Improvements for mpn_mul_fft
Jeunder Yu
gis91542 at cis.nctu.edu.tw
Sat Oct 30 05:48:18 CEST 2004
After applied my trick to the head code, here is the result.
size old new
====================
n=13 0.0108 0.0103
n=14 0.0230 0.0217
n=15 0.0492 0.0461
n=16 0.1274 0.1211
n=17 0.2653 0.2524
n=18 0.6635 0.6466
n=19 1.4387 1.3693
n=20 2.8044 2.6499
n=21 5.8088 5.4526
n=22 12.1742 11.4590
n=23 25.6633 24.1082
The improvement is small, because some of my improvements are overlapped
with the head code. The advantages of non-recursive and cache locality are
small, because k is small. For example, let k=10, the transform size is
n=2^k=1024, but in another method (ex. complex fft, 3-prime moduli with
crt...),
n is large (usually n >= 2*operand_size), therefore, non-recursive and cache
locality are very important for large n.
Since the character behavior of fft algorithm was changed (non-recursive and
better cache locality, lower overhead for large k), we can try to adjust the
default
value of MUL_FFT_TABLE and SQR_FFT_TABLE table.
Because I am not familiar with GMP, I had signed my name for every change in
the source code, so that you can check my code. After the code is checked,
you can delete (so many) redundant comments for better readable.
Unfortunately, I have poor English, you can rectify or rewrite my comments.
:-)
--
Jeunder Yu
-------------- next part --------------
/* An implementation in GMP of Scho"nhage's fast multiplication algorithm
modulo 2^N+1, by Paul Zimmermann, INRIA Lorraine, February 1998.
Revised July 2002 and January 2003, Paul Zimmermann.
THE CONTENTS OF THIS FILE ARE FOR INTERNAL USE AND THE FUNCTIONS HAVE
MUTABLE INTERFACES. IT IS ONLY SAFE TO REACH THEM THROUGH DOCUMENTED
INTERFACES. IT IS ALMOST GUARANTEED THAT THEY'LL CHANGE OR DISAPPEAR IN
A FUTURE GNU MP RELEASE.
Copyright 1998, 1999, 2000, 2001, 2002, 2003, 2004 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
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 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
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MP 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. */
/* References:
Schnelle Multiplikation grosser Zahlen, by Arnold Scho"nhage and Volker
Strassen, Computing 7, p. 281-292, 1971.
Asymptotically fast algorithms for the numerical multiplication
and division of polynomials with complex coefficients, by Arnold Scho"nhage,
Computer Algebra, EUROCAM'82, LNCS 144, p. 3-15, 1982.
Tapes versus Pointers, a study in implementing fast algorithms,
by Arnold Scho"nhage, Bulletin of the EATCS, 30, p. 23-32, 1986.
See also http://www.loria.fr/~zimmerma/bignum
Future:
It might be possible to avoid a small number of MPN_COPYs by using a
rotating temporary or two.
Multiplications of unequal sized operands can be done with this code, but
it needs a tighter test for identifying squaring (same sizes as well as
same pointers). */
#include <stdio.h>
#include "gmp.h"
#include "gmp-impl.h"
/* #define WANT_ADDSUB maybe better? In my experiment, even if don't have native
mpn_addsub_n, the generic mpn_addsub_n is still efficient than native
mpn_add_n/mpn_sub_n with a MPN_COPY. by Jeunder Yu
*/
#ifdef WANT_ADDSUB
#include "generic/addsub_n.c"
#define HAVE_NATIVE_mpn_addsub_n 1
#endif
/* Change this to "#define TRACE(x) x" for some traces. */
#define TRACE(x)
FFT_TABLE_ATTRS mp_size_t mpn_fft_table[2][MPN_FFT_TABLE_SIZE] = {
MUL_FFT_TABLE,
SQR_FFT_TABLE
};
/* Interface changed, skip the _fft_l parameter. by Jeunder Yu */
/*
static int mpn_mul_fft_internal
_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t, int, int, mp_ptr *, mp_ptr *,
mp_ptr, mp_ptr, mp_size_t, mp_size_t, mp_size_t, int **, mp_ptr,
int));
*/
static int mpn_mul_fft_internal
_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t, int, int, mp_ptr *, mp_ptr *,
mp_ptr, mp_ptr, mp_size_t, mp_size_t, mp_size_t, mp_ptr, int));
/* Find the best k to use for a mod 2^(m*GMP_NUMB_BITS)+1 FFT for m >= n.
sqr==0 if for a multiply, sqr==1 for a square.
Don't declare it static since it is needed by tuneup.
*/
int
mpn_fft_best_k (mp_size_t n, int sqr)
{
int i;
for (i = 0; mpn_fft_table[sqr][i] != 0; i++)
if (n < mpn_fft_table[sqr][i])
return i + FFT_FIRST_K;
/* treat 4*last as one further entry */
if (i == 0 || n < 4 * mpn_fft_table[sqr][i - 1])
return i + FFT_FIRST_K;
else
return i + FFT_FIRST_K + 1;
}
/* Returns smallest possible number of limbs >= pl for a fft of size 2^k,
i.e. smallest multiple of 2^k >= pl.
Don't declare static: needed by tuneup.
*/
mp_size_t
mpn_fft_next_size (mp_size_t pl, int k)
{
pl = 1 + ((pl - 1) >> k); /* ceil (pl/2^k) */
return pl << k;
}
/* We don't need _fft_l table. by Jeunder Yu */
/*
static void
mpn_fft_initl (int **l, int k)
{
int i, j, K;
int *li;
l[0][0] = 0;
for (i = 1, K = 1; i <= k; i++, K *= 2)
{
li = l[i];
for (j = 0; j < K ; j++)
{
li[j] = 2 * l[i - 1][j];
li[K + j] = 1 + li[j];
}
}
}
*/
/*
shift {up, n} of cnt bits to the left, store the complemented result
in {rp, n}, and output the shifted bits (not complemented).
Same as:
cc = mpn_lshift (rp, up, n, cnt);
mpn_com_n (rp, rp, n);
return cc;
Assumes n >= 1, 1 < cnt < GMP_NUMB_BITS, rp >= up.
Copied from mpn/generic/lshift.c.
