Wednesday, February 17, 2010

Dividing by Multiplying

Peter Ammon:

Every divisor has a magic number, and most have more than one! A magic number for d is nothing more than a precomputed quotient: a power of 2 divided by d and then rounded up. At runtime, we do the same thing, except backwards: multiply by this magic number and then divide by the power of 2, rounding down. The tricky part is finding a power big enough that the “rounding up” part doesn’t hurt anything. If we are lucky, a multiple of d will happen to be only slightly larger than a power of 2, so rounding up doesn’t change much and our magic number will fit in 32 bits. If we are unlucky, well, we can always fall back to a 33 bit number, which is almost as efficient.

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