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arXiv · 2012.12369

Integer Division by Constants: Optimal Bounds

Daniel Lemire·Colin Bartlett·Owen Kaser
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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.

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Daniel Lemire, Colin Bartlett, Owen Kaser. 2020-12-22. Integer Division by Constants: Optimal Bounds. https://doi.org/10.1016/j.heliyon.2021.e07442

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