*/
static mp_limb_t
mpn_lshift_com (mp_ptr rp, mp_srcptr up, mp_size_t n, unsigned int cnt)
{
mp_limb_t high_limb, low_limb;
unsigned int tnc;
mp_size_t i;
mp_limb_t retval;
up += n;
rp += n;
tnc = GMP_NUMB_BITS - cnt;
low_limb = *--up;
retval = low_limb >> tnc;
high_limb = (low_limb << cnt);
for (i = n - 1; i != 0; i--)
{
low_limb = *--up;
*--rp = (~(high_limb | (low_limb >> tnc))) & GMP_NUMB_MASK;
high_limb = low_limb << cnt;
}
*--rp = (~high_limb) & GMP_NUMB_MASK;
return retval;
}
/* r <- a*2^e mod 2^(n*GMP_NUMB_BITS)+1 with a = {a, n+1}
Assumes a is semi-normalized, i.e. a[n] <= 1.
r and a must have n+1 limbs, and not overlap.
*/
static void
mpn_fft_mul_2exp_modF (mp_ptr r, mp_srcptr a, int e, mp_size_t n)
{
int d, sh, negate;
mp_limb_t cc = 0, rd;
d = e % (n * GMP_NUMB_BITS); /* 2^e = (+/-) 2^d */
negate = (e / (n * GMP_NUMB_BITS)) % 2;
sh = d % GMP_NUMB_BITS;
d /= GMP_NUMB_BITS;
if (negate)
{
/* r[0..d-1] <-- lshift(a[n-d]..a[n-1], sh)
r[d..n-1] <-- -lshift(a[0]..a[n-d-1], sh) */
if (sh)
{
/* no out shift below since a[n] <= 1 */
mpn_lshift (r, a + n - d, d + 1, sh);
rd = r[d];
cc = mpn_lshift_com (r + d, a, n - d, sh);
}
else
{
MPN_COPY (r, a + n - d, d);
rd = a[n];
mpn_com_n (r + d, a, n - d);
}
/* add cc to r[0], and add rd to r[d] */
/* now add 1 in r[d], subtract 1 in r[n], i.e. add 1 in r[0] */
r[n] = 0;
/* cc < 2^sh <= 2^(GMP_NUMB_BITS-1) thus no overflow here */
cc ++; /* warning: don't put this in the mpn_incr_u call,
since it may be a macro and evaluate its arg. two times */
mpn_incr_u (r, cc);
rd ++;
/* rd might overflow when sh=GMP_NUMB_BITS-1 */
cc = (rd == 0) ? CNST_LIMB(1) : rd;
r = r + d + (rd == 0);
mpn_incr_u (r, cc);
return;
}
/* if negate=0,
r[0..d-1] <-- -lshift(a[n-d]..a[n-1], sh)
r[d..n-1] <-- lshift(a[0]..a[n-d-1], sh)
*/
if (sh != 0)
{
/* no out bits below since a[n] <= 1 */
mpn_lshift_com (r, a + n - d, d + 1, sh);
rd = ~r[d];
/* {r, d+1} = {a+n-d, d+1} << sh */
cc = mpn_lshift (r + d, a, n - d, sh); /* {r+d, n-d} = {a, n-d}<<sh */
}
else
{
/* r[d] is not used below, but we save a test for d=0 */
mpn_com_n (r, a + n - d, d + 1);
rd = a[n];
MPN_COPY (r + d, a, n - d);
}
/* now complement {r, d}, subtract cc from r[0], subtract rd from r[d] */
/* Improve by Jeunder Yu */
if (d)
{
r[0] &= ~cc; /* i.e. r[0] = ~( ~r[0] | cc ), puts cc into complement part */
r[n] = CNST_LIMB(1); /* Ensures enough value for mpn_decr_u */
++rd; /* Assumes there is a borrow from r[0..d-1] */
/* rd might overflow when sh=GMP_NUMB_BITS-1 */
cc = (rd == 0) ? CNST_LIMB(1) : rd;
a = r + d + (rd == 0);
mpn_decr_u (a, cc);
/* For complement to negative and normalization (when r[n] = 0) */
cc = CNST_LIMB(2) - r[n];
r[n] = CNST_LIMB(0);
mpn_incr_u (r, cc);
}
else if (cc |= rd)
{
r[n] = CNST_LIMB(1);
--cc; /* Warning: do not put this in the mpn_incr_u call */
mpn_decr_u (r, cc);
}
else
r[n] = CNST_LIMB(0);
/* The original code following this line can be deleted. by Jeunder Yu */
return;
/* if d=0 we just have r[0]=a[n] << sh */
if (d)
{
/* now add 1 in r[0], subtract 1 in r[d] */
if (cc-- == 0) /* then add 1 to r[0] */
cc = mpn_add_1 (r, r, n, CNST_LIMB(1));
cc = mpn_sub_1 (r, r, d, cc) + CNST_LIMB(1);
/* add 1 to cc instead of rd since rd might overflow */
}
/* now subtract cc and rd from r[d..n] */
r[n] = -mpn_sub_1 (r + d, r + d, n - d, cc);
r[n] -= mpn_sub_1 (r + d, r + d, n - d, rd);
if (r[n] & GMP_LIMB_HIGHBIT)
r[n] = mpn_add_1 (r, r, n, CNST_LIMB(1));
}
/* Skips this function. by Jeunder Yu */
/* r <- a+b mod 2^(n*GMP_NUMB_BITS)+1.
Assumes a and b are semi-normalized.
*/
/*
static void
mpn_fft_add_modF (mp_ptr r, mp_srcptr a, mp_srcptr b, int n)
{
r[n] = a[n] + b[n] + mpn_add_n (r, a, b, n);
if (r[n] > 1)
{
mp_limb_t c = r[n] - CNST_LIMB(1);
r[n] = CNST_LIMB(1);
MPN_DECR_U (r, n + 1, c);
}
}
*/
/* Skips this function. by Jeunder Yu */
/* r <- a-b mod 2^(n*GMP_NUMB_BITS)+1.
Assumes a and b are semi-normalized.
*/
/*
static void
mpn_fft_sub_modF (mp_ptr r, mp_srcptr a, mp_srcptr b, int n)
{
mp_limb_t c;
c = a[n] - b[n] - mpn_sub_n (r, a, b, n);
r[n] = (c & GMP_LIMB_HIGHBIT) ? mpn_add_1 (r, r, n, -c) : c;
}
*/
/* r <- a+b mod 2^(n*GMP_NUMB_BITS)+1, s <- a-b mod 2^(n*GMP_NUMB_BITS)+1.
