Worst cases of a periodic function for large arguments

Guillaume Hanrot; Vincent Lefèvre; Damien Stehlé; Paul Zimmermann

doi:10.1109/ARITH.2007.37

Worst cases of a periodic function for large arguments

Guillaume Hanrot, Vincent Lefèvre, Damien Stehlé, Paul Zimmermann

School of Computing

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution › peer-review

8 Citations (Scopus)

Abstract

One considers the problem of finding hard to round cases of a periodic function for large floating-point inputs, more precisely when the function cannot be efficiently approximated by a polynomial. This is one of the last few issues that prevents from guaranteeing an efficient computation of correctly rounded transcendentals for the whole IEEE-754 double precision format. The first non-naive algorithm for that problem is presented, with a heuristic complexity of O(2°.⁶⁷⁶p) for a precision of p bits. The efficiency of the algorithm is shown on the largest IEEE-754 double precision binade for the sine function, and some corresponding bad cases are given. We can hope that all the worst cases of the trigonometric functions in their whole domain will be found within a few years, a task that was considered out of reach until now.

Original language	English
Title of host publication	Proceedings
Subtitle of host publication	18th IEEE Symposium on Computer Arithmetic : ARITH 18, Montpellier, France, June 25-27, 2007
Editors	Peter Kornerup, Jean-Michel Muller
Place of Publication	Los Alamitos, Calif.
Publisher	Institute of Electrical and Electronics Engineers (IEEE)
Pages	133-140
Number of pages	8
ISBN (Print)	0769528546
DOIs	https://doi.org/10.1109/ARITH.2007.37
Publication status	Published - 2007
Event	Symposium on Computer Arithmetic (18th : 2007) - Montpellier, France Duration: 25 Jun 2007 → 27 Jun 2007

Conference

Conference	Symposium on Computer Arithmetic (18th : 2007)
City	Montpellier, France
Period	25/06/07 → 27/06/07

Access to Document

10.1109/ARITH.2007.37

Cite this

Hanrot, G., Lefèvre, V., Stehlé, D., & Zimmermann, P. (2007). Worst cases of a periodic function for large arguments. In P. Kornerup, & J.-M. Muller (Eds.), Proceedings: 18th IEEE Symposium on Computer Arithmetic : ARITH 18, Montpellier, France, June 25-27, 2007 (pp. 133-140). Institute of Electrical and Electronics Engineers (IEEE). https://doi.org/10.1109/ARITH.2007.37

Hanrot, Guillaume ; Lefèvre, Vincent ; Stehlé, Damien et al. / Worst cases of a periodic function for large arguments. Proceedings: 18th IEEE Symposium on Computer Arithmetic : ARITH 18, Montpellier, France, June 25-27, 2007. editor / Peter Kornerup ; Jean-Michel Muller. Los Alamitos, Calif. : Institute of Electrical and Electronics Engineers (IEEE), 2007. pp. 133-140

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title = "Worst cases of a periodic function for large arguments",

abstract = "One considers the problem of finding hard to round cases of a periodic function for large floating-point inputs, more precisely when the function cannot be efficiently approximated by a polynomial. This is one of the last few issues that prevents from guaranteeing an efficient computation of correctly rounded transcendentals for the whole IEEE-754 double precision format. The first non-naive algorithm for that problem is presented, with a heuristic complexity of O(2°.⁶⁷⁶p) for a precision of p bits. The efficiency of the algorithm is shown on the largest IEEE-754 double precision binade for the sine function, and some corresponding bad cases are given. We can hope that all the worst cases of the trigonometric functions in their whole domain will be found within a few years, a task that was considered out of reach until now.",

author = "Guillaume Hanrot and Vincent Lef{\`e}vre and Damien Stehl{\'e} and Paul Zimmermann",

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note = "Symposium on Computer Arithmetic (18th : 2007) ; Conference date: 25-06-2007 Through 27-06-2007",

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Hanrot, G, Lefèvre, V, Stehlé, D & Zimmermann, P 2007, Worst cases of a periodic function for large arguments. in P Kornerup & J-M Muller (eds), Proceedings: 18th IEEE Symposium on Computer Arithmetic : ARITH 18, Montpellier, France, June 25-27, 2007. Institute of Electrical and Electronics Engineers (IEEE), Los Alamitos, Calif., pp. 133-140, Symposium on Computer Arithmetic (18th : 2007), Montpellier, France, 25/06/07. https://doi.org/10.1109/ARITH.2007.37

Worst cases of a periodic function for large arguments. / Hanrot, Guillaume; Lefèvre, Vincent; Stehlé, Damien et al.
Proceedings: 18th IEEE Symposium on Computer Arithmetic : ARITH 18, Montpellier, France, June 25-27, 2007. ed. / Peter Kornerup; Jean-Michel Muller. Los Alamitos, Calif.: Institute of Electrical and Electronics Engineers (IEEE), 2007. p. 133-140.

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution › peer-review

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N2 - One considers the problem of finding hard to round cases of a periodic function for large floating-point inputs, more precisely when the function cannot be efficiently approximated by a polynomial. This is one of the last few issues that prevents from guaranteeing an efficient computation of correctly rounded transcendentals for the whole IEEE-754 double precision format. The first non-naive algorithm for that problem is presented, with a heuristic complexity of O(2°.⁶⁷⁶p) for a precision of p bits. The efficiency of the algorithm is shown on the largest IEEE-754 double precision binade for the sine function, and some corresponding bad cases are given. We can hope that all the worst cases of the trigonometric functions in their whole domain will be found within a few years, a task that was considered out of reach until now.

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Worst cases of a periodic function for large arguments

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