Monotonicity of Multi-Term Floating-Point Adders

Abstract

In the literature on algorithms for performing the multi-term addition $s_n=\sum_{i=1}^n x_i$ using floating-point arithmetic it is often shown that a hardware unit that has single normalization and rounding improves precision, area, latency, and power consumption, compared with the use of standard add or fused multiply-add units. However, non-monotonicity can appear when computing sums with a subclass of multi-term addition units, which currently is not explored in the literature. We demonstrate that common techniques for performing multi-term addition with $n\geq 4$, without normalization of intermediate quantities, can result in non-monotonicity -- increasing one of the addends $x_i$ decreases the sum $s_n$. Summation is required in dot product and matrix multiplication operations, operations that have increasingly started appearing in the hardware of supercomputers, thus knowing where monotonicity is preserved can be of interest to the users of these machines. Our results suggest that non-monotonicity of summation, in some of the commercial hardware devices that implement a specific class of multi-term adders, is a feature that may have appeared unintentionally as a consequence of design choices that reduce circuit area and other metrics. To demonstrate our findings, we use formal proofs as well as a numerical simulation of non-monotonic multi-term adders in MATLAB.