Assumes a and b are semi-normalized. If HAVE_NATIVE_mpn_addsub_n is false,
s and a,b can't be overlapped. by Jeunder Yu
*/
static inline void
mpn_fft_addsub_modF (mp_ptr r, mp_ptr s, mp_srcptr a, mp_srcptr b, int n)
{
mp_limb_t c;
#if HAVE_NATIVE_mpn_addsub_n
c = mpn_addsub_n (r, s, a, b, n + 1) & CNST_LIMB(1);
#else
ASSERT (s != a && s != b);
c = mpn_sub_n (s, a, b, n + 1);
mpn_add_n (r, a, b, n + 1);
#endif
if (c)
{
c = s[n];
s[n] = CNST_LIMB(0);
mpn_incr_u (s, -c);
}
if (r[n] > CNST_LIMB(1))
{
c = r[n] - CNST_LIMB(1);
r[n] = CNST_LIMB(1);
mpn_decr_u (r, c);
}
}
/* Tree traversal DIF FFT for forward transform, by Jeunder Yu, NCTU Taiwan, October 2004.
The forward FFT is a Decimation In Frequency (DIF) FFT, and the inverse FFT
is a Decimation In Time (DIT) FFT, the post-bit-reverse permutation of DIF
was skiped, since it can be cancelled by the pre-bit-reverse permutation of DIT.
This algorithm has following advantage:
1. It is iterative (non-recursive)
2. Unlike other iterative FFT algorithms, it has better cache locality.
In my experiment, my algorithm exceeds Bailey's six-step fft (applied on
three prime modular multiplication)
3. It doesn't need omega table anymore. A single limb addition/subtraction
for in place computation of omega^xy value is negligible
4. There is not any additional temporary storage allocation, the parameter
tp is enough for whole algorithm
The main idea for this algorithm is treats the FFT computational flow as a
binary tree. The computation of DIF FFT is top-down, and the computation of
DIT FFT is bottom-up. For cache locality, it must perform left and depth
first traversal on the binary tree. Given every tree-node a numebr (from left
to right, top to down, the root is one and so on), we empoly the relation of
node numbers, so we don't need stack for tree traversal.
This idea can also be applied to Karatsuba on a 3-ary tree (or Toom-3 on a
5-ary tree), but a special case must be considered when the size of operand
is not a multiple of 2 (or 3 for Toom-3).
This technique is my own creation, I don't know whether this technique was
appeared on any literature or not.
*/
/* Interface changed, skip ll and inc parameters. by Jeunder Yu */
/*
static void
mpn_fft_fft_sqr (mp_ptr *Ap, mp_size_t K, int **ll,
mp_size_t omega, mp_size_t n, mp_size_t inc, mp_ptr tp)
*/
static void
mpn_fft_fft_sqr (mp_ptr *Ap, mp_size_t K,
mp_size_t omega, mp_size_t n, mp_ptr tp)
{
mp_size_t node = 1, step = K >> 1;
/* Top-down traversal, set node = 1 (i.e. the root of tree) */
do
{
/* Ap[0] <- Ap[0] + Ap[step], Ap[step] <- Ap[0] - Ap[step]. */
#if HAVE_NATIVE_mpn_addsub_n
mpn_fft_addsub_modF (Ap[0], Ap[step], Ap[0], Ap[step], n);
#else
mpn_fft_addsub_modF (Ap[0], tp, Ap[0], Ap[step], n);
MPN_COPY (Ap[step], tp, n + 1);
#endif
if (step == 1)
{ /* We are on the bottom of tree, if we are on right node,
then go to parent until we meet a left node */
while (node & 1) /* FIXME: count the trailing zeros of node+1 maybe better? */
{
node >>= 1; /* Go to parent node */
step <<= 1;
omega >>= 1;
}
node |= 1; /* Go to right node from left node */
Ap += 2;
}
else
{
mp_size_t i, w;
/* Ap[i] <- Ap[i] + Ap[i + step],
Ap[i + step] <- (Ap[i] - Ap[i + step]) * 2^(-i*omega). */
for (i = 1, w = 2 * n * BITS_PER_MP_LIMB - omega; i < step; ++i, w -= omega)
{
mpn_fft_addsub_modF (Ap[i], tp, Ap[i], Ap[i + step], n);
mpn_fft_mul_2exp_modF (Ap[i + step], tp, w, n);
}
node <<= 1; /* Go to left child */
step >>= 1;
omega <<= 1;
}
}
while (node != 1);
/* Post-bit-reverse permutation was skiped */
}
/* Skips this function. by Jeunder Yu */
/* A1 <- A0 + A1 * 2^e1
A0 <- A0 + A1 * 2^e0
Same for B0, B1.
t0 and t1 must have space for (n+1) limbs each.
We have 0 <= e0, e1 < 2*n*GMP_NUMB_BITS.
*/
/*
static inline void
mpn_fft_butterfly (mp_ptr A0, mp_ptr A1, mp_ptr B0, mp_ptr B1, int e0, int e1,
mp_ptr t0, mp_ptr t1, mp_size_t n)
{
int dif;
ASSERT (e0 != e1);
dif = (e1 - e0) % (n * GMP_NUMB_BITS);
mpn_fft_mul_2exp_modF (t0, A1, e0, n);
if (dif)
{
mpn_fft_mul_2exp_modF (t1, A1, e1, n);
mpn_fft_add_modF (A1, A0, t1, n);
}
else
mpn_fft_sub_modF (A1, A0, t0, n);
mpn_fft_add_modF (A0, A0, t0, n);
mpn_fft_mul_2exp_modF (t0, B1, e0, n);
if (dif)
{
mpn_fft_mul_2exp_modF (t1, B1, e1, n);
mpn_fft_add_modF (B1, B0, t1, n);
}
else
mpn_fft_sub_modF (B1, B0, t0, n);
mpn_fft_add_modF (B0, B0, t0, n);
}
*/
/* Tree traversal DIF FFT for forward transform, by Jeunder Yu, NCTU Taiwan, October 2004.
In fact, this function was skiped, we have replaced one mpn_fft_fft call by
two mpn_fft_fft_sqr calls for better cache locality. You can delete it and
rename mpn_fft_fft_sqr to mpn_fft_fft.
Interface changed, skip ll and inc parameters.
