# Division interfaces

Niels Möller nisse at lysator.liu.se
Tue Mar 15 08:35:36 CET 2011

```nisse at lysator.liu.se (Niels Möller) writes:

> And then we have the issue of unnormalized divisors (it looks like
> mpn_sbpi1_div_qr still requires that dp[dn-1] has the high bit set.

I'm attaching a first crude version with on-the-fly normalization of the
high limbs.

If one reads the top 3 limbs of n and top 2 of d and shift left to
normalize d (shifting in zeros), and use 3/2 division, one will in
effect do a (2 limbs + k bits)/(1 limb + k bits) division, where 0 < k =
(GMP_LIMB_BITS - shift_count). Then the probability of the "unlikely"
update in schoolbook division is on the order of 2^{-k}, which for small
k (large shift count) actually isn't very unlikely.

For this reason, the code uses 4 limbs of n and 3 limbs of d (and a 4/3
division built from 3/2 division and an extra update step). This is
overkill when the shift count is small, so maybe one should have three
different variants,

1. Normalized d.

2. Shift count < (GMP_LIMB_BITS - 10). Use 3/2 division, the "unlikely"
update happens with probability < 2^{-10}.

3. Larger shift counts. Use 4/3 division (like the attached code), then
the unlikely update probability is < 2^{-GMP_LIMB_BITS}.

/Niels

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/* mpn_sbpi1_div_qr -- Schoolbook division using the MÃ¶ller-Granlund 3/2
division algorithm.

Contributed to the GNU project by Torbjorn Granlund.

THE FUNCTION IN THIS FILE IS INTERNAL WITH A MUTABLE INTERFACE.  IT IS ONLY
SAFE TO REACH IT THROUGH DOCUMENTED INTERFACES.  IN FACT, IT IS ALMOST
GUARANTEED THAT IT WILL CHANGE OR DISAPPEAR IN A FUTURE GMP RELEASE.

Copyright 2007, 2009, 2011 Free Software Foundation, Inc.

This file is part of the GNU MP Library.

The GNU MP Library is free software; you can redistribute it and/or modify
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.

The GNU MP 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

You should have received a copy of the GNU Lesser General Public License
along with the GNU MP Library.  If not, see http://www.gnu.org/licenses/.  */

#include <stdlib.h>
#include <stdio.h>

#include "gmp.h"
#include "gmp-impl.h"
#include "longlong.h"

#define udiv_qr_4by3(q, r2, r1, r0, n3, n2, n1, n0, d2, d1, d0, dinv)	\
do {									\
mp_limb_t _q, _p1, _p0, _cy;					\
udiv_qr_3by2 (_q, (r2), (r1), (n3), (n2), (n1), (d2), (d1), (dinv)); \
umul_ppmm (_p1, _p0, _q, (d0));					\
_p1 += _p0 > (n0);							\
(r0) = (n0) - _p0;							\
_cy = _p1 > (r1);							\
(r1) -= _p1;							\
if (UNLIKELY (_cy > (r2)))						\
{	/* _cy == 1, r2 == 0 */						\
(r0) += (d0);							\
_cy = (r0) < (d0);						\
(r1) += _cy;							\
_cy = (r1) < _cy;						\
add_ssaaaa ((r2), (r1), _cy - 1, (r1), (d2), (d1));		\
_q--;								\
}									\
else								\
(r2) -= _cy;							\
(q) = _q;								\
} while (0)

mp_limb_t
sb_div_qr (mp_ptr qp,
mp_ptr np, mp_size_t nn,
mp_srcptr dp, mp_size_t dn,
mp_limb_t d1, mp_limb_t d0,
mp_limb_t dinv, int shift)
{
mp_limb_t qh;

ASSERT (dn > 2);
ASSERT (nn >= dn);

qp += nn - dn;

if (shift == 0)
{
mp_size_t i;
mp_limb_t n1, n0;
mp_limb_t cy, cy1;
mp_limb_t q;

np += nn;

ASSERT ((dp[dn-1] & GMP_NUMB_HIGHBIT) != 0);

qh = mpn_cmp (np - dn, dp, dn) >= 0;
if (qh != 0)
mpn_sub_n (np - dn, np - dn, dp, dn);

dn -= 2;			/* offset dn by 2 for main division loops,
saving two iterations in mpn_submul_1.  */
np -= 2;

n1 = np[1];

for (i = nn - (dn + 2); i > 0; i--)
{
np--;
if (UNLIKELY (n1 == d1) && np[1] == d0)
{
mpn_submul_1 (np - dn, dp, dn + 2, q);
n1 = np[1];		/* update n1, last loop's value will now be invalid */
}
else
{
udiv_qr_3by2 (q, n1, n0, n1, np[1], np[0], d1, d0, dinv);

cy = mpn_submul_1 (np - dn, dp, dn, q);

cy1 = n0 < cy;
n0 = (n0 - cy) & GMP_NUMB_MASK;
cy = n1 < cy1;
n1 = (n1 - cy1) & GMP_NUMB_MASK;
np[0] = n0;

if (UNLIKELY (cy != 0))
{
n1 += d1 + mpn_add_n (np - dn, np - dn, dp, dn + 1);
q--;
}
}

*--qp = q;
}
np[1] = n1;
}
else
{
mp_limb_t n3, n2, n1, n0;
mp_limb_t cy;
mp_limb_t dl;
mp_limb_t c1, c0;

/*   +----+----+----+----+----
d: |    |    |    |    | ...
