235, 236,
236, 237, 237, 238, 238, 239, 240, 240, 241, 241, 242, 242, 243, 243,
244, 244, 245, 245, 246, 246, 247, 247, 248, 248, 249, 249, 250, 250,
251, 251, 252, 252, 253, 253, 254, 254, 255};
/* same as mpn_sqrtrem, but for size=1 and {np, 1} normalized */
static mp_size_t
mpn_sqrtrem1 (mp_ptr sp, mp_ptr rp, mp_srcptr np)
{
mp_limb_t np0, s, r, q, u;
int prec;
/* first compute a 8-bit approximation from the high 8-bits of np[0] */
np0 = np[0];
q = np0 >> (BITS_PER_MP_LIMB - 8);
/* 2^6 = 64 <= q < 256 = 2^8 */
s = approx_tab[q - 64]; /* 128 <= s < 255 */
r = (np0 >> (BITS_PER_MP_LIMB - 16)) - s * s; /* r <= 2*s */
if (r > 2 * s) { r -= 2 * s + 1; s++; }
#ifdef DEBUG
if (r > 2*s) {
fprintf(stderr, "Error: r > 2*s, np[0]=%lu r=%lu s=%lu\n", np[0], r, s); exit(1);
}
#endif
prec = 8;
np0 <<= 2 * prec;
while (2*prec < BITS_PER_MP_LIMB) {
/* invariant: s has prec bits, and r <= 2*s */
r = (r << prec) + (np0 >> (BITS_PER_MP_LIMB - prec));
np0 <<= prec;
u = 2 * s;
q = r / u;
u = r - q * u;
s = (s << prec) + q;
u = (u << prec) + (np0 >> (BITS_PER_MP_LIMB - prec));
q = q * q;
r = u - q;
if (u < q) {
r += 2 * s - 1;
s --;
}
np0 <<= prec;
prec = 2 * prec;
}
sp[0] = s;
if (rp != NULL) rp[0] = r;
return (r) ? 1 : 0;
}
#define Prec (BITS_PER_MP_LIMB >> 1)
/* same as mpn_sqrtrem, but for size=2 and {np, 2} normalized
return cc such that {np, 2} = sp[0]^2 + cc*2^BITS+PER_MP_LIMB + rp[0] */
static mp_limb_t
mpn_sqrtrem2 (mp_ptr sp, mp_ptr rp, mp_srcptr np)
{
mp_limb_t qhl, q, u, np0;
int cc;
np0 = np[0];
mpn_sqrtrem1(sp, rp, np+1);
qhl = 0;
while (rp[0] >= sp[0]) {
qhl ++;
rp[0] -= sp[0];
}
/* now rp[0] < sp[0] < 2^Prec */
rp[0] = (rp[0] << Prec) + (np0 >> Prec);
u = 2 * sp[0];
q = rp[0] / u;
u = rp[0] - q * u;
q += (qhl & 1) << (Prec-1);
qhl >>= 1; /* if qhl=1, necessary q=0 as qhl*2^Prec + q <= 2^Prec */
/* now we have (initial rp[0])<<Prec + np0>>Prec = (qhl<<Prec + q) * (2sp[0]) + u */
sp[0] = ((sp[0]+qhl) << Prec) + q;
cc = u >> Prec;
rp[0] = (u << Prec) + (np0 & ((MP_LIMB_T_ONE << Prec) - MP_LIMB_T_ONE));
/* subtract q * q or qhl*2^(2*Prec) from rp */
cc -= mpn_sub_1(rp, rp, 1, q * q) + qhl;
/* now subtract 2*q*2^Prec + 2^(2*Prec) if qhl is set */
if (cc < 0) {
cc += (sp[0]) ? mpn_add_1(rp, rp, 1, sp[0]) : 1;
cc += mpn_add_1(rp, rp, 1, --sp[0]);
}
return cc;
}
/* writes in {sp, n} the square root (rounded towards zero) of {np, 2n},
and in {np, n} the low n limbs of the remainder, returns the high
limb of the remainder (which is 0 or 1).
Assumes {np, 2n} is normalized, i.e. np[2n-1] >= B/4
where B=2^BITS_PER_MP_LIMB.
