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Proceedings Paper

Precise accounting of bit errors in floating-point computations
Author(s): Mark S. Schmalz
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Paper Abstract

Floating-point computation generates errors at the bit level through four processes, namely, overflow, underflow, truncation, and rounding. Overflow and underflow can be detected electronically, and represent systematic errors that are not of interest in this study. Truncation occurs during shifting toward the least-significant bit (herein called right-shifting), and rounding error occurs at the least significant bit. Such errors are not easy to track precisely using published means. Statistical error propagation theory typically yields conservative estimates that are grossly inadequate for deep computational cascades. Forward error analysis theory developed for image and signal processing or matrix operations can yield a more realistic typical case, but the error of the estimate tends to be high in relationship to the estimated error. In this paper, we discuss emerging technology for forward error analysis, which allows an algorithm designer to precisely estimate the output error of a given operation within a computational cascade, under a prespecified set of constraints on input error and computational precision. This technique, called bit accounting, precisely tracks the number of rounding and truncation errors in each bit position of interest to the algorithm designer. Because all errors associated with specific bit positions are tracked, and because integer addition only is involved in error estimation, the error of the estimate is zero. The technique of bit accounting is evaluated for its utility in image and signal processing. Complexity analysis emphasizes the relationship between the work and space estimates of the algorithm being analyzed, and its error estimation algorithm. Because of the significant overhead involved in error representation, it is shown that bit accounting is less useful for real-time error estimation, but is well suited to analysis in support of algorithm design.

Paper Details

Date Published: 3 September 2009
PDF: 10 pages
Proc. SPIE 7444, Mathematics for Signal and Information Processing, 744407 (3 September 2009); doi: 10.1117/12.829204
Show Author Affiliations
Mark S. Schmalz, Univ. of Florida (United States)

Published in SPIE Proceedings Vol. 7444:
Mathematics for Signal and Information Processing
Franklin T. Luk; Mark S. Schmalz; Gerhard X. Ritter; Junior Barrera; Jaakko T. Astola, Editor(s)

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