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Elementary functions, such as reciprocals and trigonometric functions, are critical in computer engineering applications like digital signal processing and graphics processing. However, their direct computation in hardware is resource-intensive. Table-based approximation methods, such as the Symmetric Table Addition Method (STAM) and the Multipartite Table Method, address this challenge by using two or more tables and an adder to approximate a target elementary function. This thesis compares the STAM and Multi-partite table methods through a rigorous analysis of their partitioning schemes, compression techniques, and error equations. In this Thesis, algorithms are developed to identify optimal partition schemes that minimize table sizes while still ensuring faithful rounding, and in some cases pushing the boundaries on previously reported partition schemes. Additionally, a Python-based GUI was implemented for exploring these methods. The findings reveal that Multipartite’s slope method o↵ers negligible error reduction over the STAM’s derivative-based approach for small partition counts, and error bounds were slightly too strict, missing optimal configurations. These insights suggest potential hybrid approaches combining Multi-partite’s notation with the STAM’s Taylor-series framework, giving us more exact constraints for finding optimal partition schemes.