Predicting practical speedups offered by future quantum computers has become a major focus of the quantum computing community. Typically, these predictions are supported by lengthy manual analyses and numerical simulations and are carried out for one specific application at a time. In this paper, we present Traq, a principled approach towards estimating the quantum speedup of classical programs fully automatically and with provable guarantees. It consists of a classical language that includes high-level primitives amenable to quantum speedups, a cost analysis, and a compilation to low-level quantum programs. Our cost analysis upper bounds the complexity of the resulting quantum program in a fine-grained way: it captures non-asymptotic information and is sensitive to the input of the program (rather than providing worst-case costs).