Tekum: Balanced Ternary Tapered Precision Real Arithmetic

In light of recent hardware advances, it is striking that real arithmetic in balanced ternary logic has received almost no attention in the literature. This is particularly surprising given ternary logic's promising properties, which could open new avenues for energy-efficient computing and offer novel strategies for overcoming the memory wall. This paper revisits the concept of tapered precision arithmetic, as used in posit and takum formats, and introduces a new scheme for balanced ternary logic: tekum arithmetic. Several fundamental design challenges are addressed along the way. The proposed format is evaluated and shown to exhibit highly promising characteristics. In many respects, it outperforms both posits and takums. As ternary hardware matures, this work represents a crucial step toward unlocking the full potential of real-number computation in ternary systems, laying the groundwork for a new class of number formats designed from the ground up for a new category of next-generation hardware.