Print a specific value in the infinite Walsh matrix
GolfScript (18 15 14 bytes)

~&amp;2base-1\0-,?


Takes input space-separated; outputs to stdout. Thanks to Howard for pointing out that the space in the 15-char solution was removable.

The way I reasoned to this solution is quite simple. Take the highest set bit of x|y: this tells you the size of the smallest Walsh matrix which contains the relevant index. Then clearing that bit in both x and y drops us down to the top-left quadrant, and we need to multiply by -1 if we were previously in the bottom-right quadrant: i.e. if the bit was set in both x and y. By induction down the bit sequence, we just need to count the number of bits set in x&amp;y.