/* Mulders' MulHigh function (short product)
Copyright 2005-2017 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU 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 3 of the License, or (at your
option) any later version.
The GNU 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 GNU MPFR Library; see the file COPYING.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
/* References:
[1] Short Division of Long Integers, David Harvey and Paul Zimmermann,
Proceedings of the 20th Symposium on Computer Arithmetic (ARITH-20),
July 25-27, 2011, pages 7-14.
*/
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
#ifndef MUL_FFT_THRESHOLD
#define MUL_FFT_THRESHOLD 8448
#endif
/* Don't use MPFR_MULHIGH_SIZE since it is handled by tuneup */
#ifdef MPFR_MULHIGH_TAB_SIZE
static short mulhigh_ktab[MPFR_MULHIGH_TAB_SIZE];
#else
static short mulhigh_ktab[] = {MPFR_MULHIGH_TAB};
#define MPFR_MULHIGH_TAB_SIZE \
((mp_size_t) (sizeof(mulhigh_ktab) / sizeof(mulhigh_ktab[0])))
#endif
/* Put in rp[n..2n-1] an approximation of the n high limbs
of {up, n} * {vp, n}. The error is less than n ulps of rp[n] (and the
approximation is always less or equal to the truncated full product).
Assume 2n limbs are allocated at rp.
Implements Algorithm ShortMulNaive from [1].
*/
static void
mpfr_mulhigh_n_basecase (mpfr_limb_ptr rp, mpfr_limb_srcptr up,
mpfr_limb_srcptr vp, mp_size_t n)
{
mp_size_t i;
rp += n - 1;
umul_ppmm (rp[1], rp[0], up[n-1], vp[0]); /* we neglect up[0..n-2]*vp[0],
which is less than B^n */
for (i = 1 ; i < n ; i++)
/* here, we neglect up[0..n-i-2] * vp[i], which is less than B^n too */
rp[i + 1] = mpn_addmul_1 (rp, up + (n - i - 1), i + 1, vp[i]);
/* in total, we neglect less than n*B^n, i.e., n ulps of rp[n]. */
}
/* Put in rp[0..n] the n+1 low limbs of {up, n} * {vp, n}.
Assume 2n limbs are allocated at rp. */
static void
mpfr_mullow_n_basecase (mpfr_limb_ptr rp, mpfr_limb_srcptr up,
mpfr_limb_srcptr vp, mp_size_t n)
{
mp_size_t i;
rp[n] = mpn_mul_1 (rp, up, n, vp[0]);
for (i = 1 ; i < n ; i++)
mpn_addmul_1 (rp + i, up, n - i + 1, vp[i]);
}
/* Put in rp[n..2n-1] an approximation of the n high limbs
of {np, n} * {mp, n}. The error is less than n ulps of rp[n] (and the
approximation is always less or equal to the truncated full product).
Implements Algorithm ShortMul from [1].
*/
void
mpfr_mulhigh_n (mpfr_limb_ptr rp, mpfr_limb_srcptr np, mpfr_limb_srcptr mp,
mp_size_t n)
{
mp_size_t k;
MPFR_ASSERTN (MPFR_MULHIGH_TAB_SIZE >= 8); /* so that 3*(n/4) > n/2 */
k = MPFR_LIKELY (n < MPFR_MULHIGH_TAB_SIZE) ? mulhigh_ktab[n] : 3*(n/4);
/* Algorithm ShortMul from [1] requires k >= (n+3)/2, which translates
into k >= (n+4)/2 in the C language. */
MPFR_ASSERTD (k == -1 || k == 0 || (k >= (n+4)/2 && k < n));
if (k < 0)
mpn_mul_basecase (rp, np, n, mp, n); /* result is exact, no error */
else if (k == 0)
mpfr_mulhigh_n_basecase (rp, np, mp, n); /* basecase error < n ulps */
else if (n > MUL_FFT_THRESHOLD)
mpn_mul_n (rp, np, mp, n); /* result is exact, no error */
else
{
mp_size_t l = n - k;
mp_limb_t cy;
mpn_mul_n (rp + 2 * l, np + l, mp + l, k); /* fills rp[2l..2n-1] */
mpfr_mulhigh_n (rp, np + k, mp, l); /* fills rp[l-1..2l-1] */
cy = mpn_add_n (rp + n - 1, rp + n - 1, rp + l - 1, l + 1);
mpfr_mulhigh_n (rp, np, mp + k, l); /* fills rp[l-1..2l-1] */
cy += mpn_add_n (rp + n - 1, rp + n - 1, rp + l - 1, l + 1);
mpn_add_1 (rp + n + l, rp + n + l, k, cy); /* propagate carry */
}
}
/* Put in rp[0..n] the n+1 low limbs of {np, n} * {mp, n}.
