@INPROCEEDINGS{ChevillardJoldesLauter2009,
  author = {S.~Chevillard and M.~Jolde\c{s} and C.~Lauter},
  title = {Certified and fast computation of supremum norms of approximation
	errors},
  booktitle = {19th IEEE SYMPOSIUM on Computer Arithmetic},
  year = {2009},
  editor = {J.~D.~Bruguera and M.~Cornea and D.~Das-Sarma and J.~Harrison},
  pages = {169--176},
  address = {Los Alamitos, CA},
  month = {June},
  publisher = {IEEE Computer Society},
  abstract = {In many numerical programs there is a need for a high-quality floating-point
	approximation of useful functions $f$, such as such as $\exp$, $\sin$,
	$\erf$.
	
	In the actual implementation, the function is replaced by a polynomial
	$p$, leading to an approximation error (absolute or relative) $\epsilon
	= p-f$ or $\epsilon = \nicefrac{p}{f}-1$. The tight yet certain bounding
	of this error is an important step towards safe implementations.
	
	
	The main difficulty of this problem is due to the fact that this approximation
	error is very small and the difference $p-f$ is highly cancellating.
	In consequence, previous approaches for computing the supremum norm
	in this degenerate case, have proven to be either unsafe, not sufficiently
	tight or too tedious in manual work.
	
	
	We present a safe and fast algorithm that computes a tight lower and
	upper bound for the supremum norms of approximation errors. The algorithm
	is based on a combination of several techniques, including enhanced
	interval arithmetic, automatic differentiation and isolation of the
	roots of a polynomial. We have implemented our algorithm and timings
	on several examples are given.},
  keywords = {supremum norm, approximation error, certified computation, elementary
	function, interval arithmetic, automatic differentiation, roots isolation
	technique},
  url = {https://hal.archives-ouvertes.fr/ensl-00334545/}
}