Merge pull request #406 from cppalliance/sqrt

Author: mborland

.drone.jsonnet

@@ -1,8 +1,9 @@
 # Copyright 2022, 2023 Peter Dimov
+# Copyright 2025 - 2026 Matt Borland
 # Distributed under the Boost Software License, Version 1.0.
 # https://www.boost.org/LICENSE_1_0.txt
 
-local library = "decimal";
+local library = "int128";
 
 local triggers =
 {

include/boost/int128/utilities.hpp

@@ -218,6 +218,41 @@ BOOST_INT128_EXPORT BOOST_INT128_HOST_DEVICE constexpr int128_t ipow(int128_t ba
     return result;
 }
 
+// Integer square root: returns floor(sqrt(n)).
+BOOST_INT128_EXPORT BOOST_INT128_HOST_DEVICE constexpr uint128_t isqrt(const uint128_t n) noexcept
+{
+    if (n < 2U)
+    {
+        return n;
+    }
+
+    // 2^ceil(bit_width(n)/2) is the smallest power of two whose square exceeds n.
+    uint128_t x {uint128_t{1} << ((bit_width(n) + 1) / 2)};
+
+    while (true)
+    {
+        const uint128_t y {(x + n / x) >> 1};
+
+        if (y >= x)
+        {
+            return x;
+        }
+
+        x = y;
+    }
+}
+
+// Signed overload. Negative inputs are documented to return 0.
+BOOST_INT128_EXPORT BOOST_INT128_HOST_DEVICE constexpr int128_t isqrt(const int128_t n) noexcept
+{
+    if (BOOST_INT128_UNLIKELY(n < 0))
+    {
+        return int128_t{0};
+    }
+
+    return static_cast<int128_t>(isqrt(static_cast<uint128_t>(n)));
+}
+
 } // namespace int128
 } // namespace boost
 

test/Jamfile

@@ -81,6 +81,7 @@ run test_gcd_lcm.cpp ;
 run test_midpoint.cpp ;
 run test_powm.cpp ;
 run test_ipow.cpp ;
+run test_isqrt.cpp ;
 
 run test_format.cpp ;
 run test_fmt_format.cpp ;