Assume 2n limbs are allocated at rp. */
void
mpfr_mullow_n (mpfr_limb_ptr rp, mpfr_limb_srcptr np, mpfr_limb_srcptr mp,
mp_size_t n)
{
mp_size_t k;
MPFR_ASSERTN (MPFR_MULHIGH_TAB_SIZE >= 8); /* so that 3*(n/4) > n/2 */
k = MPFR_LIKELY (n < MPFR_MULHIGH_TAB_SIZE) ? mulhigh_ktab[n] : 3*(n/4);
MPFR_ASSERTD (k == -1 || k == 0 || (2 * k >= n && k < n));
if (k < 0)
mpn_mul_basecase (rp, np, n, mp, n);
else if (k == 0)
mpfr_mullow_n_basecase (rp, np, mp, n);
else if (n > MUL_FFT_THRESHOLD)
mpn_mul_n (rp, np, mp, n);
else
{
mp_size_t l = n - k;
mpn_mul_n (rp, np, mp, k); /* fills rp[0..2k] */
mpfr_mullow_n (rp + n, np + k, mp, l); /* fills rp[n..n+2l] */
mpn_add_n (rp + k, rp + k, rp + n, l + 1);
mpfr_mullow_n (rp + n, np, mp + k, l); /* fills rp[n..n+2l] */
mpn_add_n (rp + k, rp + k, rp + n, l + 1);
}
}
#ifdef MPFR_SQRHIGH_TAB_SIZE
static short sqrhigh_ktab[MPFR_SQRHIGH_TAB_SIZE];
#else
static short sqrhigh_ktab[] = {MPFR_SQRHIGH_TAB};
#define MPFR_SQRHIGH_TAB_SIZE (sizeof(sqrhigh_ktab) / sizeof(sqrhigh_ktab[0]))
#endif
/* Put in rp[n..2n-1] an approximation of the n high limbs
of {np, n}^2. The error is less than n ulps of rp[n]. */
void
mpfr_sqrhigh_n (mpfr_limb_ptr rp, mpfr_limb_srcptr np, mp_size_t n)
{
mp_size_t k;
MPFR_ASSERTN (MPFR_SQRHIGH_TAB_SIZE > 2); /* ensures k < n */
k = MPFR_LIKELY (n < MPFR_SQRHIGH_TAB_SIZE) ? sqrhigh_ktab[n]
: (n+4)/2; /* ensures that k >= (n+3)/2 */
MPFR_ASSERTD (k == -1 || k == 0 || (k >= (n+4)/2 && k < n));
if (k < 0)
/* we can't use mpn_sqr_basecase here, since it requires
n <= SQR_KARATSUBA_THRESHOLD, where SQR_KARATSUBA_THRESHOLD
is not exported by GMP */
mpn_sqr_n (rp, np, n);
else if (k == 0)
mpfr_mulhigh_n_basecase (rp, np, np, n);
else
{
mp_size_t l = n - k;
mp_limb_t cy;
mpn_sqr_n (rp + 2 * l, np + l, k); /* fills rp[2l..2n-1] */
mpfr_mulhigh_n (rp, np, np + k, l); /* fills rp[l-1..2l-1] */
/* {rp+n-1,l+1} += 2 * {rp+l-1,l+1} */
cy = mpn_lshift (rp + l - 1, rp + l - 1, l + 1, 1);
cy += mpn_add_n (rp + n - 1, rp + n - 1, rp + l - 1, l + 1);
mpn_add_1 (rp + n + l, rp + n + l, k, cy); /* propagate carry */
}
}
#ifdef MPFR_DIVHIGH_TAB_SIZE
static short divhigh_ktab[MPFR_DIVHIGH_TAB_SIZE];
#else
static short divhigh_ktab[] = {MPFR_DIVHIGH_TAB};
#define MPFR_DIVHIGH_TAB_SIZE (sizeof(divhigh_ktab) / sizeof(divhigh_ktab[0]))
#endif
#ifndef __GMPFR_GMP_H__
#define mpfr_pi1_t gmp_pi1_t /* with a GMP build */
#endif
#if !(defined(WANT_GMP_INTERNALS) && defined(HAVE___GMPN_SBPI1_DIVAPPR_Q))
/* Put in Q={qp, n} an approximation of N={np, 2*n} divided by D={dp, n},
with the most significant limb of the quotient as return value (0 or 1).
