Optimizing compilers apply numerous interdependent optimizations, leading to the notoriously difficult phase-ordering problem-that of deciding which transformations to apply and in which order. Fortunately, new infrastructures such as the polyhedral compilation framework host a variety of transformations, facilitating the efficient exploration and configuration of multiple transformation sequences. Many powerful optimizations, however, remain external to the polyhedral framework, including vectorization. The low-level, target-specific aspects of vectorization for fine-grain SIMD has so far excluded it from being part of the polyhedral framework. In this paper we examine the interactions between loop transformations of the polyhedral framework and subsequent vectorization. We model the performance impact of the different loop transformations and vectorization strategies, and then show how this cost model can be integrated seamlessly into the polyhedral representation. This predictive modelling facilitates efficient exploration and educated decision making to best apply various polyhedral loop transformations while considering the subsequent effects of different vectorization schemes. Our work demonstrates the feasibility and benefit of tuning the polyhedral model in the context of vectorization. Experimental results confirm that our model has accurate predictions, providing speedups of over 2.0x on average over traditional innermost-loop vectorization on PowerPC970 and Cell-SPU SIMD platforms.

To preserve the validity of loop nest transformations and parallelization, data dependences need to be analyzed. Memory dependences come in two varieties: true dependences or false dependences. While true dependences must be satisfied in order to preserve the correct order of computations, false dependences are induced by the reuse of a single memory location to store multiple values. False dependences reduce the degrees of freedom for loop transformations. In particular, loop tiling is severely limited in the presence of these dependences. While array expansion removes all false dependences, the overhead on memory and the detrimental impact on register-level reuse can be catastrophic. We propose and evaluate a compilation technique to safely ignore a large number of false dependences in order to enable loop nest tiling in the polyhedral model. It is based on the precise characterization of interferences between live range intervals, and it does not incur any scalar or array expansion. Our algorithms have been implemented in the Pluto polyhedral compiler, and evaluated on the PolyBench suite.

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