Modulo in High-Performance Code: Strength Reduction for Modulo-Based Array Indexing in Loops

Modulo operations on array indices provide a convenient means for specifying cyclic operations on arrays, such as rotations. A naive compilation of such operations involves integer divisions, which often can be avoided by partitioning or rearranging iteration spaces. This paper proposes an optimisation to this effect. The key idea is to systematically replace modulo operations within loops by partitioning such loops and replacing the modulo operations with division-free equivalents. We present two different code transformation schemes for the proposed optimisation, provide correctness arguments for both, and present initial performance observations of the effects of these transformations in the context of a compiler for the array language SaC. We look at several benchmarks ranging from micro-benchmarks to some more realistic applications and run them on hardware ranging from multi-core CPUs to GPUs. Depending on the problem size, machine characteristics, and application context, we see direct performance benefits of up to a factor of 25 and even higher indirect benefits due to other optimisations that the proposed transformations enable.