SOAP: Structural optimization of arithmetic expressions for high-level synthesis

This paper introduces SOAP, a new tool to automatically optimize the structure of arithmetic expressions for FPGA implementation as part of a high level synthesis flow, taking into account axiomatic rules derived from real arithmetic, such as distributivity, associativity and others. We explicitly target an optimized area/accuracy trade-off, allowing arithmetic expressions to be automatically re-written for this purpose. For the first time, we bring rigorous approaches from software static analysis, specifically formal semantics and abstract interpretation, to bear on source-to-source transformation for high-level synthesis. New abstract semantics are developed to generate a computable subset of equivalent expressions from an original expression. Using formal semantics, we calculate two objectives, the accuracy of computation and an estimate of resource utilization in FPGA. The optimization of these objectives produces a Pareto frontier consisting of a set of expressions. This gives the synthesis tool the flexibility to choose an implementation satisfying constraints on both accuracy and resource usage. We thus go beyond existing literature by not only optimizing the precision requirements of an implementation, but changing the structure of the implementation itself. Using our tool to optimize the structure of a variety of real world and artificially generated examples in single precision, we improve either their accuracy or the resource utilization by up to 60%.