AK: Implement Knuth's algorithm D for dividing UFixedBigInt's

Author: https://github.com/DanShaders
Commit: https://github.com/SerenityOS/serenity/commit/2d27c98659
Pull-request: https://github.com/SerenityOS/serenity/pull/17330
Reviewed-by: https://github.com/ADKaster
Reviewed-by: https://github.com/Hendiadyoin1

AK/FloatingPointStringConversions.cpp

@@ -10,6 +10,7 @@
 #include <AK/ScopeGuard.h>
 #include <AK/StringView.h>
 #include <AK/UFixedBigInt.h>
+#include <AK/UFixedBigIntDivision.h>
 
 namespace AK {
 

AK/UFixedBigInt.h

@@ -60,6 +60,13 @@ constexpr void mul_internal(Operand1 const& operand1, Operand2 const& operand2,
     StorageOperations::baseline_mul(operand1, operand2, result, g_null_allocator);
 }
 
+template<size_t dividend_size, size_t divisor_size, bool restore_remainder>
+constexpr void div_mod_internal( // Include AK/UFixedBigIntDivision.h to use UFixedBigInt division
+    StaticStorage<false, dividend_size> const& dividend,
+    StaticStorage<false, divisor_size> const& divisor,
+    StaticStorage<false, dividend_size>& quotient,
+    StaticStorage<false, divisor_size>& remainder);
+
 template<size_t bit_size, typename Storage>
 class UFixedBigInt {
     constexpr static size_t static_size = Storage::static_size;
@@ -395,84 +402,49 @@ public:
         return result;
     }
 
-    // FIXME: Refactor out this
-    using R = UFixedBigInt<bit_size>;
-
-    static constexpr size_t my_size()
+    template<NotBuiltInUFixedInt T>
+    constexpr UFixedBigInt<bit_size> div_mod(T const& divisor, T& remainder) const
     {
-        return sizeof(Storage);
-    }
-
-    // FIXME: Do something smarter (process at least one word per iteration).
-    // FIXME: no restraints on this
-    template<Unsigned U>
-    requires(sizeof(Storage) >= sizeof(U)) constexpr R div_mod(U const& divisor, U& remainder) const
-    {
-        // FIXME: Is there a better way to raise a division by 0?
-        //        Maybe as a compiletime warning?
-#pragma GCC diagnostic push
-#pragma GCC diagnostic ignored "-Wdiv-by-zero"
-        if (!divisor) {
-            int volatile x = 1;
-            int volatile y = 0;
-            [[maybe_unused]] int volatile z = x / y;
-        }
-#pragma GCC diagnostic pop
-
-        // fastpaths
-        if (*this < divisor) {
-            remainder = static_cast<U>(*this);
-            return 0u;
-        }
-        if (*this == divisor) {
-            remainder = 0u;
-            return 1u;
-        }
-        if (divisor == 1u) {
-            remainder = 0u;
-            return *this;
-        }
-
-        remainder = 0u;
-        R quotient = 0u;
-
-        for (ssize_t i = sizeof(R) * 8 - clz() - 1; i >= 0; --i) {
-            remainder <<= 1u;
-            remainder |= static_cast<unsigned>(*this >> (size_t)i) & 1u;
-            if (remainder >= divisor) {
-                remainder -= divisor;
-                quotient |= R { 1u } << (size_t)i;
-            }
-        }
-
+        UFixedBigInt<bit_size> quotient;
+        UFixedBigInt<assumed_bit_size<T>> resulting_remainder;
+        div_mod_internal<bit_size, assumed_bit_size<T>, true>(m_data, get_storage_of(divisor), get_storage_of(quotient), get_storage_of(resulting_remainder));
+        remainder = resulting_remainder;
         return quotient;
     }
 
-    template<Unsigned U>
-    constexpr R operator/(U const& other) const
+    template<UFixedInt T>
+    constexpr auto operator/(T const& other) const
     {
-        U mod { 0u }; // unused
-        return div_mod(other, mod);
-    }
-    template<Unsigned U>
-    constexpr U operator%(U const& other) const
-    {
-        R res { 0u };
-        div_mod(other, res);
-        return res;
+        UFixedBigInt<bit_size> quotient;
+        StaticStorage<false, assumed_bit_size<T>> remainder; // unused
+        div_mod_internal<bit_size, assumed_bit_size<T>, false>(m_data, get_storage_of(other), get_storage_of(quotient), remainder);
+        return quotient;
     }
 