+----+----+----+----+----
`----'----'----'
d1   d0   dl
*/

dl = dp[dn-3] << shift;

n3 = (np[nn-1] >> (GMP_LIMB_BITS - shift));
n2 = (np[nn-1] << shift) | (np[nn-2] >> (GMP_LIMB_BITS - shift));
n1 = (np[nn-2] << shift) | (np[nn-3] >> (GMP_LIMB_BITS - shift));
n0 = (np[nn-3] << shift);

udiv_qr_4by3 (qh, n3, n2, n1, n3, n2, n1, n0, d1, d0, dl, dinv);

cy = mpn_submul_1 (np + nn - dn, dp, dn - 3, qh);
c0 = cy << shift;
c1 = cy >> (GMP_LIMB_BITS - shift);

cy = (n2 <= c1) && (n2 < c1 || n1 < c0);

sub_ddmmss (n2, n1, n2, n1, c1, c0);

if (UNLIKELY (n3 < cy))
{
n1 += dl;
cy = n1 < dl;
n2 += cy;
cy = (n2 < cy);
add_ssaaaa (n3, n2, cy, n2, d1 - 1, d0);

cy = mpn_add_n (np + nn - dn, np + nn - dn, dp, dn - 3);

cy <<= shift;
n1 += cy;
cy = n1 < cy;
n2 += cy;
n3 += (n2 < cy);

qh--;
}
else
n3 -= cy;

for (; nn > dn; nn--)
{
mp_limb_t q;
/*   +----+----+----+----+----+----
n: |    |    |    |    |    | ...
+----+----+----+----+----+----
`----'----'----'----'
n3   n2   n1   n0
*/
n1 |= np[nn-4] >> (GMP_LIMB_BITS - shift);
n0 = np[nn-4] << shift;

if (UNLIKELY (n3 == d1) && n2 == d1)
{
q = MP_LIMB_T_MAX;

/* NOTE: Not always true that n1 <= dl; then n3 may
overflow, and this overflow is cancelled when
subtracting the low part. */
/* <d1, d0, n1, n0> - (B-1) <d1, d0, dl>
= <d1, d0, dl> + <n1, n0> - B dl */

n3 = d1;
n2 = d0 + n1;
n3 += (n2 < d0);
n1 = dl + n0;
cy = n1 < dl;
n2 += cy;
n3 += (n2 < cy);

n3 -= (n2 < dl);
n2 -= dl;

/* Can't share code with the common case, because n3 may
have overflowed. */
cy = mpn_submul_1 (np + nn - dn - 1, dp, dn - 3, q);
c0 = cy << shift;
c1 = cy >> (GMP_LIMB_BITS - shift);
cy = (n2 <= c1) && (n2 < c1 || n1 < c0);

sub_ddmmss (n2, n1, n2, n1, c1, c0);

n3 -= cy;
}
else
{
udiv_qr_4by3 (q, n3, n2, n1, n3, n2, n1, n0, d1, d0, dl, dinv);

cy = mpn_submul_1 (np + nn - dn - 1, dp, dn - 3, q);
c0 = cy << shift;
c1 = cy >> (GMP_LIMB_BITS - shift);
cy = (n2 <= c1) && (n2 < c1 || n1 < c0);

sub_ddmmss (n2, n1, n2, n1, c1, c0);

if (UNLIKELY (n3 < cy))
{
n1 += dl;
cy = n1 < dl;
n2 += cy;
cy = (n2 < cy);
add_ssaaaa (n3, n2, cy, n2, d1 - 1, d0);

cy = mpn_add_n (np + nn - dn - 1, np + nn - dn - 1, dp, dn - 3);

