Strength Reduction of Integer Division and Modulo Operations

Abstract

Integer division, modulo, and remainder operations are expressive and useful operations. They are logical candidates to express complex data accesses such as the wrap-around behavior in queues using ring buffers. In addition, they appear frequently in address computations as a result of compiler optimizations that improve data locality, perform data distribution, or enable parallelization. Experienced application programmers, however, avoid them because they are slow. Furthermore, while advances in both hardware and software have improved the performance of many parts of a program, few are applicable to division and modulo operations. This trend makes these operations increasingly detrimental to program performance. This paper describes a suite of optimizations for eliminating division, modulo, and remainder operations from programs. These techniques are analogous to strength reduction techniques used for multiplications. In addition to some algebraic simplifications, we present a set of optimization techniques that eliminates division and modulo operations that are functions of loop induction variables and loop constants. The optimizations rely on algebra, integer programming, and loop transformations.