-    template<Unsigned U>
-    constexpr R& operator/=(U const& other)
+    template<UFixedInt T>
+    constexpr auto operator%(T const& other) const
     {
-        *this = *this / other;
-        return *this;
+        StaticStorage<false, bit_size> quotient; // unused
+        UFixedBigInt<assumed_bit_size<T>> remainder;
+        div_mod_internal<bit_size, assumed_bit_size<T>, true>(m_data, get_storage_of(other), quotient, get_storage_of(remainder));
+        return remainder;
     }
+
+    constexpr auto operator/(IntegerWrapper const& other) const { return *this / static_cast<UFixedBigInt<32>>(other); }
+    constexpr auto operator%(IntegerWrapper const& other) const { return *this % static_cast<UFixedBigInt<32>>(other); }
+
+    template<UFixedInt T>
+    constexpr auto& operator/=(T const& other) { return *this = *this / other; }
+    constexpr auto& operator/=(IntegerWrapper const& other) { return *this = *this / other; }
+
     template<Unsigned U>
-    constexpr R& operator%=(U const& other)
+    constexpr auto& operator%=(U const& other) { return *this = *this % other; }
+    constexpr auto& operator%=(IntegerWrapper const& other) { return *this = *this % other; }
+
+    // FIXME: Replace uses with more general `assumed_bit_size<T>`.
+    static constexpr size_t my_size()
     {
-        *this = *this % other;
-        return *this;
+        return sizeof(Storage);
     }
 
     // Note: If there ever be need for non side-channel proof sqrt/pow/pow_mod of UFixedBigInt, you

AK/UFixedBigIntDivision.h

@@ -0,0 +1,136 @@
+/*
+ * Copyright (c) 2023, Dan Klishch <danilklishch@gmail.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#pragma once
+
+#include <AK/Diagnostics.h>
+#include <AK/UFixedBigInt.h>
+
+namespace AK {
+namespace Detail {
+
+template<size_t dividend_bit_size, size_t divisor_bit_size, bool restore_remainder>
+constexpr void div_mod_internal(
+    StaticStorage<false, dividend_bit_size> const& operand1,
+    StaticStorage<false, divisor_bit_size> const& operand2,
+    StaticStorage<false, dividend_bit_size>& quotient,
+    StaticStorage<false, divisor_bit_size>& remainder)
+{
+    size_t dividend_len = operand1.size(), divisor_len = operand2.size();
+    while (divisor_len > 0 && !operand2[divisor_len - 1])
+        --divisor_len;
+    while (dividend_len > 0 && !operand1[dividend_len - 1])
+        --dividend_len;
+
+    // FIXME: Should raise SIGFPE instead
+    VERIFY(divisor_len); // VERIFY(divisor != 0)
+
+    // Fast paths
+    if (divisor_len == 1 && operand2[0] == 1) { // divisor == 1
+        quotient = operand1;
+        if constexpr (restore_remainder)
+            StorageOperations::set(0, remainder);
+        return;
+    }
+
+    if (dividend_len < divisor_len) { // dividend < divisor
+        StorageOperations::set(0, quotient);
+        if constexpr (restore_remainder)
+            remainder = operand1;
+        return;
+    }
+
+    if (divisor_len == 1 && dividend_len == 1) { // NativeWord / NativeWord
+        StorageOperations::set(operand1[0] / operand2[0], quotient);
+        if constexpr (restore_remainder)
+            StorageOperations::set(operand1[0] % operand2[0], remainder);
+        return;
+    }
+
+    if (divisor_len == 1) { // BigInt by NativeWord
+        auto u = (static_cast<DoubleWord>(operand1[dividend_len - 1]) << word_size) + operand1[dividend_len - 2];
+        auto divisor = operand2[0];
+
+        auto top = u / divisor;
+        quotient[dividend_len - 1] = static_cast<NativeWord>(top >> word_size);
+        quotient[dividend_len - 2] = static_cast<NativeWord>(top);
+
+        auto carry = static_cast<NativeWord>(u % divisor);
+        for (size_t i = dividend_len - 2; i--;)
+            quotient[i] = div_mod_words(operand1[i], carry, divisor, carry);
+        for (size_t i = dividend_len; i < quotient.size(); ++i)
+            quotient[i] = 0;
+        if constexpr (restore_remainder)
+            StorageOperations::set(carry, remainder);
+        return;
+    }
+
+    // Knuth's algorithm D
+    StaticStorage<false, dividend_bit_size + word_size> dividend;
+    StorageOperations::copy(operand1, dividend);
+    auto divisor = operand2;
+
+    // D1. Normalize
+    // FIXME: Investigate GCC producing bogus -Warray-bounds when dividing u128 by u32. This code
+    //        should not be reachable at all in this case because fast paths above cover all cases
+    //        when `operand2.size() == 1`.
+    AK_IGNORE_DIAGNOSTIC("-Warray-bounds", size_t shift = count_leading_zeroes(divisor[divisor_len - 1]);)
+    StorageOperations::shift_left(dividend, shift, dividend);
+    StorageOperations::shift_left(divisor, shift, divisor);
+
+    auto divisor_approx = divisor[divisor_len - 1];
+
+    for (size_t i = dividend_len + 1; i-- > divisor_len;) {
+        // D3. Calculate qhat
+        NativeWord qhat;
+        VERIFY(dividend[i] <= divisor_approx);
+        if (dividend[i] == divisor_approx) {
+            qhat = max_word;
+        } else {
+            NativeWord rhat;
+            qhat = div_mod_words(dividend[i - 1], dividend[i], divisor_approx, rhat);
+
+            auto is_qhat_too_large = [&] {
+                return UFixedBigInt<word_size> { qhat }.wide_multiply(divisor[divisor_len - 2]) > u128 { dividend[i - 2], rhat };
+            };
+            if (is_qhat_too_large()) {
+                --qhat;
+                bool carry = false;
+                rhat = add_words(rhat, divisor_approx, carry);
+                if (!carry && is_qhat_too_large())
+                    --qhat;
+            }
+        }
+
+        // D4. Multiply & subtract
+        NativeWord mul_carry = 0;
+        bool sub_carry = false;
+        for (size_t j = 0; j < divisor_len; ++j) {
+            auto mul_result = UFixedBigInt<word_size> { qhat }.wide_multiply(divisor[j]) + mul_carry;
+            auto& output = dividend[i + j - divisor_len];
+            output = sub_words(output, mul_result.low(), sub_carry);
+            mul_carry = mul_result.high();
+        }
+        dividend[i] = sub_words(dividend[i], mul_carry, sub_carry);
+
+        if (sub_carry) {
+            // D6. Add back
+            auto dividend_part = UnsignedStorageSpan { dividend.data() + i - divisor_len, divisor_len + 1 };
+            VERIFY(StorageOperations::add<false>(dividend_part, divisor, dividend_part));
+        }
+
+        quotient[i - divisor_len] = qhat - sub_carry;
+    }
+
+    for (size_t i = dividend_len - divisor_len + 1; i < quotient.size(); ++i)
+        quotient[i] = 0;
+
+    // D8. Unnormalize
+    if constexpr (restore_remainder)
+        StorageOperations::shift_right(UnsignedStorageSpan { dividend.data(), remainder.size() }, shift, remainder);
+}
+}
+}

