cpython/Modules/_decimal/libmpdec/umodarith.h

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/*
* Copyright (c) 2008-2016 Stefan Krah. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS "AS IS" AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#ifndef UMODARITH_H
#define UMODARITH_H
#include "constants.h"
#include "mpdecimal.h"
#include "typearith.h"
/* Bignum: Low level routines for unsigned modular arithmetic. These are
used in the fast convolution functions for very large coefficients. */
/**************************************************************************/
/* ANSI modular arithmetic */
/**************************************************************************/
/*
* Restrictions: a < m and b < m
* ACL2 proof: umodarith.lisp: addmod-correct
*/
static inline mpd_uint_t
addmod(mpd_uint_t a, mpd_uint_t b, mpd_uint_t m)
{
mpd_uint_t s;
s = a + b;
s = (s < a) ? s - m : s;
s = (s >= m) ? s - m : s;
return s;
}
/*
* Restrictions: a < m and b < m
* ACL2 proof: umodarith.lisp: submod-2-correct
*/
static inline mpd_uint_t
submod(mpd_uint_t a, mpd_uint_t b, mpd_uint_t m)
{
mpd_uint_t d;
d = a - b;
d = (a < b) ? d + m : d;
return d;
}
/*
* Restrictions: a < 2m and b < 2m
* ACL2 proof: umodarith.lisp: section ext-submod
*/
static inline mpd_uint_t
ext_submod(mpd_uint_t a, mpd_uint_t b, mpd_uint_t m)
{
mpd_uint_t d;
a = (a >= m) ? a - m : a;
b = (b >= m) ? b - m : b;
d = a - b;
d = (a < b) ? d + m : d;
return d;
}
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/*
* Reduce double word modulo m.
* Restrictions: m != 0
* ACL2 proof: umodarith.lisp: section dw-reduce
*/
static inline mpd_uint_t
dw_reduce(mpd_uint_t hi, mpd_uint_t lo, mpd_uint_t m)
{
mpd_uint_t r1, r2, w;
_mpd_div_word(&w, &r1, hi, m);
_mpd_div_words(&w, &r2, r1, lo, m);
return r2;
}
/*
* Subtract double word from a.
* Restrictions: a < m
* ACL2 proof: umodarith.lisp: section dw-submod
*/
static inline mpd_uint_t
dw_submod(mpd_uint_t a, mpd_uint_t hi, mpd_uint_t lo, mpd_uint_t m)
{
mpd_uint_t d, r;
r = dw_reduce(hi, lo, m);
d = a - r;
d = (a < r) ? d + m : d;
return d;
}
#ifdef CONFIG_64
/**************************************************************************/
/* 64-bit modular arithmetic */
/**************************************************************************/
/*
* A proof of the algorithm is in literature/mulmod-64.txt. An ACL2
* proof is in umodarith.lisp: section "Fast modular reduction".
*
* Algorithm: calculate (a * b) % p:
*
* a) hi, lo <- a * b # Calculate a * b.
*
* b) hi, lo <- R(hi, lo) # Reduce modulo p.
*
* c) Repeat step b) until 0 <= hi * 2**64 + lo < 2*p.
*
* d) If the result is less than p, return lo. Otherwise return lo - p.