Assumes the most significant bit of D is set. Clobbers N.
The approximate quotient Q satisfies - 2(n-1) < N/D - Q <= 4.
*/
static mp_limb_t
mpfr_divhigh_n_basecase (mpfr_limb_ptr qp, mpfr_limb_ptr np,
mpfr_limb_srcptr dp, mp_size_t n)
{
mp_limb_t qh, d1, d0, dinv, q2, q1, q0;
mpfr_pi1_t dinv2;
np += n;
if ((qh = (mpn_cmp (np, dp, n) >= 0)))
mpn_sub_n (np, np, dp, n);
/* now {np, n} is less than D={dp, n}, which implies np[n-1] <= dp[n-1] */
d1 = dp[n - 1];
if (n == 1)
{
invert_limb (dinv, d1);
umul_ppmm (q1, q0, np[0], dinv);
qp[0] = np[0] + q1;
return qh;
}
/* now n >= 2 */
d0 = dp[n - 2];
invert_pi1 (dinv2, d1, d0);
/* dinv2.inv32 = floor ((B^3 - 1) / (d0 + d1 B)) - B */
while (n > 1)
{
/* Invariant: it remains to reduce n limbs from N (in addition to the
initial low n limbs).
Since n >= 2 here, necessarily we had n >= 2 initially, which means
that in addition to the limb np[n-1] to reduce, we have at least 2
extra limbs, thus accessing np[n-3] is valid. */
/* Warning: we can have np[n-1]>d1 or (np[n-1]=d1 and np[n-2]>=d0) here,
since we truncate the divisor at each step, but since {np,n} < D
originally, the largest possible partial quotient is B-1. */
if (MPFR_UNLIKELY(np[n-1] > d1 || (np[n-1] == d1 && np[n-2] >= d0)))
q2 = ~ (mp_limb_t) 0;
else
udiv_qr_3by2 (q2, q1, q0, np[n - 1], np[n - 2], np[n - 3],
d1, d0, dinv2.inv32);
/* since q2 = floor((np[n-1]*B^2+np[n-2]*B+np[n-3])/(d1*B+d0)),
we have q2 <= (np[n-1]*B^2+np[n-2]*B+np[n-3])/(d1*B+d0),
thus np[n-1]*B^2+np[n-2]*B+np[n-3] >= q2*(d1*B+d0)
and {np-1, n} >= q2*D - q2*B^(n-2) >= q2*D - B^(n-1)
thus {np-1, n} - (q2-1)*D >= D - B^(n-1) >= 0
which proves that at most one correction is needed */
q0 = mpn_submul_1 (np - 1, dp, n, q2);
if (MPFR_UNLIKELY(q0 > np[n - 1]))
{
mpn_add_n (np - 1, np - 1, dp, n);
q2 --;
}
qp[--n] = q2;
dp ++;
}
/* we have B+dinv2 = floor((B^3-1)/(d1*B+d0)) < B^2/d1
q1 = floor(np[0]*(B+dinv2)/B) <= floor(np[0]*B/d1)
<= floor((np[0]*B+np[1])/d1)
thus q1 is not larger than the true quotient.
q1 > np[0]*(B+dinv2)/B - 1 > np[0]*(B^3-1)/(d1*B+d0)/B - 2
For d1*B+d0 <> B^2/2, we have B+dinv2 = floor(B^3/(d1*B+d0))
thus q1 > np[0]*B^2/(d1*B+d0) - 2, i.e.,
(d1*B+d0)*q1 > np[0]*B^2 - 2*(d1*B+d0)
d1*B*q1 > np[0]*B^2 - 2*d1*B - 2*d0 - d0*q1 >= np[0]*B^2 - 2*d1*B - B^2
thus q1 > np[0]*B/d1 - 2 - B/d1 > np[0]*B/d1 - 4.
For d1*B+d0 = B^2/2, dinv2 = B-1 thus q1 > np[0]*(2B-1)/B - 1 >
np[0]*B/d1 - 2.
In all cases, if q = floor((np[0]*B+np[1])/d1), we have:
q - 4 <= q1 <= q
*/
umul_ppmm (q1, q0, np[0], dinv2.inv32);
qp[0] = np[0] + q1;
return qh;
}
#endif
/* Put in {qp, n} an approximation of N={np, 2*n} divided by D={dp, n},
with the most significant limb of the quotient as return value (0 or 1).
Assumes the most significant bit of D is set. Clobbers N.
