Improved error bounds for floating-point quotients

Publikationstyp
Journal Article
Date Issued
2025-07-10
Sprache
English
Author(s)
Bünger, Florian  
Mathematik E-10  
TORE-URI
https://hdl.handle.net/11420/56598
Journal
IEEE transactions on computers  
Volume
74
Issue
11
Start Page
3559
End Page
3567
Citation
IEEE Transactions on Computers 74 (11): 3559-3567 (2025)
Publisher DOI
10.1109/TC.2025.3585341
Scopus ID
2-s2.0-105010324165
Publisher
IEEE Computer Soc.
Let x<inf>0</inf>, y<inf>1</inf>, ...., y<inf>k</inf> be nonzero floating-point numbers in base β ≥ 2 and precision p ≥ 1. Let z := x<inf>0</inf>/ y<inf>1</inf> /.../ y<inf>k</inf>, whereby the divisions are evaluated from left to right, and let ẑ be the corresponding floating-point evaluation according to the IEEE 754 standard in rounding to nearest. We prove that, in absence of underflow and overflow,(Formula presented) provided that (Formula presented). Here (Formula presented) denotes the relative rounding error unit and ω := 2 if β is even and ω := 1 if β is odd. Thus, the relative rounding error of k consecutive floating-point divisions is bounded by ku. This improves on the classical Wilkinson-type bound γk := ku/(1 - ku).
Subjects
Floating-point quotients
IEEE 754 standard
Wilkinson-type relative error estimates
DDC Class
510: Mathematics