This paper presents a computational investigation that searches for arithmetic expressions approximating a given target constant using a finite set of base expressions and allowed operators. In our approach, the search is organized by a notion of DEPTH (i.e., the number of operations applied) and is limited to a maximum depth (typically 10 or 11) due to computational constraints. We describe the algorithm rigorously, introduce precise definitions and notation, present pseudocode in the algorithmic style, and discuss sample solutions—including approximations for e, φ (the golden ratio), and π with their respectivedepths and computed absolute errors. We also include an example run showing the number ofcandidate expressions generated at each depth (with depth 10 evaluating approximately 2.5 million sequences). Finally, we discuss the inherent limitations, including memory (RAM) requirements for deeper searches, and how increased computational power could extend the search depth.