Cascading GEMM: High Precision from Low Precision

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.