cy <<= shift;
n1 += cy;
cy = n1 < cy;
n2 += cy;
n3 += (n2 < cy);

q--;
}
else
n3 -= cy;
}

*--qp = q;
}

np[nn-1] = (n3 >> shift);
np[nn-2] = (n2 >> shift) | (n3 << (GMP_LIMB_BITS - shift));
np[nn-3] = (n1 >> shift) | (n2 << (GMP_LIMB_BITS - shift));

}
return qh;
}

static mp_limb_t
random_word (gmp_randstate_ptr rs)
{
mpz_t x;
mp_limb_t r;
TMP_DECL;
TMP_MARK;

MPZ_TMP_INIT (x, 2);
mpz_urandomb (x, rs, 32);
r = mpz_get_ui (x);
TMP_FREE;
return r;
}

static void
check (const mpz_t n, const mpz_t d, const mpz_t q, const mpz_t r)
{
mpz_t t;
mpz_init (t);
mpz_mul (t, q, d);

if (mpz_cmp (t, n))
{
gmp_fprintf (stderr, "n = %Zx\n"
"d = %Zx\n"
"q = %Zx\n"
"r = %Zx\n"
"q d + r = %Zx\n",
n, d, q, r, t);
abort();
}
mpz_clear (t);
}

/* Maximum d and q */
#define MAXBITS 1000
#define MAXLIMBS (MAXBITS / GMP_NUMB_BITS + 10)

int
main (int argc, char **argv)
{
gmp_randstate_ptr rands;
unsigned nbits;
unsigned qbits;
unsigned dbits;
unsigned i;
mpz_t n, d, q, r;
mp_size_t qn;
mp_size_t rn;

rands = RANDS;

mpz_init (n);
mpz_init (d);
mpz_init (q);
mpz_init (r);

_mpz_realloc (q, MAXLIMBS);

for (i = 0; i < 500; i++)
{
mp_limb_t d1;
mp_limb_t d0;
gmp_pi1_t dinv;
mp_limb_t qh;
int shift;
mp_ptr dp;
mp_size_t dn;

dbits = 3 * GMP_NUMB_BITS + random_word (rands) % 200;
qbits = 1 + random_word (rands) % 200;
nbits = dbits + qbits;

mpz_rrandomb (d, rands, dbits);
mpz_rrandomb (n, rands, nbits);

dp = PTR(d);
dn = SIZ(d);

d1 = dp[dn-1];
if (! (GMP_NUMB_HIGHBIT & d1))
{
d1 = (d1 << shift) | (dp[dn-2] >> (GMP_LIMB_BITS - shift));
d0 = (dp[dn-2] << shift) | (dp[dn-3] >> (GMP_LIMB_BITS - shift));
}
else
{
shift = 0;
d0 = dp[dn-2];
}
invert_pi1(dinv, d1, d0);

qn = SIZ(n) - SIZ(d);
ASSERT (rn <= MAXLIMBS);
ASSERT (qn < MAXLIMBS);

mpz_set (r, n);
qh = sb_div_qr (PTR(q), PTR(r), SIZ(r), PTR(d), SIZ(d),
d1, d0, dinv.inv32, shift);
if (qh > 0)
PTR(q)[qn++] = qh;
SIZ(q) = qn;
rn = SIZ(d);
MPN_NORMALIZE (PTR(r), rn);
SIZ(r) = rn;

check (n, d, q, r);
}
return EXIT_SUCCESS;
}
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--
Niels Möller. PGP-encrypted email is preferred. Keyid C0B98E26.
Internet email is subject to wholesale government surveillance.
```