This implements the ShortDiv algorithm from reference [1].
*/
#if 1
mp_limb_t
mpfr_divhigh_n (mpfr_limb_ptr qp, mpfr_limb_ptr np, mpfr_limb_ptr dp,
mp_size_t n)
{
mp_size_t k, l;
mp_limb_t qh, cy;
mpfr_limb_ptr tp;
MPFR_TMP_DECL(marker);
MPFR_ASSERTN (MPFR_MULHIGH_TAB_SIZE >= 15); /* so that 2*(n/3) >= (n+4)/2 */
k = MPFR_LIKELY (n < MPFR_DIVHIGH_TAB_SIZE) ? divhigh_ktab[n] : 2*(n/3);
if (k == 0)
#if defined(WANT_GMP_INTERNALS) && defined(HAVE___GMPN_SBPI1_DIVAPPR_Q)
{
mpfr_pi1_t dinv2;
invert_pi1 (dinv2, dp[n - 1], dp[n - 2]);
return __gmpn_sbpi1_divappr_q (qp, np, n + n, dp, n, dinv2.inv32);
}
#else /* use our own code for base-case short division */
return mpfr_divhigh_n_basecase (qp, np, dp, n);
#endif
else if (k == n)
/* for k=n, we use a division with remainder (mpn_divrem),
which computes the exact quotient */
return mpn_divrem (qp, 0, np, 2 * n, dp, n);
MPFR_ASSERTD ((n+4)/2 <= k && k < n); /* bounds from [1] */
MPFR_TMP_MARK (marker);
l = n - k;
/* first divide the most significant 2k limbs from N by the most significant
k limbs of D */
qh = mpn_divrem (qp + l, 0, np + 2 * l, 2 * k, dp + l, k); /* exact */
/* it remains {np,2l+k} = {np,n+l} as remainder */
/* now we have to subtract high(Q1)*D0 where Q1=qh*B^k+{qp+l,k} and
D0={dp,l} */
tp = MPFR_TMP_LIMBS_ALLOC (2 * l);
mpfr_mulhigh_n (tp, qp + k, dp, l);
/* we are only interested in the upper l limbs from {tp,2l} */
cy = mpn_sub_n (np + n, np + n, tp + l, l);
if (qh)
cy += mpn_sub_n (np + n, np + n, dp, l);
while (cy > 0) /* Q1 was too large: subtract 1 to Q1 and add D to np+l */
{
qh -= mpn_sub_1 (qp + l, qp + l, k, MPFR_LIMB_ONE);
cy -= mpn_add_n (np + l, np + l, dp, n);
}
/* now it remains {np,n+l} to divide by D */
cy = mpfr_divhigh_n (qp, np + k, dp + k, l);
qh += mpn_add_1 (qp + l, qp + l, k, cy);
MPFR_TMP_FREE(marker);
return qh;
}
#else /* below is the FoldDiv(K) algorithm from [1] */
mp_limb_t
mpfr_divhigh_n (mpfr_limb_ptr qp, mpfr_limb_ptr np, mpfr_limb_ptr dp,
mp_size_t n)
{
mp_size_t k, r;
mpfr_limb_ptr ip, tp, up;
mp_limb_t qh = 0, cy, cc;
int count;
MPFR_TMP_DECL(marker);
#define K 3
if (n < K)
return mpn_divrem (qp, 0, np, 2 * n, dp, n);
k = (n - 1) / K + 1; /* ceil(n/K) */
MPFR_TMP_MARK (marker);
ip = MPFR_TMP_LIMBS_ALLOC (k + 1);
tp = MPFR_TMP_LIMBS_ALLOC (n + k);
up = MPFR_TMP_LIMBS_ALLOC (2 * (k + 1));
mpn_invert (ip, dp + n - (k + 1), k + 1, NULL); /* takes about 13% for n=1000 */
/* {ip, k+1} = floor((B^(2k+2)-1)/D - B^(k+1) where D = {dp+n-(k+1),k+1} */
for (r = n, cc = 0UL; r > 0;)
{
/* cc is the carry at np[n+r] */
MPFR_ASSERTD(cc <= 1);
/* FIXME: why can we have cc as large as say 8? */
count = 0;
while (cc > 0)
{
count ++;
MPFR_ASSERTD(count <= 1);
/* subtract {dp+n-r,r} from {np+n,r} */
cc -= mpn_sub_n (np + n, np + n, dp + n - r, r);
/* add 1 at qp[r] */
qh += mpn_add_1 (qp + r, qp + r, n - r, 1UL);
}
/* it remains r limbs to reduce, i.e., the remainder is {np, n+r} */
if (r < k)
{
ip += k - r;
k = r;
}
/* now r >= k */
