gistfile1.cs

private unsafe static BigInteger Sqrt(BigInteger num, BigInteger n)
{
    var squaresUnder100 = new Dictionary<int, int> { { 1, 1 }, { 2, 4 }, { 3, 9 }, { 4, 16 }, { 5, 25 }, { 6, 36 }, { 7, 49 }, { 8, 64 }, { 9, 81 } };
    var pairCount = 0;
    var pairs = GetSqrtPairs(num, out pairCount);

    // Find the largest integer whose square is smaller than or equal to the current pair.
    var pairSq = (BigInteger)1;
    foreach (var sq in squaresUnder100)
    {
        if (sq.Value <= pairs[0])
        {
            pairSq = sq.Key;
        }
        else
        {
            break;
        }
    }

    // Check if we only want the first digit.
    if (n == 1)
    {
        return pairSq;
    }

    var currentRes = (BigInteger)pairSq;
    var i = 0;
    var z = squaresUnder100[(int)pairSq] - currentRes; z = z == 0 ? 1 : 0;
    var currentDigits = 1;
    while (true)
    {
        var a = (z * 100) + pairs[i + 1];
        var y = currentRes * 20;
        var x = 0;

        for (var k = 1; k < 10; k++)
        {
            if ((y + k) * k <= a)
            {
                x = k;
            }
            else
            {
                break;
            }
        }
        
        z = a - ((y + x) * x);

        // Update current output.
        currentRes *= 10;
        currentRes += x;
        currentDigits++;

        // Check if we've reached the desired accuracy, and return if so. Otherwise, continue calculating...
        if (currentDigits == n)
        {
            Marshal.FreeHGlobal((IntPtr)pairs);
            return currentRes;
        }

        if (i + 2 != pairCount)
        {
            i++;
        }
    }
}

private unsafe static byte* GetSqrtPairs(BigInteger num, out int pairCount)
{
    var data = num.ToString();

    if (data.Length % 2 != 0)
    {
        data = "0" + data;
    }

    var pairsStn = Regex.Split(data, @"(\d{2})").Where(pair => !String.IsNullOrEmpty(pair)).ToArray();
    var ptr = (byte*)Marshal.AllocHGlobal(pairsStn.Length + 2 + 4);

    for (var i = 0; i < pairsStn.Length; i++)
    {
        ptr[i] = Byte.Parse(pairsStn[i]);
    }

    ptr[pairsStn.Length] = 0;
    ptr[pairsStn.Length + 1] = 0;

    pairCount = pairsStn.Length + 2;
    return ptr;
}