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// Copyright 2021 the V8 project authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
// Toom-Cook multiplication.
// Reference: https://en.wikipedia.org/wiki/Toom%E2%80%93Cook_multiplication
#include <algorithm>
#include "src/bigint/bigint-internal.h"
#include "src/bigint/digit-arithmetic.h"
#include "src/bigint/vector-arithmetic.h"
namespace v8 {
namespace bigint {
namespace {
void TimesTwo(RWDigits X) {
digit_t carry = 0;
for (int i = 0; i < X.len(); i++) {
digit_t d = X[i];
X[i] = (d << 1) | carry;
carry = d >> (kDigitBits - 1);
}
}
void DivideByTwo(RWDigits X) {
digit_t carry = 0;
for (int i = X.len() - 1; i >= 0; i--) {
digit_t d = X[i];
X[i] = (d >> 1) | carry;
carry = d << (kDigitBits - 1);
}
}
void DivideByThree(RWDigits X) {
digit_t remainder = 0;
for (int i = X.len() - 1; i >= 0; i--) {
digit_t d = X[i];
digit_t upper = (remainder << kHalfDigitBits) | (d >> kHalfDigitBits);
digit_t u_result = upper / 3;
remainder = upper - 3 * u_result;
digit_t lower = (remainder << kHalfDigitBits) | (d & kHalfDigitMask);
digit_t l_result = lower / 3;
remainder = lower - 3 * l_result;
X[i] = (u_result << kHalfDigitBits) | l_result;
}
}
} // namespace
#if DEBUG
// Set {len_} to 1 rather than 0 so that attempts to access the first digit
// will crash.
#define MARK_INVALID(D) D = RWDigits(nullptr, 1)
#else
#define MARK_INVALID(D) (void(0))
#endif
void ProcessorImpl::Toom3Main(RWDigits Z, Digits X, Digits Y) {
DCHECK(Z.len() >= X.len() + Y.len());
// Phase 1: Splitting.
int i = DIV_CEIL(std::max(X.len(), Y.len()), 3);
Digits X0(X, 0, i);
Digits X1(X, i, i);
Digits X2(X, 2 * i, i);
Digits Y0(Y, 0, i);
Digits Y1(Y, i, i);
Digits Y2(Y, 2 * i, i);
// Temporary storage.
int p_len = i + 1; // For all px, qx below.
int r_len = 2 * p_len; // For all r_x, Rx below.
Storage temp_storage(4 * r_len);
// We will use the same variable names as the Wikipedia article, as much as
// C++ lets us: our "p_m1" is their "p(-1)" etc. For consistency with other
// algorithms, we use X and Y where Wikipedia uses m and n.
// We will use and re-use the temporary storage as follows:
//
// chunk | -------- time ----------->
// [0 .. i] |( po )( p_m1 ) ( r_m2 )
// [i+1 .. rlen-1] |( qo )( q_m1 ) ( r_m2 )
// [rlen .. rlen+i] | (p_1 ) ( p_m2 ) (r_inf)
// [rlen+i+1 .. 2*rlen-1] | (q_1 ) ( q_m2 ) (r_inf)
// [2*rlen .. 3*rlen-1] | ( r_1 )
// [3*rlen .. 4*rlen-1] | ( r_m1 )
//
// This requires interleaving phases 2 and 3 a bit: after computing
// r_1 = p_1 * q_1, we can re-use p_1's storage for p_m2, and so on.
digit_t* t = temp_storage.get();
RWDigits po(t, p_len);
RWDigits qo(t + p_len, p_len);
RWDigits p_1(t + r_len, p_len);
RWDigits q_1(t + r_len + p_len, p_len);
RWDigits r_1(t + 2 * r_len, r_len);
RWDigits r_m1(t + 3 * r_len, r_len);
// We can also share the backing stores of Z, r_0, R0.
DCHECK(Z.len() >= r_len);
RWDigits r_0(Z, 0, r_len);
// Phase 2a: Evaluation, steps 0, 1, m1.
// po = X0 + X2
Add(po, X0, X2);
// p_0 = X0
// p_1 = po + X1
Add(p_1, po, X1);
// p_m1 = po - X1
RWDigits p_m1 = po;
bool p_m1_sign = SubtractSigned(p_m1, po, false, X1, false);
MARK_INVALID(po);
// qo = Y0 + Y2
Add(qo, Y0, Y2);
// q_0 = Y0
// q_1 = qo + Y1
Add(q_1, qo, Y1);
// q_m1 = qo - Y1
RWDigits q_m1 = qo;
bool q_m1_sign = SubtractSigned(q_m1, qo, false, Y1, false);
MARK_INVALID(qo);
// Phase 3a: Pointwise multiplication, steps 0, 1, m1.