test/test_isqrt.cpp

@@ -0,0 +1,278 @@
+// Copyright 2026 Matt Borland
+// Distributed under the Boost Software License, Version 1.0.
+// https://www.boost.org/LICENSE_1_0.txt
+
+#if defined(__GNUC__) && __GNUC__ == 7 && defined(__i386__)
+
+// 32-bit GCC-7 fails with: "error: constexpr loop iteration count exceeds limit of 262144"
+
+int main() { return 0; }
+
+#else
+
+#ifndef BOOST_INT128_BUILD_MODULE
+
+#include <boost/int128.hpp>
+
+#else
+
+import boost.int128;
+
+#endif
+
+#include <boost/core/lightweight_test.hpp>
+#include <cstdint>
+#include <limits>
+
+using namespace boost::int128;
+
+namespace {
+
+// Naive bit-by-bit reference. Independent of the Newton implementation under
+// test, so it serves as a ground truth for cross-checking.
+constexpr uint128_t isqrt_ref(uint128_t n) noexcept
+{
+    if (n < 2U)
+    {
+        return n;
+    }
+
+    uint128_t res {0};
+    uint128_t bit {uint128_t{1} << 126};
+
+    while (bit > n)
+    {
+        bit >>= 2;
+    }
+
+    while (bit != 0U)
+    {
+        if (n >= res + bit)
+        {
+            n -= res + bit;
+            res = (res >> 1) + bit;
+        }
+        else
+        {
+            res >>= 1;
+        }
+
+        bit >>= 2;
+    }
+
+    return res;
+}
+
+// Verify the defining invariant: r == isqrt(n) iff r*r <= n < (r+1)^2.
+// Computed with overflow-safe comparisons so callers can pass values near the
+// uint128 upper bound.
+void check_invariant(const uint128_t n)
+{
+    const uint128_t r {isqrt(n)};
+
+    BOOST_TEST(r * r <= n);
+
+    const uint128_t r_plus_1 {r + 1U};
+
+    // (r+1)^2 may overflow uint128 when r is close to 2^64; in that case the
+    // invariant n < (r+1)^2 is trivially satisfied since n fits in 128 bits.
+    if (r_plus_1 != 0U && r_plus_1 <= ((std::numeric_limits<uint128_t>::max)() / r_plus_1))
+    {
+        BOOST_TEST(n < r_plus_1 * r_plus_1);
+    }
+}
+
+} // namespace
+
+void test_uint128_isqrt_small()
+{
+    BOOST_TEST_EQ(isqrt(uint128_t{0}), uint128_t{0});
+    BOOST_TEST_EQ(isqrt(uint128_t{1}), uint128_t{1});
+    BOOST_TEST_EQ(isqrt(uint128_t{2}), uint128_t{1});
+    BOOST_TEST_EQ(isqrt(uint128_t{3}), uint128_t{1});
+    BOOST_TEST_EQ(isqrt(uint128_t{4}), uint128_t{2});
+    BOOST_TEST_EQ(isqrt(uint128_t{5}), uint128_t{2});
+    BOOST_TEST_EQ(isqrt(uint128_t{8}), uint128_t{2});
+    BOOST_TEST_EQ(isqrt(uint128_t{9}), uint128_t{3});
+    BOOST_TEST_EQ(isqrt(uint128_t{10}), uint128_t{3});
+    BOOST_TEST_EQ(isqrt(uint128_t{15}), uint128_t{3});
+    BOOST_TEST_EQ(isqrt(uint128_t{16}), uint128_t{4});
+    BOOST_TEST_EQ(isqrt(uint128_t{99}), uint128_t{9});
+    BOOST_TEST_EQ(isqrt(uint128_t{100}), uint128_t{10});
+    BOOST_TEST_EQ(isqrt(uint128_t{101}), uint128_t{10});
+
+    // Exhaustive cross-check against the reference for every small n.
+    for (std::uint64_t i {0}; i < 200; ++i)
+    {
+        BOOST_TEST_EQ(isqrt(uint128_t{i}), isqrt_ref(uint128_t{i}));
+    }
+}
+
+void test_uint128_isqrt_perfect_squares()
+{
+    // Squares spanning the full 64-bit range, including the largest k whose
+    // square still fits in 128 bits (k = 2^64 - 1).
+    for (std::uint64_t k {0}; k < 10000; ++k)
+    {
+        const uint128_t kk {k};
+        BOOST_TEST_EQ(isqrt(kk * kk), kk);
+
+        if (k > 0)
+        {
+            BOOST_TEST_EQ(isqrt(kk * kk - 1U), kk - 1U);
+            BOOST_TEST_EQ(isqrt(kk * kk + 2U * kk), kk);  // (k+1)^2 - 1
+        }
+    }
+
+    // Powers of two as bases: k = 2^i for i in [0, 63] - largest exact square
+    // is (2^63)^2 = 2^126.
+    for (int i {0}; i < 64; ++i)
+    {
+        const uint128_t k {uint128_t{1} << i};
+        BOOST_TEST_EQ(isqrt(k * k), k);
+    }
+
+    // Largest representable perfect square: (2^64 - 1)^2 = 2^128 - 2^65 + 1.
+    const uint128_t k_max {(std::numeric_limits<std::uint64_t>::max)()};
+    BOOST_TEST_EQ(isqrt(k_max * k_max), k_max);
+}
+
+void test_uint128_isqrt_bit_boundaries()
+{
+    // 2^(2k) has integer square root 2^k.
+    for (int k {0}; k < 64; ++k)
+    {
+        const uint128_t n {uint128_t{1} << (2 * k)};
+        BOOST_TEST_EQ(isqrt(n), uint128_t{1} << k);
+    }
+
+    // 2^(2k+1) has integer square root floor(2^(k+0.5)) = floor(sqrt(2) * 2^k).
+    // Check the invariant holds rather than hard-coding the value.
+    for (int k {0}; k < 63; ++k)
+    {