*/
/*
static void
mpn_fft_fft (mp_ptr *Ap, mp_ptr *Bp, mp_size_t K, int **ll,
mp_size_t omega, mp_size_t n, mp_size_t inc, mp_ptr tp)
*/
static void
mpn_fft_fft (mp_ptr *Ap, mp_ptr *Bp, mp_size_t K,
mp_size_t omega, mp_size_t n, mp_ptr tp)
{
mp_size_t node = 1, step = K >> 1;
/* Top-down traversal, set node = 1 (i.e. the root of tree) */
do
{
/* Ap[0] <- Ap[0] + Ap[step], Ap[step] <- Ap[0] - Ap[step], Bp is same as Ap. */
#if HAVE_NATIVE_mpn_addsub_n
mpn_fft_addsub_modF (Ap[0], Ap[step], Ap[0], Ap[step], n);
mpn_fft_addsub_modF (Bp[0], Bp[step], Bp[0], Bp[step], n);
#else
mpn_fft_addsub_modF (Ap[0], tp, Ap[0], Ap[step], n);
MPN_COPY (Ap[step], tp, n + 1);
mpn_fft_addsub_modF (Bp[0], tp, Bp[0], Bp[step], n);
MPN_COPY (Bp[step], tp, n + 1);
#endif
if (step == 1)
{ /* We are on the bottom of tree, if we are on right node,
then go to parent until we meet a left node */
while (node & 1) /* FIXME: count the trailing zeros of node+1 maybe better? */
{
node >>= 1; /* Go to parent node */
step <<= 1;
omega >>= 1;
}
node |= 1; /* Go to right node from left node */
Ap += 2;
Bp += 2;
}
else
{
mp_size_t i, w;
/* Ap[i] <- Ap[i] + Ap[i + step],
Ap[i + step] <- (Ap[i] - Ap[i + step]) * 2^(-i*omega), Bp is same as Ap. */
for (i = 1, w = 2 * n * BITS_PER_MP_LIMB - omega; i < step; ++i, w -= omega)
{
mpn_fft_addsub_modF (Ap[i], tp, Ap[i], Ap[i + step], n);
mpn_fft_mul_2exp_modF (Ap[i + step], tp, w, n);
mpn_fft_addsub_modF (Bp[i], tp, Bp[i], Bp[i + step], n);
mpn_fft_mul_2exp_modF (Bp[i + step], tp, w, n);
}
node <<= 1; /* go to left child */
step >>= 1;
omega <<= 1;
}
}
while (node != 1);
/* Post-bit-reverse permutation was skiped */
}
/* Given ap[0..n] with ap[n]<=1, reduce it modulo 2^(n*GMP_NUMB_BITS)+1,
by subtracting that modulus if necessary.
If ap[0..n] is exactly 2^(n*GMP_NUMB_BITS) then mpn_sub_1 produces a
borrow and the limbs must be zeroed out again. This will occur very
infrequently. */
static void
mpn_fft_normalize (mp_ptr ap, mp_size_t n)
{
if (ap[n])
{
if ((ap[n] = mpn_sub_1 (ap, ap, n, CNST_LIMB(1))))
MPN_ZERO (ap, n);
}
}
/* a[i] <- a[i]*b[i] mod 2^(n*GMP_NUMB_BITS)+1 for 0 <= i < K */
static void
mpn_fft_mul_modF_K (mp_ptr *ap, mp_ptr *bp, mp_size_t n, int K)
{
int i;
int sqr = (ap == bp);
TMP_DECL(marker);
TMP_MARK(marker);
if (n >= (sqr ? SQR_FFT_MODF_THRESHOLD : MUL_FFT_MODF_THRESHOLD))
{
int k, K2, nprime2, Nprime2, M2, maxLK, l, Mp2;
/* int **_fft_l; */ /* We don't need _fft_l table. by Jeunder Yu */
mp_ptr *Ap, *Bp, A, B, T;
k = mpn_fft_best_k (n, sqr);
K2 = 1 << k;
ASSERT_ALWAYS(n % K2 == 0);
maxLK = (K2 > GMP_NUMB_BITS) ? K2 : GMP_NUMB_BITS;
M2 = n * GMP_NUMB_BITS / K2;
l = n / K2;
/* Additional two bits can be skiped, only one bit for signed
representation is enough. by Jeunder Yu
*/
/* Nprime2 = ((2 * M2 + k + 2 + maxLK) / maxLK) * maxLK; */
/* Nprime2 = ceil((2*M2+k+3)/maxLK)*maxLK*/
Nprime2 = ((2 * M2 + k + maxLK) / maxLK) * maxLK;
/* Nprime2 = ceil((2*M2+k+1)/maxLK)*maxLK*/
nprime2 = Nprime2 / GMP_NUMB_BITS;
/* we should ensure that nprime2 is a multiple of the next K */
if (nprime2 >= (sqr ? SQR_FFT_MODF_THRESHOLD : MUL_FFT_MODF_THRESHOLD))
{
unsigned long K3;
while (nprime2 % (K3 = 1 << mpn_fft_best_k (nprime2, sqr)))
{
nprime2 = ((nprime2 + K3 - 1) / K3) * K3;
Nprime2 = nprime2 * BITS_PER_MP_LIMB;
/* warning: since nprime2 changed, K3 may change too! */
}
ASSERT(nprime2 % K3 == 0);
}
ASSERT_ALWAYS(nprime2 < n); /* otherwise we'll loop */
Mp2 = Nprime2 / K2;
Ap = TMP_ALLOC_MP_PTRS (K2);
Bp = TMP_ALLOC_MP_PTRS (K2);
A = TMP_ALLOC_LIMBS (2 * K2 * (nprime2 + 1));
/* T have space for only nprime2+1 limbs is enough. by Jeunder Yu */
/* T = TMP_ALLOC_LIMBS (2 * (nprime2 + 1)); */
T = TMP_ALLOC_LIMBS (nprime2 + 1);
B = A + K2 * (nprime2 + 1);
/* We don't need _fft_l table. by Jeunder Yu */
/*
_fft_l = TMP_ALLOC_TYPE (k + 1, int*);
for (i = 0; i <= k; i++)
