Extended Abstract: Towards a Performance Comparison of Syntax and Type-Directed NbE

A key part of any dependent type-checker is the method for checking whether two types are equal. A common claim is that syntax-directed equality is more performant, although type-directed equality is more expressive. However, this claim is difficult to make precise, since implementations choose only one or the other approach, making a direct comparison impossible. We present some work-in-progress developing a realistic platform for direct, apples-to-apples, comparison of the two approaches, quantifying how much slower type-directed equality checking is, and analyzing why and how it can be improved.