Tests/AK/TestUFixedBigInt.cpp

@@ -9,6 +9,7 @@
 #include <AK/NumericLimits.h>
 #include <AK/Random.h>
 #include <AK/UFixedBigInt.h>
+#include <AK/UFixedBigIntDivision.h>
 
 constexpr int test_iterations = 32;
 
@@ -78,6 +79,62 @@ TEST_CASE(div_mod)
     }
 }
 
+TEST_CASE(div_anti_knuth)
+{
+    EXPECT_EQ((u256 { { 0ull, 0xffffffffffffffffull, 1ull, 0ull } } / u128 { 0x8000000000000001ull, 0xffffffffffffffffull }), 1u);
+    EXPECT_EQ((u128 { { 0xffffffff00000000ull, 1ull } } / u128 { 0xffffffff80000001ull }), 1u);
+
+    srand(0);
+
+    auto generate_u512 = [] {
+        using namespace AK::Detail;
+
+        u512 number;
+        auto& storage = get_storage_of(number);
+
+        static constexpr u32 interesting_words_count = 14;
+        static constexpr NativeWord interesting_words[interesting_words_count] = {
+            0,
+            0,
+            1,
+            2,
+            3,
+            max_word / 4 - 1,
+            max_word / 4,
+            max_word / 2 - 1,
+            max_word / 2,
+            max_word / 2 + 1,
+            max_word / 2 + 2,
+            max_word - 3,
+            max_word - 2,
+            max_word - 1,
+        };
+        for (size_t i = 0; i < storage.size(); ++i) {
+            u32 type = get_random_uniform(interesting_words_count + 1);
+            NativeWord& next_word = storage[i];
+            if (type == interesting_words_count)
+                next_word = get_random<NativeWord>();
+            else
+                next_word = interesting_words[type];
+        }
+
+        return number;
+    };
+
+    for (int i = 0; i < 16384; ++i) {
+        u512 a = generate_u512(), b = generate_u512();
+        if (b == 0)
+            continue;
+
+        u512 mod;
+        u512 div = a.div_mod(b, mod);
+
+        EXPECT_EQ(div * b + mod, a);
+        EXPECT_EQ(div.wide_multiply(b) + mod, a);
+        EXPECT(0 <= mod && mod < b);
+    }
+}
+
 TEST_CASE(shifts)
 {
     u128 val { 0x1234ULL };