Strength Reduction of Integer Division and Modulo Operations

Integer division, modulo, and remainder operations are expressive and useful operations. They are logical candidates to express many complex data accesses such as the wrap-around behavior in queues using ring buffers and array address calculations in data distribution and cache locality compiler-optimizations. 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 analagous to strength reduction techiques used for multiplications. In addition to some algebraic simplifications, we present a set of optimization techniques which eliminates division and modulo operations that are functions of loop induction variables and loop constants. The optimizations rely on number theory, integer programming and loop transformations.