I've got a (let's say nonnegative) rational number expressed as the ratio of two integers a/b, where 0 <= a < 2^m and 0 < b < 2^n. I'd like to round that to a smaller representation with only p bits in the denominator; that is, find the largest number c/d such that c/d <= a/b, where 0 < d < 2^p.
Example: if m=3, n=4, p=2, I'd like to round 4/7 down to 1/2, and 5/7 down to 2/3.
My first impulse was to right-shift the denominator by n-p, adding 1 if a 1 was popped off, and right-shift the numerator by the same amount. That's guaranteed to produce a result less than a/b, but it doesn't guarantee optimality of the results. For instance, 3/1 rounds to 2/1.
Ideally, I'd like to do this without division or modulus, but I suspect that might not be practical. m, n, and p may run into the hundreds, and this is going to be in an inner loop, so I'd like something as blazingly fast as possible.