_fft_l[i] = TMP_ALLOC_TYPE (1<<i, int);
mpn_fft_initl (_fft_l, k);
*/
TRACE (printf ("recurse: %dx%d limbs -> %d times %dx%d (%1.2f)\n", n,
n, K2, nprime2, nprime2, 2.0*(double)n/nprime2/K2));
for (i = 0; i < K; i++, ap++, bp++)
{
mpn_fft_normalize (*ap, n);
if (!sqr)
mpn_fft_normalize (*bp, n);
/* Skips the _fft_l parameter. by Jeunder Yu */
/*
mpn_mul_fft_internal (*ap, *ap, *bp, n, k, K2, Ap, Bp, A, B, nprime2,
l, Mp2, _fft_l, T, 1);
*/
mpn_mul_fft_internal (*ap, *ap, *bp, n, k, K2, Ap, Bp, A, B, nprime2,
l, Mp2, T, 1);
}
}
else
{
mp_ptr a, b, tp, tpn;
mp_limb_t cc;
int n2 = 2 * n;
tp = TMP_ALLOC_LIMBS (n2);
tpn = tp + n;
TRACE (printf (" mpn_mul_n %d of %d limbs\n", K, n));
for (i = 0; i < K; i++)
{
a = *ap++;
b = *bp++;
if (sqr)
mpn_sqr_n (tp, a, n);
else
mpn_mul_n (tp, b, a, n);
if (a[n] != 0)
cc = mpn_add_n (tpn, tpn, b, n);
else
cc = 0;
if (b[n] != 0)
cc += mpn_add_n (tpn, tpn, a, n) + a[n];
if (cc != 0)
{
cc = mpn_add_1 (tp, tp, n2, cc);
ASSERT_NOCARRY (mpn_add_1 (tp, tp, n2, cc));
}
a[n] = mpn_sub_n (a, tp, tpn, n) && mpn_add_1 (a, a, n, CNST_LIMB(1));
}
}
TMP_FREE(marker);
}
/* Tree traversal DIT FFT for inverse transform, by Jeunder Yu, NCTU Taiwan, October 2004. */
static void
mpn_fft_fftinv (mp_ptr *Ap, int K, mp_size_t omega, mp_size_t n, mp_ptr tp)
{
/* Pre-bit-reverse permutation was skiped */
mp_size_t node = K >> 1, step = 1, level = 0;
omega *= K >> 1;
/* Bottom-up traversal, set node = K / 2 (most left-bottom node).
The corresponding 2^omega value on bottom is -1 (i.e. primitive_root^(K/2)) */
do
{
mp_size_t i, w;
if (step != 1)
Ap -= step << 1;
/* Ap[0] <- Ap[0] + Ap[step], Ap[step] <- Ap[0] - Ap[step] */
#if HAVE_NATIVE_mpn_addsub_n
mpn_fft_addsub_modF (Ap[0], Ap[step], Ap[0], Ap[step], n);
#else
mpn_fft_addsub_modF (Ap[0], tp, Ap[0], Ap[step], n);
MPN_COPY (Ap[step], tp, n + 1);
#endif
/* Ap[i] <- Ap[i] + 2^(i*omega) * Ap[i + step],
Ap[i + step] <- Ap[i] - 2^(i*omega) * Ap[i + step] */
for (i = 1, w = omega; i < step; ++i, w += omega)
{
mpn_fft_mul_2exp_modF (tp, Ap[i + step], w, n);
mpn_fft_addsub_modF (Ap[i], Ap[i + step], Ap[i], tp, n);
}
Ap += step << 1;
if (node & 1)
{ /* We are on right node */
node >>= 1; /* Go to parent */
step <<= 1;
omega >>= 1;
++level;
}
else
{ /* We are on left node */
node = (node | 1) << level; /* Go to right node and down to left bottom */
step = 1;
omega <<= level;
level = 0;
}
}
while (node != 0);
}
/* A <- A/2^k mod 2^(n*GMP_NUMB_BITS)+1 */
static void
mpn_fft_div_2exp_modF (mp_ptr r, mp_srcptr a, int k, mp_size_t n)
{
int i;
ASSERT (r != a);
i = 2 * n * GMP_NUMB_BITS;
i = (i - k) % i;
mpn_fft_mul_2exp_modF (r, a, i, n);
/* 1/2^k = 2^(2nL-k) mod 2^(n*GMP_NUMB_BITS)+1 */
/* normalize so that R < 2^(n*GMP_NUMB_BITS)+1 */
mpn_fft_normalize (r, n);
}
/* {rp,n} <- {ap,an} mod 2^(n*GMP_NUMB_BITS)+1, n <= an <= 3*n.
Returns carry out, i.e. 1 iff {ap,an} = -1 mod 2^(n*GMP_NUMB_BITS)+1,
then {rp,n}=0.
*/
static int
mpn_fft_norm_modF (mp_ptr rp, mp_size_t n, mp_ptr ap, mp_size_t an)
{
mp_size_t l;
long int m;
mp_limb_t cc;
int rpn;
ASSERT ((n <= an) && (an <= 3 * n));
m = an - 2 * n;
if (m > 0)
{
l = n;
/* add {ap, m} and {ap+2n, m} in {rp, m} */
cc = mpn_add_n (rp, ap, ap + 2 * n, m);
/* copy {ap+m, n-m} to {rp+m, n-m} */
rpn = mpn_add_1 (rp + m, ap + m, n - m, cc);
}
else
{
l = an - n; /* l <= n */
MPN_COPY (rp, ap, n);
rpn = 0;
}
/* remains to subtract {ap+n, l} from {rp, n+1} */
cc = mpn_sub_n (rp, rp, ap + n, l);
rpn -= mpn_sub_1 (rp + l, rp + l, n - l, cc);
if (rpn < 0) /* necessarily rpn = -1 */
rpn = mpn_add_1 (rp, rp, n, CNST_LIMB(1));
return rpn;
}
/* store in A[0..nprime] the first M bits from {n, nl},
in A[nprime+1..] the following M bits, ...
Assumes M is a multiple of GMP_NUMB_BITS (M = l * GMP_NUMB_BITS).
T must have space for at least (nprime + 1) limbs.
We must have nl <= 2*K*l.
