Integer Division by Constants: Optimal Bounds
Data Structures and Algorithms
2021-11-16 v3
Abstract
The integer division of a numerator n by a divisor d gives a quotient q and a remainder r. Optimizing compilers accelerate software by replacing the division of n by d with the division of c * n (or c * n + c) by m for convenient integers c and m chosen so that they approximate the reciprocal: c/m ~= 1/d. Such techniques are especially advantageous when m is chosen to be a power of two and when d is a constant so that c and m can be precomputed. The literature contains many bounds on the distance between c/m and the divisor d. Some of these bounds are optimally tight, while others are not. We present optimally tight bounds for quotient and remainder computations.
Keywords
Cite
@article{arxiv.2012.12369,
title = {Integer Division by Constants: Optimal Bounds},
author = {Daniel Lemire and Colin Bartlett and Owen Kaser},
journal= {arXiv preprint arXiv:2012.12369},
year = {2021}
}