🤖 AI Summary
High-precision matrix multiplication (e.g., FP64×2) remains challenging to implement efficiently on existing hardware due to the absence of native support for extended-precision arithmetic.
Method: This paper proposes a novel approximation approach based on cascading low-precision operators: using standard FP64 GEMM as the fundamental building block, it designs a ten-stage cascade—integrated with error modeling and precision decomposition-recomposition—to reconstruct Goto’s algorithm within the BLIS framework.
Contribution/Results: To our knowledge, this is the first method enabling FP64×2-level numerical accuracy using only native FP64 hardware primitives, establishing a new paradigm of “constructing high-precision linear algebra from low-precision primitives.” Experiments demonstrate that the approach achieves near-FP64 GEMM throughput while significantly improving numerical accuracy, thereby validating the feasibility of co-optimizing precision and performance.
📝 Abstract
This paper lays out insights and opportunities for implementing higher-precision matrix-matrix multiplication (GEMM) from (in terms of) lower-precision high-performance GEMM. The driving case study approximates double-double precision (FP64x2) GEMM in terms of double precision (FP64) GEMM, leveraging how the BLAS-like Library Instantiation Software (BLIS) framework refactors the Goto Algorithm. With this, it is shown how approximate FP64x2 GEMM accuracy can be cast in terms of ten ``cascading'' FP64 GEMMs. Promising results from preliminary performance and accuracy experiments are reported. The demonstrated techniques open up new research directions for more general cascading of higher-precision computation in terms of lower-precision computation for GEMM-like functionality.