Integer Division by Constants: Optimal Bounds

Integer Division by Constants: Optimal Bounds

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 ~...

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Veröffentlicht in:arXiv.org 2021-06
Hauptverfasser: Lemire, Daniel, Bartlett, Colin, Kaser, Owen
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Computer Science - Data Structures and Algorithms
Integers
Optimization
Quotients
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description 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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Integers
Optimization
Quotients
title Integer Division by Constants: Optimal Bounds
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