Computing the high bits of a multiplication in C#

Question

I'm trying to convert an open source library from .Net 4.0 to 3.5 and cannot easily convert the following long multiplication code:

    /// <summary>
    /// Calculate the most significant 64 bits of the 128-bit
        product x * y, where x and y are 64-bit integers.
    /// </summary>
    /// <returns>Returns the most significant 64 bits of the product x * y.</returns>
    public static long mul64hi(long x, long y)
    {
 #if !NET35
        BigInteger product = BigInteger.Multiply(x, y);
        product = product >> 64;
        long l = (long)product;
        return l;
 #else
        throw new NotSupportedException(); //TODO!
 #endif
    }

As you can see the author didn't find a way to do this. BigInteger does not exist in .NET 3.5.

How can I compute the high bits 64 bits of a 64*64 multiplication on .NET 3.5?

Answer 1

You can build a 2N-bit multiplier from multiple N-bit multipliers.

public static ulong mul64hi(ulong x, ulong y)
{
    ulong accum = ((ulong)(uint)x) * ((ulong)(uint)y);
    accum >>= 32;
    ulong term1 = (x >> 32) * ((ulong)(uint)y);
    ulong term2 = (y >> 32) * ((ulong)(uint)x);
    accum += (uint)term1;
    accum += (uint)term2;
    accum >>= 32;
    accum += (term1 >> 32) + (term2 >> 32);
    accum += (x >> 32) * (y >> 32);
    return accum;
}

It's just elementary-school long multiplication, really.

With signed numbers, it's a bit harder, because if intermediate results carry into the sign bit everything goes wrong. A long can't hold the result of a 32-bit by 32-bit multiply without that happening, so we have to do it in smaller chunks:

public static long mul64hi(long x, long y)
{
    const long thirtybitmask = 0x3FFFFFFF;
    const long fourbitmask = 0x0F;
    long accum = (x & thirtybitmask) * (y & thirtybitmask);
    accum >>= 30;
    accum += ((x >> 30) & thirtybitmask) * (y & thirtybitmask);
    accum += ((y >> 30) & thirtybitmask) * (x & thirtybitmask);
    accum >>= 30;
    accum += ((x >> 30) & thirtybitmask) * ((y >> 30) & thirtybitmask);
    accum += (x >> 60) * (y & fourbitmask);
    accum += (y >> 60) * (x & fourbitmask);
    accum >>= 4;
    accum += (x >> 60) * (y >> 4);
    accum += (y >> 60) * (x >> 4);
    return accum;
}

Inspired by harold's comment about Hacker's Delight, the signed version can be made just as efficient as the other, by carefully controlling whether intermediate results are or are not signed:

public static long mul64hi(long x, long y)
{
    ulong u = ((ulong)(uint)x) * ((ulong)(uint)y);
    long s = u >> 32;
    s += (x >> 32) * ((long)(uint)y);
    s += (y >> 32) * ((long)(uint)x);
    s >>= 32;
    s += (x >> 32) * (y >> 32);
    return s;
}