Mikaitis, M. orcid.org/0000-0001-8706-1436 (2024) Monotonicity of Multi-Term Floating-Point Adders. IEEE Transactions on Computers. ISSN 1557-9956
Abstract
In the literature on algorithms for computing multi-term addition sn=∑ni=1xi 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 xi decreases the sum sn . 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2024 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. |
Keywords: | Monotonicity, floating-point arithmetic, multi-term addition, dot product, matrix multiply |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 22 Feb 2024 14:28 |
Last Modified: | 16 Apr 2024 11:40 |
Status: | Published online |
Publisher: | Institute of Electrical and Electronics Engineers |
Identification Number: | https://doi.org/10.1109/TC.2024.3371783 |