mpir/mpn/generic/invert.c

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/* floating-point Newton, with inversion in 3M(n) */
/* mpn_invert
Copyright 2009 Paul Zimmermann
Copyright 2009 William Hart
This file is part of the MPIR Library.
The MPIR 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 MPIR 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 MPIR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
MA 02110-1301, USA. */
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
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#include "mpir.h"
#include "gmp-impl.h"
#include "longlong.h"
#define ZERO (mp_limb_t) 0
#define ONE (mp_limb_t) 1
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#define WRAP_AROUND_BOUND 1500
int
mpn_is_invert (mp_srcptr xp, mp_srcptr ap, mp_size_t n)
{
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int res = 1;
mp_size_t i;
mp_ptr tp, up;
mp_limb_t cy;
TMP_DECL;
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TMP_MARK;
tp = TMP_ALLOC_LIMBS (2 * n);
up = TMP_ALLOC_LIMBS (2 * n);
/* first check X*A < B^(2*n) */
mpn_mul_n (tp, xp, ap, n);
cy = mpn_add_n (tp + n, tp + n, ap, n); /* A * msb(X) */
if (cy != 0)
res = 0;
/* now check B^(2n) - X*A <= A */
mpn_com_n (tp, tp, 2 * n);
mpn_add_1 (tp, tp, 2 * n, 1); /* B^(2n) - X*A */
MPN_ZERO (up, 2 * n);
MPN_COPY (up, ap, n);
res = mpn_cmp (tp, up, 2 * n) <= 0;
TMP_FREE;
return res;
}
/* Input: A = {ap, n} with most significant bit set.
Output: X = B^n + {xp, n} where B = 2^GMP_NUMB_BITS.
X is a lower approximation of B^(2n)/A with implicit msb.
More precisely, one has:
A*X < B^(2n) <= A*(X+1)
or X = ceil(B^(2n)/A) - 1.
*/
void
mpn_invert (mp_ptr xp, mp_srcptr ap, mp_size_t n)
{
if (n == 1)
{
/* invert_limb returns min(B-1, floor(B^2/ap[0])-B),
which is B-1 when ap[0]=B/2, and 1 when ap[0]=B-1.
For X=B+xp[0], we have A*X < B^2 <= A*(X+1) where
the equality holds only when A=B/2.
We thus have A*X < B^2 <= A*(X+1).
*/
invert_limb (xp[0], ap[0]);
}
else if (n == 2)
{
mp_limb_t tp[4], up[2], sp[2], cy;
tp[0] = ZERO;
invert_limb (xp[1], ap[1]);
tp[3] = mpn_mul_1 (tp + 1, ap, 2, xp[1]);
cy = mpn_add_n (tp + 2, tp + 2, ap, 2);
while (cy) /* Xh is too large */
{
xp[1] --;
cy -= mpn_sub (tp + 1, tp + 1, 3, ap, 2);
}
/* tp[3] should be 111...111 */
mpn_com_n (sp, tp + 1, 2);
cy = mpn_add_1 (sp, sp, 2, ONE);
/* cy should be 0 */
up[1] = mpn_mul_1 (up, sp + 1, 1, xp[1]);
cy = mpn_add_1 (up + 1, up + 1, 1, sp[1]);
/* cy should be 0 */
xp[0] = up[1];
/* update tp */
cy = mpn_addmul_1 (tp, ap, 2, xp[0]);
cy = mpn_add_1 (tp + 2, tp + 2, 2, cy);
do
{
cy = mpn_add (tp, tp, 4, ap, 2);
if (cy == ZERO)
mpn_add_1 (xp, xp, 2, ONE);
}
while (cy == ZERO);
