Can the standard Ackermann be optimized?

Question

The standard Ackermann formula as written in Java:

public static int ack(int x, int y) {

        if (x == 0) {
            return y + 1;
        } else if (y == 0) {
            return ack(x-1, 1); 
        } else {
            // perforce (x > 0) && (y > 0)
            return ack(x-1, ack(x,y-1));
        }
    }

I've been wondering - is there a faster version to implement this? I'm thinking maybe there is by using an accumulator or a loop.

Answer 1

Yes, for example by "cheating". If m is 5 or higher, none of the results can be represented by an int. For m = 4, only the n < 2 cases can be represented. For m < 4, there are simple closed formula's based on n.

Everything else would overflow anyway, so let's pretend those cases don't even happen (or you could throw an error or whatever).

Not tested:

int Ackerman(int m, int n) {
    switch (m) {
    case 0:
        return n + 1;
    case 1:
        return n + 2;
    case 2:
        return n * 2 + 3;
    case 3:
        return (int)((1L << (n + 3)) - 3);
    case 4:
        return n == 0 ? 13 : 65533;
    }
}