<p> This thesis details the application of computational methods to the problem of determining custom data representations when building hardware accelerators for numerical computations. A majority of scientific applications which require hardware acceleration are implemented in IEEE-754 double precision. However, in many cases the error tolerance requirements of the application are much less than the accuracy which IEEE-754 double precision provides. By leveraging custom data representations, a more resource efficient hardware implementation arises thereby enabling greater parallelism and thus higher performance of the accelerator. </p> <p> The existing custom representation methods are unable to guarantee robust representations while at the same time adequately supporting ill-conditioned operators. Support for both of these scenarios is necessary for accelerating scientific calculations. To address this, we propose the use of a computational method based on Satisfiability-Modulo Theory (SMT). By capturing a calculation as a set of constraints, an SMT instance can be formulated which provides meaningful bounds even in the presence of ill-conditioned operators. At the same time, the analytical nature of SMT satisfies the need for robustness. Utilizing block vector arithmetic, our SMT approach is extended to provide scalability to large instances involving vector calculus which arise in scientific calculations. Atop this foundation, a unified error model is proposed which deals simultaneously with absolute and relative error, thereby providing the means of supporting both fixed-point and custom floating-point data types. Iterative algorithm analysis is leveraged to derive constraints for the SMT method. The application of the method to several scientific algorithms is discussed by way of case studies. </p> / Thesis / Doctor of Philosophy (PhD)