/* now A*X < B^4 <= A*(X+1) */
}
else
{
mp_size_t l, h;
mp_ptr tp, up;
mp_limb_t cy, th;
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int special = 0;
TMP_DECL;
l = (n - 1) / 2;
h = n - l;
mpn_invert (xp + l, ap + l, h);
TMP_MARK;
tp = TMP_ALLOC_LIMBS (n + h);
up = TMP_ALLOC_LIMBS (2 * h);
if (n <= WRAP_AROUND_BOUND)
{
mpn_mul (tp, ap, n, xp + l, h);
cy = mpn_add_n (tp + h, tp + h, ap, n);
}
else
{
mp_size_t m = n + 1;
unsigned long k;
int cc;
k = mpn_fft_best_k (m, 0);
m = mpn_fft_next_size (m, k);
/* we have m >= n + 1 by construction, thus m > h */
ASSERT(m < n + h);
cy = mpn_mul_fft (tp, m, ap, n, xp + l, h, k);
/* cy, {tp, m} = A * {xp + l, h} mod (B^m+1) */
cy += mpn_add_n (tp + h, tp + h, ap, m - h);
cc = mpn_sub_n (tp, tp, ap + m - h, n + h - m);
cc = mpn_sub_1 (tp + n + h - m, tp + n + h - m, 2 * m - n - h, cc);
if (cc > cy) /* can only occur if cc=1 and cy=0 */
cy = mpn_add_1 (tp, tp, m, ONE);
else
cy -= cc;
/* cy, {tp, m} = A * Xh */
/* add B^(n+h) + B^(n+h-m) */
MPN_ZERO (tp + m, n + h - m);
tp[m] = cy;
/* note: since tp[n+h-1] is either 0, or cy<=1 if m=n+h-1,
the mpn_incr_u() below cannot produce a carry */
mpn_incr_u (tp + n + h - m, ONE);
cy = 1;
do /* check if T >= B^(n+h) + 2*B^n */
{
mp_size_t i;
if (cy == ZERO)
break; /* surely T < B^(n+h) */
if (cy == ONE)
{
for (i = n + h - 1; tp[i] == ZERO && i > n; i--);
if (i == n && tp[i] < (mp_limb_t) 2)
break;
}
/* subtract B^m+1 */
cy -= mpn_sub_1 (tp, tp, n + h, ONE);
cy -= mpn_sub_1 (tp + m, tp + m, n + h - m, ONE);
}
while (1);
}
while (cy)
{
mpn_sub_1 (xp + l, xp + l, h, ONE);
cy -= mpn_sub (tp, tp, n + h, ap, n);
}
mpn_com_n (tp, tp, n);
th = ~tp[n] + mpn_add_1 (tp, tp, n, ONE);
mpn_mul_n (up, tp + l, xp + l, h);
cy = mpn_add_n (up + h, up + h, tp + l, h);
if (th != ZERO)
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{
cy += ONE + mpn_add_n (up + h, up + h, xp + l, h);
}
if (up[2*h-l-1] + 4 <= CNST_LIMB(3)) special = 1;
MPN_COPY (xp, up + 2 * h - l, l);
mpn_add_1 (xp + l, xp + l, h, cy);
TMP_FREE;
if ((special) && !mpn_is_invert(xp, ap, n))
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mpn_add_1 (xp, xp, n, 1);
}
}
void mpn_invert_truncate(mp_ptr x_new, mp_size_t m, mp_srcptr xp, mp_size_t n, mp_srcptr ap)
{
mp_ptr tp;
mp_limb_t cy;
TMP_DECL;
TMP_MARK;
tp = TMP_ALLOC_LIMBS (2 * m);
MPN_COPY(x_new, xp + n - m, m);
ap += (n - m);
mpn_mul_n (tp, x_new, ap, m);
mpn_add_n (tp + m, tp + m, ap, m); /* A * msb(X) */
/* now check B^(2n) - X*A <= A */
mpn_com_n (tp, tp, 2 * m);
mpn_add_1 (tp, tp, 2 * m, 1); /* B^(2m) - X*A */
while (tp[m] || mpn_cmp (tp, ap, m) > 0)
{
mpn_add_1(x_new, x_new, m, 1);
tp[m] -= mpn_sub_n(tp, tp, ap, m);
}
TMP_FREE;
}