Evaluation of Bfloat16, Posit, and Takum Arithmetics in Sparse Linear Solvers

Our experiments demonstrate that tapered-precision arithmetic consistently outperforms IEEE 754 floating-point numbers in all tested scenarios. Among the tapered-precision formats, takums exhibited exceptional performance, outperforming bfloat16 in every case. While occasionally marginally less accurate than posits, takums delivered comparable results overall and demonstrated superior numerical stability. Notably, we successfully introduced the application of 8-bit posits and takums in mixed-precision iterative refinement, marking a possibly significant milestone in numerical computing. Additionally, GMRES exhibited particular benefit from tapered-precision formats, with takums delivering outstanding results, surpassing posits in all cases.

Impressive results from Takums (https://arxiv.org/abs/2404.18603).

They also seem better then Posits for implementation in hardware. https://arxiv.org/pdf/2408.10594

Amazing.