/* qp + r - 2 * k -> up */
mpfr_mulhigh_n (up, np + n + r - (k + 1), ip, k + 1);
/* take into account the term B^k in the inverse: B^k * {np+n+r-k, k} */
cy = mpn_add_n (qp + r - k, up + k + 2, np + n + r - k, k);
/* since we need only r limbs of tp (below), it suffices to consider
r high limbs of dp */
if (r > k)
{
#if 0
mpn_mul (tp, dp + n - r, r, qp + r - k, k);
#else /* use a short product for the low k x k limbs */
/* we know the upper k limbs of the r-limb product cancel with the
remainder, thus we only need to compute the low r-k limbs */
if (r - k >= k)
mpn_mul (tp + k, dp + n - r + k, r - k, qp + r - k, k);
else /* r-k < k */
{
/* #define LOW */
#ifndef LOW
mpn_mul (tp + k, qp + r - k, k, dp + n - r + k, r - k);
#else
mpfr_mullow_n_basecase (tp + k, qp + r - k, dp + n - r + k, r - k);
/* take into account qp[2r-2k] * dp[n - r + k] */
tp[r] += qp[2*r-2*k] * dp[n - r + k];
#endif
/* tp[k..r] is filled */
}
#if 0
mpfr_mulhigh_n (up, dp + n - r, qp + r - k, k);
#else /* compute one more limb. FIXME: we could add one limb of dp in the
above, to save one mpn_addmul_1 call */
mpfr_mulhigh_n (up, dp + n - r, qp + r - k, k - 1); /* {up,2k-2} */
/* add {qp + r - k, k - 1} * dp[n-r+k-1] */
up[2*k-2] = mpn_addmul_1 (up + k - 1, qp + r - k, k-1, dp[n-r+k-1]);
/* add {dp+n-r, k} * qp[r-1] */
up[2*k-1] = mpn_addmul_1 (up + k - 1, dp + n - r, k, qp[r-1]);
#endif
#ifndef LOW
cc = mpn_add_n (tp + k, tp + k, up + k, k);
mpn_add_1 (tp + 2 * k, tp + 2 * k, r - k, cc);
#else
/* update tp[k..r] */
if (r - k + 1 <= k)
mpn_add_n (tp + k, tp + k, up + k, r - k + 1);
else /* r - k >= k */
{
cc = mpn_add_n (tp + k, tp + k, up + k, k);
mpn_add_1 (tp + 2 * k, tp + 2 * k, r - 2 * k + 1, cc);
}
#endif
#endif
}
else /* last step: since we only want the quotient, no need to update,
just propagate the carry cy */
{
MPFR_ASSERTD(r < n);
if (cy > 0)
qh += mpn_add_1 (qp + r, qp + r, n - r, cy);
break;
}
/* subtract {tp, n+k} from {np+r-k, n+k}; however we only want to
update {np+n, n} */
/* we should have tp[r] = np[n+r-k] up to 1 */
MPFR_ASSERTD(tp[r] == np[n + r - k] || tp[r] + 1 == np[n + r - k]);
#ifndef LOW
cc = mpn_sub_n (np + n - 1, np + n - 1, tp + k - 1, r + 1); /* borrow at np[n+r] */
#else
cc = mpn_sub_n (np + n - 1, np + n - 1, tp + k - 1, r - k + 2);
#endif
/* if cy = 1, subtract {dp, n} from {np+r, n}, thus
{dp+n-r,r} from {np+n,r} */
if (cy)
{
if (r < n)
cc += mpn_sub_n (np + n - 1, np + n - 1, dp + n - r - 1, r + 1);
else
cc += mpn_sub_n (np + n, np + n, dp + n - r, r);
/* propagate cy */
if (r == n)
qh = cy;
else
qh += mpn_add_1 (qp + r, qp + r, n - r, cy);
}
/* cc is the borrow at np[n+r] */
count = 0;
while (cc > 0) /* quotient was too large */
{
count++;
MPFR_ASSERTD (count <= 1);
cy = mpn_add_n (np + n, np + n, dp + n - (r - k), r - k);
cc -= mpn_add_1 (np + n + r - k, np + n + r - k, k, cy);
qh -= mpn_sub_1 (qp + r - k, qp + r - k, n - (r - k), 1UL);
}
r -= k;
cc = np[n + r];
}
MPFR_TMP_FREE(marker);
return qh;
}
#endif