Algorithms and Arithmetic: Choose Wisely

I will introduce a semi-formalism to allow us to conceptually reason about the differences between customised arithmetic design, as one might see in FPGA-based compute, and general purpose arithmetic, as one might find in microprocessor design. This framework will, I hope, expose to the reader the reason that we should be thinking carefully about appropriate data representations when designing custom hardware for compute, as well as clearly showing the link between these decisions and algorithmic ones. I will then provide a concrete example from the literature on matrix computation where some careful algorithmic tweaking results in the ability to use fixed-point arithmetic and, hence, far higher performance than would otherwise be achieved.