*/
static inline mpd_uint_t
x64_mulmod(mpd_uint_t a, mpd_uint_t b, mpd_uint_t m)
{
mpd_uint_t hi, lo, x, y;
_mpd_mul_words(&hi, &lo, a, b);
if (m & (1ULL<<32)) { /* P1 */
/* first reduction */
x = y = hi;
hi >>= 32;
x = lo - x;
if (x > lo) hi--;
y <<= 32;
lo = y + x;
if (lo < y) hi++;
/* second reduction */
x = y = hi;
hi >>= 32;
x = lo - x;
if (x > lo) hi--;
y <<= 32;
lo = y + x;
if (lo < y) hi++;
return (hi || lo >= m ? lo - m : lo);
}
else if (m & (1ULL<<34)) { /* P2 */
/* first reduction */
x = y = hi;
hi >>= 30;
x = lo - x;
if (x > lo) hi--;
y <<= 34;
lo = y + x;
if (lo < y) hi++;
/* second reduction */
x = y = hi;
hi >>= 30;
x = lo - x;
if (x > lo) hi--;
y <<= 34;
lo = y + x;
if (lo < y) hi++;
/* third reduction */
x = y = hi;
hi >>= 30;
x = lo - x;
if (x > lo) hi--;
y <<= 34;
lo = y + x;
if (lo < y) hi++;
return (hi || lo >= m ? lo - m : lo);
}
else { /* P3 */
/* first reduction */
x = y = hi;
hi >>= 24;
x = lo - x;
if (x > lo) hi--;
y <<= 40;
lo = y + x;
if (lo < y) hi++;
/* second reduction */
x = y = hi;
hi >>= 24;
x = lo - x;
if (x > lo) hi--;
y <<= 40;
lo = y + x;
if (lo < y) hi++;
/* third reduction */
x = y = hi;
hi >>= 24;
x = lo - x;
if (x > lo) hi--;
y <<= 40;
lo = y + x;
if (lo < y) hi++;
return (hi || lo >= m ? lo - m : lo);
}
}
static inline void
x64_mulmod2c(mpd_uint_t *a, mpd_uint_t *b, mpd_uint_t w, mpd_uint_t m)
{
*a = x64_mulmod(*a, w, m);
*b = x64_mulmod(*b, w, m);
}
static inline void
x64_mulmod2(mpd_uint_t *a0, mpd_uint_t b0, mpd_uint_t *a1, mpd_uint_t b1,
mpd_uint_t m)
{
*a0 = x64_mulmod(*a0, b0, m);
*a1 = x64_mulmod(*a1, b1, m);
}
static inline mpd_uint_t
x64_powmod(mpd_uint_t base, mpd_uint_t exp, mpd_uint_t umod)
{
mpd_uint_t r = 1;
while (exp > 0) {
if (exp & 1)
r = x64_mulmod(r, base, umod);
base = x64_mulmod(base, base, umod);
exp >>= 1;
}
return r;
}
/* END CONFIG_64 */
#else /* CONFIG_32 */
/**************************************************************************/
/* 32-bit modular arithmetic */
/**************************************************************************/
#if defined(ANSI)
#if !defined(LEGACY_COMPILER)
/* HAVE_UINT64_T */
static inline mpd_uint_t
std_mulmod(mpd_uint_t a, mpd_uint_t b, mpd_uint_t m)
{
return ((mpd_uuint_t) a * b) % m;
}
static inline void
std_mulmod2c(mpd_uint_t *a, mpd_uint_t *b, mpd_uint_t w, mpd_uint_t m)
{
*a = ((mpd_uuint_t) *a * w) % m;
*b = ((mpd_uuint_t) *b * w) % m;
}
static inline void
std_mulmod2(mpd_uint_t *a0, mpd_uint_t b0, mpd_uint_t *a1, mpd_uint_t b1,
mpd_uint_t m)
{
*a0 = ((mpd_uuint_t) *a0 * b0) % m;
*a1 = ((mpd_uuint_t) *a1 * b1) % m;
}
/* END HAVE_UINT64_T */
#else
/* LEGACY_COMPILER */
static inline mpd_uint_t
std_mulmod(mpd_uint_t a, mpd_uint_t b, mpd_uint_t m)
{
mpd_uint_t hi, lo, q, r;
_mpd_mul_words(&hi, &lo, a, b);
_mpd_div_words(&q, &r, hi, lo, m);
return r;
}
static inline void
std_mulmod2c(mpd_uint_t *a, mpd_uint_t *b, mpd_uint_t w, mpd_uint_t m)
{
*a = std_mulmod(*a, w, m);
*b = std_mulmod(*b, w, m);
}
static inline void
std_mulmod2(mpd_uint_t *a0, mpd_uint_t b0, mpd_uint_t *a1, mpd_uint_t b1,
mpd_uint_t m)
{
*a0 = std_mulmod(*a0, b0, m);
*a1 = std_mulmod(*a1, b1, m);
}
/* END LEGACY_COMPILER */
#endif
static inline mpd_uint_t
std_powmod(mpd_uint_t base, mpd_uint_t exp, mpd_uint_t umod)
{
mpd_uint_t r = 1;
while (exp > 0) {
if (exp & 1)
r = std_mulmod(r, base, umod);
base = std_mulmod(base, base, umod);
exp >>= 1;
}
return r;
}
#endif /* ANSI CONFIG_32 */
/**************************************************************************/
/* Pentium Pro modular arithmetic */
/**************************************************************************/
/*
* A proof of the algorithm is in literature/mulmod-ppro.txt. The FPU
* control word must be set to 64-bit precision and truncation mode
* prior to using these functions.
