Just.PreciseMath/1-tests/Just.PreciseMath.Tests/PreciseMathSqrtTests.cs

using System.Numerics;
using Shouldly;
using Xunit;

namespace Just.PreciseMath.Tests;

public class PreciseMathSqrtTests
{
    [Theory]
    [InlineData(1.0, 1.0)]
    [InlineData(4.0, 2.0)]
    [InlineData(9.0, 3.0)]
    [InlineData(0.25, 0.5)]
    [InlineData(2.25, 1.5)]
    public void ExactBinarySquaresHaveExactRoots(double square, double root)
    {
        DoubleDouble actual = DDMath.Sqrt(new DoubleDouble(square));
        actual.High.ShouldBe(root);
        BitConverter.DoubleToInt64Bits(actual.Low).ShouldBe(0L);
    }

    [Fact]
    public void PowersOfFourHaveExactRootsAcrossTheFiniteRange()
    {
        // sqrt(2^(2k)) = 2^k exactly, including the minimum subnormal input.
        for (int exponent = -1074; exponent <= 1022; exponent += 2)
        {
            DoubleDouble actual = DDMath.Sqrt(new DoubleDouble(Math.ScaleB(1.0, exponent)));
            actual.High.ShouldBe(Math.ScaleB(1.0, exponent / 2));
            BitConverter.DoubleToInt64Bits(actual.Low).ShouldBe(0L);
        }
    }

    [Theory]
    [InlineData(2.0)]
    [InlineData(3.0)]
    [InlineData(5.0)]
    [InlineData(1e-308)]
    [InlineData(1e308)]
    public void IrrationalRootsRetainMoreThanBinary64Precision(double value)
    {
        DoubleDouble input = new(value);
        DoubleDouble actual = DDMath.Sqrt(input);
        actual.Low.ShouldNotBe(0.0);
        AssertSqrtBound(input, actual);
    }

    [Fact]
    public void LowComponentsOnEitherSideOfAnExactSquareAffectTheRoot()
    {
        foreach (double low in new[] { Math.ScaleB(1.0, -54), -Math.ScaleB(1.0, -54) })
        {
            DoubleDouble input = DoubleDouble.FromComponents(1.0, low);
            DoubleDouble actual = DDMath.Sqrt(input);
            actual.High.ShouldBe(1.0);
            (Math.Sign(actual.Low) == Math.Sign(low)).ShouldBeTrue();
            AssertSqrtBound(input, actual);
        }
    }

    [Fact]
    public void SpecialValuesMatchBinary64AndRemainCanonical()
    {
        double[] values = [0.0, -0.0, double.PositiveInfinity, double.NegativeInfinity,
            double.NaN, -1.0, -double.Epsilon, double.MinValue];
        foreach (double value in values)
        {
            DoubleDouble actual = DDMath.Sqrt(new DoubleDouble(value));
            double expected = Math.Sqrt(value);
            // Construction canonicalizes all NaNs to double.NaN, including
            // domain errors; zeros must instead be compared by their sign bits.
            BitConverter.DoubleToInt64Bits(actual.High).ShouldBe(
                BitConverter.DoubleToInt64Bits(double.IsNaN(expected) ? double.NaN : expected));
            BitConverter.DoubleToInt64Bits(actual.Low).ShouldBe(0L);
            DoubleDouble.IsCanonical(actual).ShouldBeTrue();
        }

        DoubleDouble negative = DoubleDouble.FromComponents(-1.0, Math.ScaleB(1.0, -54));
        DoubleDouble.IsNaN(DDMath.Sqrt(negative)).ShouldBeTrue();
    }

    [Fact]
    public void PositiveFiniteInputsMeetExactRelativeErrorBound()
    {
        // Every binary64 exponent, including subnormal input exponents. Test zero,
        // dense, and sparse lows of either sign, without relying on DD arithmetic
        // to compute the expected square root or its square.
        Random random = new(314159);
        for (int exponent = -1074; exponent <= 1023; ++exponent)
        {
            double high = Math.ScaleB(1.0 + random.NextDouble(), exponent);
            double low = Math.ScaleB(random.NextDouble(), exponent - 54);
            double power = Math.ScaleB(1.0, exponent);
            foreach (double boundary in new[] { Math.BitDecrement(power), power, Math.BitIncrement(power) })
            {
                if (boundary > 0.0)
                {
                    DoubleDouble input = new(boundary);
                    AssertSqrtBound(input, DDMath.Sqrt(input));
                }
            }
            foreach (double residual in new[] { 0.0, low, -low, double.Epsilon, -double.Epsilon })
            {
                DoubleDouble input = DoubleDouble.FromComponents(high, residual);
                if (input.High > 0.0)
                {
                    AssertSqrtBound(input, DDMath.Sqrt(input));
                }
            }
        }

        // Pin the tiny-input precision loss described in the optional legacy review,
        // and guard against intermediate squaring overflow near the finite limit.
        DoubleDouble[] boundaries = [new(1e-308), new(double.Epsilon),
            new(Math.BitDecrement(Math.ScaleB(1.0, -1022))), new(Math.ScaleB(1.0, -1022)),
            new(Math.BitIncrement(Math.ScaleB(1.0, -1022))), new(double.MaxValue),
            DoubleDouble.FromComponents(double.MaxValue, Math.ScaleB(1.0, 969)),
            DoubleDouble.FromComponents(double.MaxValue, -Math.ScaleB(1.0, 969)),
            DoubleDouble.FromComponents(1.0, Math.ScaleB(1.0, -53)),
            DoubleDouble.FromComponents(1.0, -Math.ScaleB(1.0, -54)),
            DoubleDouble.FromComponents(4.0, Math.ScaleB(1.0, -51)),
            DoubleDouble.FromComponents(4.0, -Math.ScaleB(1.0, -52))];
        foreach (DoubleDouble input in boundaries)
        {
            AssertSqrtBound(input, DDMath.Sqrt(input));
        }
    }

    private static void AssertSqrtBound(DoubleDouble input, DoubleDouble actual)
    {
        DoubleDouble.IsFinite(actual).ShouldBeTrue();
        DoubleDouble.IsCanonical(actual).ShouldBeTrue();
        (actual.High > 0.0).ShouldBeTrue();

        // Independent exact rational oracle: x = X*2^-1074, y = Y*2^-1074.
        // For positive x and y, |y/sqrt(x) - 1| <= t iff
        // x*(1-t)^2 <= y^2 <= x*(1+t)^2. With t=2^-100 we
        // cross-multiply to integers: no rounded sqrt, product, or DD conversion.
        BigInteger x = (Units(input.High) + Units(input.Low)) << 1074;
        BigInteger y = Units(actual.High) + Units(actual.Low);
        BigInteger scale = BigInteger.One << 100;
        BigInteger squared = (y * y) << 200;
        BigInteger lower = x * (scale - 1) * (scale - 1);
        BigInteger upper = x * (scale + 1) * (scale + 1);
        (squared >= lower && squared <= upper).ShouldBeTrue(
            $"Sqrt bound failed for ({input.High:R}, {input.Low:R}): ({actual.High:R}, {actual.Low:R})");
    }

    private static BigInteger Units(double value)
    {
        long bits = BitConverter.DoubleToInt64Bits(value);
        int exponent = (int)((bits >> 52) & 0x7ff);
        BigInteger significand = bits & 0xfffffffffffffL;
        if (exponent != 0)
        {
            significand += BigInteger.One << 52;
            significand <<= exponent - 1;
        }
        return bits < 0 ? -significand : significand;
    }
}