Monotonicity of Multi-Term Floating-Point Adders

In the literature on algorithms for computing multi-term addition
$s_n=\sum_{i=1}^n x_i$
in 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 is currently not explored in the literature. We prove that computing multi-term floating-point addition with
n
≥ 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 are increasingly appearing in the hardware of high-performance computers, and knowing where monotonicity is preserved can be of interest to the developers and users. Non-monotonicity of summation in existent hardware devices that implement a specific class of multi-term adders may have appeared unintentionally as a consequence of design choices that reduce circuit area and other metrics. To demonstrate our findings we simulate non-monotonic multi-term adders in MATLAB using the
CPFloat
custom-precision floating-point simulator.