*
* Algorithm: calculate (a * b) % p:
*
* p := prime < 2**31
* pinv := (long double)1.0 / p (precalculated)
*
* a) n = a * b # Calculate exact product.
* b) qest = n * pinv # Calculate estimate for q = n / p.
* c) q = (qest+2**63)-2**63 # Truncate qest to the exact quotient.
* d) r = n - q * p # Calculate remainder.
*
* Remarks:
*
* - p = dmod and pinv = dinvmod.
* - dinvmod points to an array of three uint32_t, which is interpreted
* as an 80 bit long double by fldt.
* - Intel compilers prior to version 11 do not seem to handle the
* __GNUC__ inline assembly correctly.
* - random tests are provided in tests/extended/ppro_mulmod.c
*/
#if defined(PPRO)
#if defined(ASM)
/* Return (a * b) % dmod */
static inline mpd_uint_t
ppro_mulmod(mpd_uint_t a, mpd_uint_t b, double *dmod, uint32_t *dinvmod)
{
mpd_uint_t retval;
__asm__ (
"fildl %2\n\t"
"fildl %1\n\t"
"fmulp %%st, %%st(1)\n\t"
"fldt (%4)\n\t"
"fmul %%st(1), %%st\n\t"
"flds %5\n\t"
"fadd %%st, %%st(1)\n\t"
"fsubrp %%st, %%st(1)\n\t"
"fldl (%3)\n\t"
"fmulp %%st, %%st(1)\n\t"
"fsubrp %%st, %%st(1)\n\t"
"fistpl %0\n\t"
: "=m" (retval)
: "m" (a), "m" (b), "r" (dmod), "r" (dinvmod), "m" (MPD_TWO63)
: "st", "memory"
);
return retval;
}
/*
* Two modular multiplications in parallel:
* *a0 = (*a0 * w) % dmod
* *a1 = (*a1 * w) % dmod
*/
static inline void
ppro_mulmod2c(mpd_uint_t *a0, mpd_uint_t *a1, mpd_uint_t w,
double *dmod, uint32_t *dinvmod)
{
__asm__ (
"fildl %2\n\t"
"fildl (%1)\n\t"
"fmul %%st(1), %%st\n\t"
"fxch %%st(1)\n\t"
"fildl (%0)\n\t"
"fmulp %%st, %%st(1) \n\t"
"fldt (%4)\n\t"
"flds %5\n\t"
"fld %%st(2)\n\t"
"fmul %%st(2)\n\t"
"fadd %%st(1)\n\t"
"fsub %%st(1)\n\t"
"fmull (%3)\n\t"
"fsubrp %%st, %%st(3)\n\t"
"fxch %%st(2)\n\t"
"fistpl (%0)\n\t"
"fmul %%st(2)\n\t"
"fadd %%st(1)\n\t"
"fsubp %%st, %%st(1)\n\t"
"fmull (%3)\n\t"
"fsubrp %%st, %%st(1)\n\t"
"fistpl (%1)\n\t"
: : "r" (a0), "r" (a1), "m" (w),
"r" (dmod), "r" (dinvmod),
"m" (MPD_TWO63)
: "st", "memory"
);
}
/*
* Two modular multiplications in parallel:
* *a0 = (*a0 * b0) % dmod
* *a1 = (*a1 * b1) % dmod
*/
static inline void
ppro_mulmod2(mpd_uint_t *a0, mpd_uint_t b0, mpd_uint_t *a1, mpd_uint_t b1,
double *dmod, uint32_t *dinvmod)
{
__asm__ (
"fildl %3\n\t"
"fildl (%2)\n\t"
"fmulp %%st, %%st(1)\n\t"
"fildl %1\n\t"
"fildl (%0)\n\t"
"fmulp %%st, %%st(1)\n\t"
"fldt (%5)\n\t"
"fld %%st(2)\n\t"
"fmul %%st(1), %%st\n\t"
"fxch %%st(1)\n\t"
"fmul %%st(2), %%st\n\t"
"flds %6\n\t"
"fldl (%4)\n\t"
"fxch %%st(3)\n\t"
"fadd %%st(1), %%st\n\t"
"fxch %%st(2)\n\t"
"fadd %%st(1), %%st\n\t"
"fxch %%st(2)\n\t"
"fsub %%st(1), %%st\n\t"
"fxch %%st(2)\n\t"
"fsubp %%st, %%st(1)\n\t"
"fxch %%st(1)\n\t"
"fmul %%st(2), %%st\n\t"
"fxch %%st(1)\n\t"
"fmulp %%st, %%st(2)\n\t"
"fsubrp %%st, %%st(3)\n\t"
"fsubrp %%st, %%st(1)\n\t"
"fxch %%st(1)\n\t"
"fistpl (%2)\n\t"
"fistpl (%0)\n\t"
: : "r" (a0), "m" (b0), "r" (a1), "m" (b1),
"r" (dmod), "r" (dinvmod),
"m" (MPD_TWO63)
: "st", "memory"
);
}
/* END PPRO GCC ASM */
#elif defined(MASM)
/* Return (a * b) % dmod */
static inline mpd_uint_t __cdecl
