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div: Use div 3by2 in Knuth's division #134

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Mar 10, 2020
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div: Use div 3by2 in Knuth's division #134

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div: Use div 3by2 in Knuth's division
chfast Mar 10, 2020
e62e027
div: Reuse remainder for 2 submul words
chfast Mar 10, 2020
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43 changes: 13 additions & 30 deletions lib/intx/div.cpp
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Original file line number Diff line number Diff line change
Expand Up @@ -93,7 +93,7 @@ inline uint64_t submul(
uint64_t r[], const uint64_t x[], const uint64_t y[], int len, uint64_t multiplier) noexcept
{
// OPT: Add MinLen template parameter and unroll first loop iterations.
REQUIRE(len >= 3);
REQUIRE(len >= 1);

uint64_t borrow = 0;
for (int i = 0; i < len; ++i)
Expand All @@ -113,54 +113,37 @@ void udivrem_knuth(uint64_t q[], uint64_t u[], int ulen, const uint64_t d[], int
REQUIRE(ulen >= dlen);

const auto divisor = uint128{d[dlen - 1], d[dlen - 2]};
const auto reciprocal = reciprocal_2by1(divisor.hi);
const auto reciprocal = reciprocal_3by2(divisor);
for (int j = ulen - dlen - 1; j >= 0; --j)
{
const auto u2 = u[j + dlen];
const auto u1 = u[j + dlen - 1];
const auto u0 = u[j + dlen - 2];

uint64_t qhat;
uint128 rhat;
const auto numerator = uint128{u2, u1};
if (UNLIKELY(numerator.hi >= divisor.hi)) // Division overflows.
if (uint128{u2, u1} == divisor) // Division overflows.
{
qhat = ~uint64_t{0};
rhat = numerator - uint128{divisor.hi, 0};
rhat += divisor.hi;

// Adjustment (not needed for correctness, but helps avoiding "add back" case).
if (rhat.hi == 0 && umul(qhat, divisor.lo) > uint128{rhat.lo, u0})
--qhat;
u[j + dlen] = u2 - submul(&u[j], &u[j], d, dlen, qhat);
}
else
{
const auto res = udivrem_2by1(numerator, divisor.hi, reciprocal);
qhat = res.quot;
rhat = res.rem;
uint128 rhat;
std::tie(qhat, rhat) = udivrem_3by2(u2, u1, u0, divisor, reciprocal);

bool carry;
const auto overflow = submul(&u[j], &u[j], d, dlen - 2, qhat);
std::tie(u[j + dlen - 2], carry) = sub_with_carry(rhat.lo, overflow);
std::tie(u[j + dlen - 1], carry) = sub_with_carry(rhat.hi, carry);

const auto p = umul(qhat, divisor.lo);
if (p > uint128{rhat.lo, u0})
if (carry)
{
--qhat;
rhat += divisor.hi;

// Adjustment (not needed for correctness, but helps avoiding "add back" case).
if (rhat.hi == 0 && (p - divisor.lo) > uint128{rhat.lo, u0})
--qhat;
u[j + dlen - 1] += divisor.hi + add(&u[j], &u[j], d, dlen - 1);
}
}

// Multiply and subtract.
bool carry;
const auto overflow = submul(&u[j], &u[j], d, dlen, qhat);
std::tie(u[j + dlen], carry) = sub_with_carry(u2, overflow);
if (carry) // Too much subtracted, add back.
{
--qhat;
u[j + dlen - 1] += divisor.hi + add(&u[j], &u[j], d, dlen - 1);
}

q[j] = qhat; // Store quotient digit.
}
}
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