*/
static void
mpn_mul_fft_decompose (mp_ptr A, mp_ptr *Ap, int K, int nprime, mp_srcptr n,
mp_size_t nl, int l, int Mp, mp_ptr T)
{
int i, j, cc;
mp_ptr tmp;
mp_size_t Kl = K * l;
TMP_DECL(marker);
TMP_MARK(marker);
ASSERT_ALWAYS (nl <= 2 * Kl);
if (nl > Kl) /* normalize {n, nl} mod 2^(Kl*GMP_NUMB_BITS)+1 */
{
int dif = nl - Kl;
/* FIXME: we could subtract the "high" part from the low part in place,
while doing the decomposition below */
tmp = TMP_ALLOC_LIMBS (Kl + 1);
MPN_COPY (tmp, n, Kl);
cc = mpn_sub_n (tmp, tmp, n + Kl, dif);
cc = mpn_sub_1 (tmp + dif, tmp + dif, Kl - dif, (mp_limb_t) cc);
tmp[Kl] = mpn_add_1 (tmp, tmp, Kl, (mp_limb_t) cc);
nl = Kl + 1;
n = tmp;
}
for (i = 0; i < K; i++)
{
Ap[i] = A;
/* store the next M bits of n into A[0..nprime] */
if (nl > 0) /* nl is the number of remaining limbs */
{
j = (l <= nl && i < K - 1) ? l : nl; /* store j next limbs */
nl -= j;
MPN_COPY (T, n, j);
MPN_ZERO (T + j, nprime + 1 - j);
n += l;
mpn_fft_mul_2exp_modF (A, T, i * Mp, nprime);
}
else
MPN_ZERO (A, nprime + 1);
A += nprime + 1;
}
ASSERT_ALWAYS (nl == 0);
TMP_FREE(marker);
}
/* op <- n*m mod 2^N+1 with fft of size 2^k where N=pl*GMP_NUMB_BITS
n and m have respectively nl and ml limbs
op must have space for pl+1 limbs if rec=1 (and pl limbs if rec=0).
One must have pl = mpn_fft_next_size (pl, k).
T must have space for 2 * (nprime + 1) limbs.
If rec=0, then store only the pl low bits of the result, and return
the out carry.
*/
/* Interface changed, skips _fft_l parameter. by Jeunder Yu */
/*
static int
mpn_mul_fft_internal (mp_ptr op, mp_srcptr n, mp_srcptr m, mp_size_t pl,
int k, int K,
mp_ptr *Ap, mp_ptr *Bp,
mp_ptr A, mp_ptr B,
mp_size_t nprime, mp_size_t l, mp_size_t Mp,
int **_fft_l,
mp_ptr T, int rec)
*/
static int
mpn_mul_fft_internal (mp_ptr op, mp_srcptr n, mp_srcptr m, mp_size_t pl,
int k, int K,
mp_ptr *Ap, mp_ptr *Bp,
mp_ptr A, mp_ptr B,
mp_size_t nprime, mp_size_t l, mp_size_t Mp,
mp_ptr T, int rec)
{
/* Variable j is unnecessary. by Jeunder Yu */
/* int i, sqr, pla, lo, sh, j; */
int i, sqr, pla, lo, sh;
mp_ptr p;
sqr = n == m;
TRACE (printf ("pl=%d k=%d K=%d np=%d l=%d Mp=%d rec=%d sqr=%d\n",
pl,k,K,nprime,l,Mp,rec,sqr));
/* decomposition of inputs into arrays Ap[i] and Bp[i] */
if (rec)
{
mpn_mul_fft_decompose (A, Ap, K, nprime, n, K * l + 1, l, Mp, T);
if (!sqr)
mpn_mul_fft_decompose (B, Bp, K, nprime, m, K * l + 1, l, Mp, T);
}
/* direct fft's */
/* 1. Skip _fft_l and inc parameters on fft function call.
2. Since the fft function is iterative (non-recursive), the overhead of
flow control is quite small. We replace single mpn_fft_fft call by two
mpn_fft_fft_sqr calls for better cache locality.
Therefore the mpn_fft_fft function is unnecessary, I recommend you to
delete mpn_fft_fft and rename mpn_fft_fft_sqr to mpn_fft_fft. by Jeunder Yu
*/
mpn_fft_fft_sqr (Ap, K, 2 * Mp, nprime, T);
if (!sqr)
mpn_fft_fft_sqr (Bp, K, 2 * Mp, nprime, T);
/*
if (sqr)
mpn_fft_fft_sqr (Ap, K, _fft_l + k, 2 * Mp, nprime, 1, T);
else
mpn_fft_fft (Ap, Bp, K, _fft_l + k, 2 * Mp, nprime, 1, T);
*/
/* term to term multiplications */
mpn_fft_mul_modF_K (Ap, (sqr) ? Ap : Bp, nprime, K);
/* inverse fft's */
mpn_fft_fftinv (Ap, K, 2 * Mp, nprime, T);
/* division of terms after inverse fft */
/* Bp[0] = T + nprime + 1; */ /* T have space for only nprime+1 limbs. by Jeunder Yu */
Bp[0] = T;
mpn_fft_div_2exp_modF (Bp[0], Ap[0], k, nprime);
for (i = 1; i < K; i++)
{
Bp[i] = Ap[i-1];
/* Our algorithm is not reverse order. by Jeunder Yu */
/* mpn_fft_div_2exp_modF (Bp[i], Ap[i], k + ((K - i) % K) * Mp, nprime); */
mpn_fft_div_2exp_modF (Bp[i], Ap[i], k + i * Mp, nprime);
}
/* addition of terms in result p */
/* MPN_ZERO (T, nprime + 1); */ /* T is unnecessary, see below. by Jeunder Yu */
pla = l * (K - 1) + nprime + 1; /* number of required limbs for p */
p = B; /* B has K*(n' + 1) limbs, which is >= pla, i.e. enough */
MPN_ZERO (p, pla);
sqr = 0; /* will accumulate the (signed) carry at p[pla] */
for (i = K - 1, lo = l * i + nprime,sh = l * i; i >= 0; i--,lo -= l,sh -= l)
{
mp_ptr n = p + sh;
/* Our algorithm is not reverse order, so variable j is unnecessary. by Jeunder Yu */
/* j = (K - i) % K; */
if (mpn_add_n (n, n, Bp[i], nprime + 1))
sqr += mpn_add_1 (n + nprime + 1, n + nprime + 1,
pla - sh - nprime - 1, CNST_LIMB(1));
/* The temp. T and mpn_cmp function call can be skiped, it can be done by
simplely checks the Ap[i][nprime] for -1 or the msb of Ap[i][nprime - 1]
for the negative term of negacyclic convolution, this is the reason for
only one additional bit on Nprime is enough, and T have space for only
nprime+1 limbs is enough. by Jeunder Yu.