+        check_invariant(uint128_t{1} << (2 * k + 1));
+    }
+
+    // Just below and just above bit boundaries.
+    for (int k {2}; k < 128; ++k)
+    {
+        const uint128_t boundary {uint128_t{1} << k};
+        check_invariant(boundary - 1U);
+        check_invariant(boundary);
+        check_invariant(boundary + 1U);
+    }
+}
+
+void test_uint128_isqrt_extreme()
+{
+    // (uint128 max). (2^64)^2 = 2^128 wraps, so isqrt(2^128 - 1) = 2^64 - 1.
+    const uint128_t u128_max {(std::numeric_limits<uint128_t>::max)()};
+    const uint128_t u64_max {(std::numeric_limits<std::uint64_t>::max)()};
+    BOOST_TEST_EQ(isqrt(u128_max), u64_max);
+
+    // 2^128 - 2^65 + 1 = (2^64 - 1)^2 - exact square at the very top.
+    BOOST_TEST_EQ(isqrt(u64_max * u64_max), u64_max);
+
+    // One above the largest representable square: still has isqrt = 2^64 - 1.
+    BOOST_TEST_EQ(isqrt(u64_max * u64_max + 1U), u64_max);
+
+    // A handful of large hand-picked values, cross-checked against the bit-by-
+    // bit reference.
+    const uint128_t big_a {UINT64_C(0x0123456789ABCDEF), UINT64_C(0xFEDCBA9876543210)};
+    const uint128_t big_b {UINT64_C(0xDEADBEEFCAFEBABE), UINT64_C(0x0123456789ABCDEF)};
+    const uint128_t big_c {UINT64_C(0x8000000000000000), 0U};
+
+    BOOST_TEST_EQ(isqrt(big_a), isqrt_ref(big_a));
+    BOOST_TEST_EQ(isqrt(big_b), isqrt_ref(big_b));
+    BOOST_TEST_EQ(isqrt(big_c), isqrt_ref(big_c));
+
+    check_invariant(big_a);
+    check_invariant(big_b);
+    check_invariant(big_c);
+    check_invariant(u128_max);
+}
+
+void test_int128_isqrt()
+{
+    BOOST_TEST_EQ(isqrt(int128_t{0}), int128_t{0});
+    BOOST_TEST_EQ(isqrt(int128_t{1}), int128_t{1});
+    BOOST_TEST_EQ(isqrt(int128_t{2}), int128_t{1});
+    BOOST_TEST_EQ(isqrt(int128_t{100}), int128_t{10});
+    BOOST_TEST_EQ(isqrt(int128_t{144}), int128_t{12});
+    BOOST_TEST_EQ(isqrt(int128_t{INT64_C(1000000000000000000)}), int128_t{INT64_C(1000000000)});
+
+    // int128 max = 2^127 - 1. floor(sqrt) = floor(2^63.5) = 6074001000.7e9
+    // Use the unsigned implementation as the source of truth.
+    const int128_t i128_max {(std::numeric_limits<int128_t>::max)()};
+    BOOST_TEST_EQ(isqrt(i128_max), static_cast<int128_t>(isqrt(static_cast<uint128_t>(i128_max))));
+
+    // Negative inputs are documented to return 0.
+    BOOST_TEST_EQ(isqrt(int128_t{-1}), int128_t{0});
+    BOOST_TEST_EQ(isqrt(int128_t{-100}), int128_t{0});
+    BOOST_TEST_EQ(isqrt((std::numeric_limits<int128_t>::min)()), int128_t{0});
+}
+
+void test_isqrt_against_ipow()
+{
+    // isqrt(ipow(k, 2)) == k for any k whose square fits.
+    for (std::uint64_t k {0}; k < 1000; ++k)
+    {
+        BOOST_TEST_EQ(isqrt(ipow(uint128_t{k}, 2U)), uint128_t{k});
+    }
+
+    // isqrt is monotonically non-decreasing.
+    uint128_t prev {0};
+
+    for (std::uint64_t i {0}; i < 1000; ++i)
+    {
+        const uint128_t curr {isqrt(uint128_t{i})};
+        BOOST_TEST(curr >= prev);
+        prev = curr;
+    }
+}
+
+#ifdef _MSC_VER
+#  pragma warning(push)
+#  pragma warning(disable : 4307) // integral constant overflow
+#  pragma warning(disable : 4308) // negative integral constant converted to unsigned type
+#endif
+
+void test_constexpr_isqrt()
+{
+    constexpr uint128_t r1 {isqrt(uint128_t{0})};
+    static_assert(r1 == uint128_t{0}, "isqrt(0) constexpr");
+
+    constexpr uint128_t r2 {isqrt(uint128_t{1})};
+    static_assert(r2 == uint128_t{1}, "isqrt(1) constexpr");
+
+    constexpr uint128_t r3 {isqrt(uint128_t{100})};
+    static_assert(r3 == uint128_t{10}, "isqrt(100) constexpr");
+
+    constexpr uint128_t r4 {isqrt(uint128_t{UINT64_C(1000000000000000000)})};
+    static_assert(r4 == uint128_t{UINT64_C(1000000000)}, "isqrt(10^18) constexpr");
+
+    constexpr int128_t r5 {isqrt(int128_t{-5})};
+    static_assert(r5 == int128_t{0}, "isqrt negative constexpr");
+
+    constexpr int128_t r6 {isqrt(int128_t{12321})};
+    static_assert(r6 == int128_t{111}, "isqrt(12321) constexpr");
+}
+
+#ifdef _MSC_VER
+#  pragma warning(pop)
+#endif
+
+int main()
+{
+    test_uint128_isqrt_small();
+    test_uint128_isqrt_perfect_squares();
+    test_uint128_isqrt_bit_boundaries();
+    test_uint128_isqrt_extreme();
+    test_int128_isqrt();
+    test_isqrt_against_ipow();
+    test_constexpr_isqrt();
+
+    return boost::report_errors();
+}
+
+#endif