*/
static mp_limb_t
mpn_dq_sqrtrem (mp_ptr sp, mp_ptr np, mp_size_t n)
{
mp_limb_t q; /* carry out of {sp, n} */
int c, b; /* carry out of remainder */
mp_size_t l, h;
if (n == 1) return mpn_sqrtrem2 (sp, np, np);
l = n / 2;
h = n - l;
q = mpn_dq_sqrtrem (sp + l, np + 2 * l, h);
if (q) mpn_sub_n (np + 2 * l, np + 2 * l, sp + l, h);
q += mpn_divrem (sp, 0, np + l, n, sp + l, h);
c = sp[0] & 1;
mpn_rshift (sp, sp, l, 1);
sp[l-1] |= q << (BITS_PER_MP_LIMB - 1);
q >>= 1;
if (c) c = mpn_add_n (np + l, np + l, sp + l, h);
mpn_sqr_n (np + n, sp, l);
b = q + mpn_sub_n (np, np, np + n, 2 * l);
c -= (l == h) ? b : mpn_sub_1 (np + 2 * l, np + 2 * l, 1, b);
q = mpn_add_1 (sp + l, sp + l, h, q);
if (c < 0) {
c += mpn_addmul_1 (np, sp, n, 2) + 2 * q;
c -= mpn_sub_1 (np, np, n, 1);
q -= mpn_sub_1 (sp, sp, n, 1);
}
return c;
}
/* main function */
mp_size_t
mpn_sqrtrem_new (mp_ptr sp, mp_ptr rp, mp_srcptr np, mp_size_t nn)
{
mp_limb_t *tp, s0[1], cc, high, rl;
int c;
mp_size_t rn, tn;
TMP_DECL (marker);
/* If OP is zero, both results are zero. */
if (nn == 0)
return 0;
high = np[nn-1];
if (nn == 1 && (high & MP_LIMB_T_HIGHBIT))
return mpn_sqrtrem1 (sp, rp, np);
count_leading_zeros(c, high);
c = c / 2; /* we have to shift left by 2c bits to normalize {np, nn} */
tn = (nn+1) / 2; /* 2*tn is the smallest even integer >= nn */
TMP_MARK (marker);
if ((nn % 2) || (c > 0)) {
tp = (mp_limb_t*) TMP_ALLOC (2 * tn * BYTES_PER_MP_LIMB);
tp[0] = 0; /* needed only when 2*tn > nn, but saves a test */
if (c) mpn_lshift(tp + 2*tn - nn, np, nn, 2 * c);
else MPN_COPY (tp + 2*tn - nn, np, nn);
rl = mpn_dq_sqrtrem (sp, tp, tn);
/* we have 2^(2k)*N = S^2 + R where k = c + (2tn-nn)*BITS_PER_MP_LIMB/2,
thus 2^(2k)*N = (S-s0)^2 + 2*S*s0 - s0^2 + R where s0=S mod 2^k */
c += (nn % 2) * BITS_PER_MP_LIMB / 2; /* c now represents k */
s0[0] = sp[0] & ((MP_LIMB_T_ONE << c) - MP_LIMB_T_ONE); /* S mod 2^k */
rl += mpn_addmul_1 (tp, sp, tn, 2 * s0[0]); /* R = R + 2*s0*S */
cc = mpn_submul_1 (tp, s0, 1, s0[0]);
rl -= (tn > 1) ? mpn_sub_1(tp + 1, tp + 1, tn - 1, cc) : cc;
mpn_rshift (sp, sp, tn, c);
tp[tn] = rl;
if (rp == NULL) rp = tp;
c = c << 1;
if (c < BITS_PER_MP_LIMB) tn++; else { tp++; c -= BITS_PER_MP_LIMB; }
if (c) mpn_rshift (rp, tp, tn, c); else MPN_COPY(rp, tp, tn);
rn = tn;
}
else {
if (rp == NULL)
rp = (mp_limb_t*) TMP_ALLOC (nn * BYTES_PER_MP_LIMB);
if (rp != np) MPN_COPY (rp, np, nn);
rn = tn + (rp[tn] = mpn_dq_sqrtrem (sp, rp, tn));
}
MPN_NORMALIZE (rp, rn);
TMP_FREE (marker);
return rn;
}
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