*/
/*
T[2 * l] = i + 1;
if (mpn_cmp (Bp[j], T, nprime + 1) > 0)
*/
if (Bp[i][nprime - 1] & GMP_LIMB_HIGHBIT || Bp[i][nprime])
{ /* subtract 2^N'+1 */
sqr -= mpn_sub_1 (n, n, pla - sh, CNST_LIMB(1));
sqr -= mpn_sub_1 (p + lo, p + lo, pla - lo, CNST_LIMB(1));
}
}
if (sqr == -1)
{
if ((sqr = mpn_add_1 (p + pla - pl,p + pla - pl, pl, CNST_LIMB(1))))
{
/* p[pla-pl]...p[pla-1] are all zero */
mpn_sub_1 (p + pla - pl - 1, p + pla - pl - 1, pl + 1, CNST_LIMB(1));
mpn_sub_1 (p + pla - 1, p + pla - 1, 1, CNST_LIMB(1));
}
}
else if (sqr == 1)
{
if (pla >= 2 * pl)
{
while ((sqr = mpn_add_1 (p + pla - 2 * pl, p + pla - 2 * pl, 2 * pl, (mp_limb_t) sqr)))
;
}
else
{
sqr = mpn_sub_1 (p + pla - pl, p + pla - pl, pl, (mp_limb_t) sqr);
ASSERT (sqr == 0);
}
}
else
ASSERT (sqr == 0);
/* here p < 2^(2M) [K 2^(M(K-1)) + (K-1) 2^(M(K-2)) + ... ]
< K 2^(2M) [2^(M(K-1)) + 2^(M(K-2)) + ... ]
< K 2^(2M) 2^(M(K-1))*2 = 2^(M*K+M+k+1) */
i = mpn_fft_norm_modF (op, pl, p, pla);
if (rec) /* store the carry out */
op[pl] = i;
return i;
}
/* return the lcm of a and 2^k */
static unsigned long int
mpn_mul_fft_lcm (unsigned long int a, unsigned int k)
{
unsigned long int l = 1;
while ((a % 2 == 0) && (k > 0))
{
a >>= 1;
k --;
l <<= 1;
}
return (l * a) << k;
}
/* For interface consistency on GMP 4.1.4. by Jeunder Yu */
/* int */
void
mpn_mul_fft (mp_ptr op, mp_size_t pl,
mp_srcptr n, mp_size_t nl,
mp_srcptr m, mp_size_t ml,
int k)
{
int K, maxLK, i;
mp_size_t N, Nprime, nprime, M, Mp, l;
mp_ptr *Ap, *Bp, A, T, B;
/* int **_fft_l; */ /* We don't need _fft_l table. by Jeunder Yu */
int sqr = (n == m && nl == ml);
TMP_DECL(marker);
TRACE (printf ("\nmpn_mul_fft pl=%ld nl=%ld ml=%ld k=%d\n", pl, nl, ml, k));
ASSERT_ALWAYS (mpn_fft_next_size (pl, k) == pl);
TMP_MARK(marker);
N = pl * GMP_NUMB_BITS;
/* We don't need _fft_l table. by Jeunder Yu */
/*
_fft_l = TMP_ALLOC_TYPE (k + 1, int*);
for (i = 0; i <= k; i++)
_fft_l[i] = TMP_ALLOC_TYPE (1 << i, int);
mpn_fft_initl (_fft_l, k);
*/
K = 1 << k;
M = N / K; /* N = 2^k M */
l = 1 + (M - 1) / GMP_NUMB_BITS;
maxLK = mpn_mul_fft_lcm ((unsigned long) GMP_NUMB_BITS, k); /* lcm (GMP_NUMB_BITS, 2^k) */
/* Additional two bits can be skiped, only one bit for signed
representation is enough. by Jeunder Yu
*/
/* Nprime = ((2 * M + k + 2 + maxLK) / maxLK) * maxLK; */
/* Nprime = ceil((2*M+k+3)/maxLK)*maxLK; */
Nprime = ((2 * M + k + maxLK) / maxLK) * maxLK;
/* Nprime = ceil((2*M+k+1)/maxLK)*maxLK; */
nprime = Nprime / GMP_NUMB_BITS;
TRACE (printf ("N=%d K=%d, M=%d, l=%d, maxLK=%d, Np=%d, np=%d\n",
N, K, M, l, maxLK, Nprime, nprime));
/* we should ensure that recursively, nprime is a multiple of the next K */
if (nprime >= (sqr ? SQR_FFT_MODF_THRESHOLD : MUL_FFT_MODF_THRESHOLD))
{
unsigned long K2;
while (nprime % (K2 = 1 << mpn_fft_best_k (nprime, sqr)))
{
nprime = ((nprime + K2 - 1) / K2) * K2;
Nprime = nprime * BITS_PER_MP_LIMB;
/* warning: since nprime changed, K2 may change too! */
}
TRACE (printf ("new maxLK=%d, Np=%d, np=%d\n", maxLK, Nprime, nprime));
ASSERT(nprime % K2 == 0);
}
ASSERT_ALWAYS (nprime < pl); /* otherwise we'll loop */
/* T have space for only nprime+1 limbs is enough. by Jeudner Yu */
/* T = TMP_ALLOC_LIMBS (2 * (nprime + 1)); */
T = TMP_ALLOC_LIMBS (nprime + 1);
Mp = Nprime / K;
TRACE (printf ("%dx%d limbs -> %d times %dx%d limbs (%1.2f)\n",
pl, pl, K, nprime, nprime, 2.0 * (double) N / Nprime / K);
printf (" temp space %ld\n", 2 * K * (nprime + 1)));
A = __GMP_ALLOCATE_FUNC_LIMBS (2 * K * (nprime + 1));
B = A + K * (nprime + 1);
Ap = TMP_ALLOC_MP_PTRS (K);
Bp = TMP_ALLOC_MP_PTRS (K);
/* special decomposition for main call */
/* nl is the number of significant limbs in n */
mpn_mul_fft_decompose (A, Ap, K, nprime, n, nl, l, Mp, T);
if (n != m)
mpn_mul_fft_decompose (B, Bp, K, nprime, m, ml, l, Mp, T);
/* Skips _fft_l parameter. by Jeunder Yu /
/* i = mpn_mul_fft_internal (op, n, m, pl, k, K, Ap, Bp, A, B, nprime, l, Mp, _fft_l, T, 0); */
i = mpn_mul_fft_internal (op, n, m, pl, k, K, Ap, Bp, A, B, nprime, l, Mp, T, 0);
TMP_FREE(marker);
__GMP_FREE_FUNC_LIMBS (A, 2 * K * (nprime + 1));
/* For interface consistency on GMP 4.1.4. by Jeunder Yu */
/* return i; */
}
/* multiply {n, nl} by {m, ml}, and put the result in {op, nl+ml} */
void
mpn_mul_fft_full (mp_ptr op,
mp_srcptr n, mp_size_t nl,
mp_srcptr m, mp_size_t ml)
{
mp_ptr pad_op;
mp_size_t pl, pl2, pl3, l;
int k2, k3;
int sqr = (n == m && nl == ml);
int cc, c2, oldcc;
pl = nl + ml; /* total number of limbs of the result */
/* perform a fft mod 2^(2N)+1 and one mod 2^(3N)+1.
