COFFEE is a COmpiler For Fast Expression Evaluation. Given mathematical expressions embedded in a loop nest, COFFEE can be driven in the application of two classes of transformations:

• Expression rewriting, for flops minimisation. A small set of rewrite operators, such as expansion, factorization, and generalized code motion, can be composed with the aim of reducing the operation count.
• Code specialisation, to maximise the impact of low level optimisation. For example, COFFEE can analyse the arrays, loops, and memory access pattern in a kernel, and then autonomously decide on the application of padding and data alignment, a transformation for enhancing the effectiveness of compiler auto-vectorisation.

COFFEE is currently used in Firedrake, an automated system for the solution of partial differential equations using the finite element method. An optimisation pipeline in COFFEE was designed to exploit a fundamental mathematical property of finite element integration kernels, namely linearity of operators in test and trial functions.