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In recent years fused-multiply-add (FMA) units with lower-precisionmultiplications and higher-precision accumulation have proven useful in machinelearning/artificial intelligence applications, most notably in training deepneural networks due to their extreme computational intensity. Compared toclassical IEEE-754 32 bit (FP32) and 64 bit (FP64) arithmetic, these reducedprecision arithmetic can naturally be sped up disproportional to theirshortened width. The common strategy of all major hardware vendors is toaggressively further enhance their performance disproportionately. Oneparticular FMA operation that multiplies two BF16 numbers while accumulating inFP32 has been found useful in deep learning, where BF16 is the 16-bit floatingpoint datatype with IEEE FP32 numerical range but 8 significant bits ofprecision. In this paper, we examine the use this FMA unit to implementhigher-precision matrix routines in terms of potential performance gain andimplications on accuracy. We demonstrate how a decomposition into multiplesmaller datatypes can be used to assemble a high-precision result, leveragingthe higher precision accumulation of the FMA unit. We first demonstrate thatcomputations of vector inner products and by natural extension, matrix-matrixproducts can be achieved by decomposing FP32 numbers in several BF16 numbersfollowed by appropriate computations that can accommodate the dynamic range andpreserve accuracy compared to standard FP32 computations, while projecting upto 5.2x speed-up. Furthermore, we examine solution of linear equationsformulated in the residual form that allows for iterative refinement. Wedemonstrate that the solution obtained to be comparable to those offered byFP64 under a large range of linear system condition numbers.

Leveraging the bfloat16 Artificial Intelligence Datatype For Higher-Precision Computations