In this paper, we present HELIX, a formally verified operator language and rewriting engine for generation of high-performance implementation for a variety of linear algebra algorithms. Based on the existing SPIRAL system, HELIX adds the rigor of formal verification of its correctness using Coq proof assistant. It formally defines two domain-specific languages: HCOL, which represents a computation data flow and Σ-HCOL, which extends HCOL with iterative computations. A framework for automatically proving semantic preservation of expression rewriting for both languages is presented. The structural properties of the dataflow graph which allow efficient compilation are formalized, and a monadic approach to tracking them and to reasoning about structural correctness of Σ-HCOL expressions is presented.
This case study examines the data-parallel functional implementation of three algorithms: generation of quasi-random Sobol numbers, breadth-first search, and calibration of Heston market parameters via a least-squares procedure. We show that while all these problems permit elegant functional implementations, good performance depends on subtle issues that must be confronted in both the implementations of the algorithms themselves, as well as the compiler that is responsible for ultimately generating high-performance code. In particular, we demonstrate a modular technique for generating quasi-random Sobol numbers in an efficient manner, study the efficient implementation of an irregular graph algorithm without sacrificing parallelism, and argue for the utility of nested regular data parallelism in the context of nonlinear parameter calibration.