Reduced-Precision and Reduced-Exponent Formats for Accelerating Adaptive Precision Sparse Matrix–Vector Product

Mixed precision algorithms aim at taking advantage of the performance of low precisions while maintaining the accuracy of high precision. In particular adaptive precision algorithms dynamically adapt at runtime the precisions used for different variables or operations. For example Graillat et al. (2023) have proposed an adaptive precision sparse matrix–vector product (SpMV) which stores the matrix elements in a precision inversely proportional to their magnitude. In theory, this algorithm can therefore make use of a large number of different precisions, but the practical results previously obtained only achieved high performance using natively supported double and single precisions. In this work we combine this algorithm with an efficient memory accessor for custom reduced precision formats (Mukunoki et al. 2016). This allows us to experiment with a large set of different precision formats with fine variations of the number of bits dedicated to the significand. Moreover we also explore the possibility to reduce the number of bits dedicated to the exponent using the fact that the elements that share the same precision format are of similar magnitude. We experimentally evaluate the performance of using four or seven different custom formats using reduced precision and possibly reduced exponent, and demonstrate their effectiveness compared with the existing version only using double and single precisions.