[dlemire]

(I am the author of the blog post and I am answering here at the request of nkurz.)

So nkurz is correct that, in the paper, N is always equal to the maximum bitwidth of the numerator. I was not careful enough to make sure that the notation of the paper matches the notation of the blog post. So the N from the blog post is the F from the paper. I am sorry about this.

I added the following footnote.

"What is N? If both the numerator n and the divisor d are 32-bit unsigned integers, then you can pick N=64. This is not the smallest possible value. The smallest possible value is given by Algorithm 2 in our paper and it involves a bit of mathematics (note: the notation in my blog post differs from the paper, N becomes F)."

To make it super simple, if you have 32-bit values, then pick N (from the blog post) to be 64 and you are all set. You can do better, but this requires knowing what the divisor is.

[mzs]

I have not used it in many years, but I have foggy memory of IBM's Power cpu compiler doing similar trick for remainder.