ppro_mulmod(mpd_uint_t a, mpd_uint_t b, double *dmod, uint32_t *dinvmod)
{
mpd_uint_t retval;
__asm {
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mov eax, dinvmod
mov edx, dmod
fild b
fild a
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fmulp st(1), st
fld TBYTE PTR [eax]
fmul st, st(1)
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fld MPD_TWO63
fadd st(1), st
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fsubp st(1), st
fld QWORD PTR [edx]
fmulp st(1), st
fsubp st(1), st
fistp retval
}
return retval;
}
/*
* Two modular multiplications in parallel:
* *a0 = (*a0 * w) % dmod
* *a1 = (*a1 * w) % dmod
*/
static inline mpd_uint_t __cdecl
ppro_mulmod2c(mpd_uint_t *a0, mpd_uint_t *a1, mpd_uint_t w,
double *dmod, uint32_t *dinvmod)
{
__asm {
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mov ecx, dmod
mov edx, a1
mov ebx, dinvmod
mov eax, a0
fild w
fild DWORD PTR [edx]
fmul st, st(1)
fxch st(1)
fild DWORD PTR [eax]
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fmulp st(1), st
fld TBYTE PTR [ebx]
fld MPD_TWO63
fld st(2)
fmul st, st(2)
fadd st, st(1)
fsub st, st(1)
fmul QWORD PTR [ecx]
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fsubp st(3), st
fxch st(2)
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fistp DWORD PTR [eax]
fmul st, st(2)
fadd st, st(1)
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fsubrp st(1), st
fmul QWORD PTR [ecx]
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fsubp st(1), st
fistp DWORD PTR [edx]
}
}
/*
* Two modular multiplications in parallel:
* *a0 = (*a0 * b0) % dmod
* *a1 = (*a1 * b1) % dmod
*/
static inline void __cdecl
ppro_mulmod2(mpd_uint_t *a0, mpd_uint_t b0, mpd_uint_t *a1, mpd_uint_t b1,
double *dmod, uint32_t *dinvmod)
{
__asm {
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mov ecx, dmod
mov edx, a1
mov ebx, dinvmod
mov eax, a0
fild b1
fild DWORD PTR [edx]
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fmulp st(1), st
fild b0
fild DWORD PTR [eax]
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fmulp st(1), st
fld TBYTE PTR [ebx]
fld st(2)
fmul st, st(1)
fxch st(1)
fmul st, st(2)
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fld DWORD PTR MPD_TWO63
fld QWORD PTR [ecx]
fxch st(3)
fadd st, st(1)
fxch st(2)
fadd st, st(1)
fxch st(2)
fsub st, st(1)
fxch st(2)
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fsubrp st(1), st
fxch st(1)
fmul st, st(2)
fxch st(1)
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fmulp st(2), st
fsubp st(3), st
fsubp st(1), st
fxch st(1)
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fistp DWORD PTR [edx]
fistp DWORD PTR [eax]
}
}
#endif /* PPRO MASM (_MSC_VER) */
/* Return (base ** exp) % dmod */
static inline mpd_uint_t
ppro_powmod(mpd_uint_t base, mpd_uint_t exp, double *dmod, uint32_t *dinvmod)
{
mpd_uint_t r = 1;
while (exp > 0) {
if (exp & 1)
r = ppro_mulmod(r, base, dmod, dinvmod);
base = ppro_mulmod(base, base, dmod, dinvmod);
exp >>= 1;
}
return r;
}
#endif /* PPRO */
#endif /* CONFIG_32 */
#endif /* UMODARITH_H */