We must have pl3 = 3/2 * pl2, with pl2 a multiple of 2^k2, and
pl3 a multiple of 2^k3. Since k3 >= k2, both are multiples of 2^k2,
and pl2 must be an even multiple of 2^k2. Thus (pl2,pl3) =
(2*j*2^k2,3*j*2^k2), which works for 3*j <= pl/2^k2 <= 5*j.
We need that consecutive intervals overlap, i.e. 5*j >= 3*(j+1),
which requires j>=2. Thus this scheme requires pl >= 6 * 2^FFT_FIRST_K. */
/* ASSERT_ALWAYS(pl >= 6 * (1 << FFT_FIRST_K)); */
pl2 = (2 * pl - 1) / 5; /* ceil (2pl/5) - 1 */
do
{
pl2 ++;
k2 = mpn_fft_best_k (pl2, sqr); /* best fft size for pl2 limbs */
pl2 = mpn_fft_next_size (pl2, k2);
pl3 = 3 * pl2 / 2; /* since k>=FFT_FIRST_K=4, pl2 is a multiple of 2^4,
thus pl2 / 2 is exact */
/* k3 = mpn_fft_best_k (pl3, sqr); */
k3 = k2;
}
while (mpn_fft_next_size (pl3, k3) != pl3);
TRACE (printf ("mpn_mul_fft_full nl=%ld ml=%ld -> pl2=%ld pl3=%ld k=%d\n",
nl, ml, pl2, pl3, k2));
ASSERT_ALWAYS(pl3 <= pl);
mpn_mul_fft (op, pl3, n, nl, m, ml, k3); /* mu */
pad_op = __GMP_ALLOCATE_FUNC_LIMBS (pl2);
mpn_mul_fft (pad_op, pl2, n, nl, m, ml, k2); /* lambda */
cc = mpn_sub_n (pad_op, pad_op, op, pl2); /* lambda - low(mu) */
/* 0 <= cc <= 1 */
l = pl3 - pl2; /* l = pl2 / 2 since pl3 = 3/2 * pl2 */
c2 = mpn_add_n (pad_op, pad_op, op + pl2, l);
cc = mpn_add_1 (pad_op + l, pad_op + l, l, (mp_limb_t) c2) - cc; /* -1 <= cc <= 1 */
if (cc < 0)
cc = mpn_add_1 (pad_op, pad_op, pl2, (mp_limb_t) -cc);
/* 0 <= cc <= 1 */
/* now lambda-mu = {pad_op, pl2} - cc mod 2^(pl2*GMP_NUMB_BITS)+1 */
oldcc = cc;
#if HAVE_NATIVE_mpn_addsub_n
c2 = mpn_addsub_n (pad_op + l, pad_op, pad_op, pad_op + l, l);
/* c2 & 1 is the borrow, c2 & 2 is the carry */
cc += c2 >> 1; /* carry out from high <- low + high */
c2 = c2 & 1; /* borrow out from low <- low - high */
#else
{
mp_ptr tmp;
TMP_DECL(marker);
TMP_MARK(marker);
tmp = TMP_ALLOC_LIMBS (l);
MPN_COPY (tmp, pad_op, l);
c2 = mpn_sub_n (pad_op, pad_op, pad_op + l, l);
cc += mpn_add_n (pad_op + l, tmp, pad_op + l, l);
TMP_FREE(marker);
}
#endif
c2 += oldcc;
/* first normalize {pad_op, pl2} before dividing by 2: c2 is the borrow
at pad_op + l, cc is the carry at pad_op + pl2 */
/* 0 <= cc <= 2 */
cc -= mpn_sub_1 (pad_op + l, pad_op + l, l, (mp_limb_t) c2);
/* -1 <= cc <= 2 */
if (cc > 0)
cc = -mpn_sub_1 (pad_op, pad_op, pl2, (mp_limb_t) cc);
/* now -1 <= cc <= 0 */
if (cc < 0)
cc = mpn_add_1 (pad_op, pad_op, pl2, (mp_limb_t) -cc);
/* now {pad_op, pl2} is normalized, with 0 <= cc <= 1 */
if (pad_op[0] & 1) /* if odd, add 2^(pl2*GMP_NUMB_BITS)+1 */
cc += 1 + mpn_add_1 (pad_op, pad_op, pl2, CNST_LIMB(1));
/* now 0 <= cc <= 2, but cc=2 cannot occur since it would give a carry
out below */
mpn_rshift (pad_op, pad_op, pl2, 1); /* divide by two */
if (cc) /* then cc=1 */
pad_op [pl2 - 1] |= (mp_limb_t) 1 << (GMP_NUMB_BITS - 1);
/* now {pad_op,pl2}-cc = (lambda-mu)/(1-2^(l*GMP_NUMB_BITS))
mod 2^(pl2*GMP_NUMB_BITS) + 1 */
c2 = mpn_add_n (op, op, pad_op, pl2); /* no need to add cc (is 0) */
/* since pl2+pl3 >= pl, necessary the extra limbs (including cc) are zero */
MPN_COPY (op + pl3, pad_op, pl - pl3);
ASSERT_MPN_ZERO_P (pad_op + pl - pl3, pl2 + pl3 - pl);
__GMP_FREE_FUNC_LIMBS (pad_op, pl2);
/* since the final result has at most pl limbs, no carry out below */
mpn_add_1 (op + pl2, op + pl2, pl - pl2, (mp_limb_t) c2);
}
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