This thesis develops a system for automatically analyzing and improving dynamic programs, such as those that have driven progress in natural language processing and computer science, more generally, for decades. Finding a correct program with the optimal asymptotic runtime can be unintuitive, time-consuming, and error-prone. This thesis aims to automate this laborious process. To this end, we develop an approach based on 1. a high-level, domain-specific language called Dyna for concisely specifying dynamic programs 2. a general-purpose solver to efficiently execute these programs 3. a static analysis system that provides type analysis and worst-case time/space complexity analyses 4. a rich collection of meaning-preserving transformations to programs, which systematizes the repeated insights of numerous authors when speeding up algorithms in the literature 5. a search algorithm for identifying a good sequence of transformations that reduce the runtime complexity, given an initial, correct program We show that, in practice, automated search -- like the mental search performed by human programmers -- can find substantial improvements to the initial program. Empirically, we show that many speed-ups described in the NLP literature could have been discovered automatically by our system. We provide a freely available prototype system at https://github.com/timvieira/dyna-pi.