Multiply(r_0, X0, Y0);
Multiply(r_1, p_1, q_1);
Multiply(r_m1, p_m1, q_m1);
bool r_m1_sign = p_m1_sign != q_m1_sign;
// Phase 2b: Evaluation, steps m2 and inf.
// p_m2 = (p_m1 + X2) * 2 - X0
RWDigits p_m2 = p_1;
MARK_INVALID(p_1);
bool p_m2_sign = AddSigned(p_m2, p_m1, p_m1_sign, X2, false);
TimesTwo(p_m2);
p_m2_sign = SubtractSigned(p_m2, p_m2, p_m2_sign, X0, false);
// p_inf = X2
// q_m2 = (q_m1 + Y2) * 2 - Y0
RWDigits q_m2 = q_1;
MARK_INVALID(q_1);
bool q_m2_sign = AddSigned(q_m2, q_m1, q_m1_sign, Y2, false);
TimesTwo(q_m2);
q_m2_sign = SubtractSigned(q_m2, q_m2, q_m2_sign, Y0, false);
// q_inf = Y2
// Phase 3b: Pointwise multiplication, steps m2 and inf.
RWDigits r_m2(t, r_len);
MARK_INVALID(p_m1);
MARK_INVALID(q_m1);
Multiply(r_m2, p_m2, q_m2);
bool r_m2_sign = p_m2_sign != q_m2_sign;
RWDigits r_inf(t + r_len, r_len);
MARK_INVALID(p_m2);
MARK_INVALID(q_m2);
Multiply(r_inf, X2, Y2);
// Phase 4: Interpolation.
Digits R0 = r_0;
Digits R4 = r_inf;
// R3 <- (r_m2 - r_1) / 3
RWDigits R3 = r_m2;
bool R3_sign = SubtractSigned(R3, r_m2, r_m2_sign, r_1, false);
DivideByThree(R3);
// R1 <- (r_1 - r_m1) / 2
RWDigits R1 = r_1;
bool R1_sign = SubtractSigned(R1, r_1, false, r_m1, r_m1_sign);
DivideByTwo(R1);
// R2 <- r_m1 - r_0
RWDigits R2 = r_m1;
bool R2_sign = SubtractSigned(R2, r_m1, r_m1_sign, R0, false);
// R3 <- (R2 - R3) / 2 + 2 * r_inf
R3_sign = SubtractSigned(R3, R2, R2_sign, R3, R3_sign);
DivideByTwo(R3);
// TODO(jkummerow): Would it be a measurable improvement to write an
// "AddTwice" helper?
R3_sign = AddSigned(R3, R3, R3_sign, r_inf, false);
R3_sign = AddSigned(R3, R3, R3_sign, r_inf, false);
// R2 <- R2 + R1 - R4
R2_sign = AddSigned(R2, R2, R2_sign, R1, R1_sign);
R2_sign = SubtractSigned(R2, R2, R2_sign, R4, false);
// R1 <- R1 - R3
R1_sign = SubtractSigned(R1, R1, R1_sign, R3, R3_sign);
#if DEBUG
R1.Normalize();
R2.Normalize();
R3.Normalize();
DCHECK(R1_sign == false || R1.len() == 0);
DCHECK(R2_sign == false || R2.len() == 0);
DCHECK(R3_sign == false || R3.len() == 0);
#endif
// Phase 5: Recomposition. R0 is already in place. Overflow can't happen.
for (int j = R0.len(); j < Z.len(); j++) Z[j] = 0;
AddAndReturnOverflow(Z + i, R1);
AddAndReturnOverflow(Z + 2 * i, R2);
AddAndReturnOverflow(Z + 3 * i, R3);
AddAndReturnOverflow(Z + 4 * i, R4);
}
void ProcessorImpl::MultiplyToomCook(RWDigits Z, Digits X, Digits Y) {
DCHECK(X.len() >= Y.len());
int k = Y.len();
// TODO(jkummerow): Would it be a measurable improvement to share the
// scratch memory for several invocations?
Digits X0(X, 0, k);
Toom3Main(Z, X0, Y);
if (X.len() > Y.len()) {
ScratchDigits T(2 * k);
for (int i = k; i < X.len(); i += k) {
Digits Xi(X, i, k);
// TODO(jkummerow): would it be a measurable improvement to craft a
// "ToomChunk" method in the style of {KaratsubaChunk}?
Toom3Main(T, Xi, Y);
AddAndReturnOverflow(Z + i, T); // Can't overflow.
}
}
}
} // namespace bigint
} // namespace v8
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