@inproceedings{nin-rev-14-aa-fastnodir,
  author = {Jordan Ninin and Nathalie Revol},
  title = {Accurate and Effcient Implementation of Affine Arithmetic Using Floating-Point Arithmetic},
  booktitle = {Abstracts of 16th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2014)},
  location = {W{\"u}rzburg, Germany},
  pages = {125-126},
  year = 2014,
  month = sep,
  publisher = {University of W{\"u}rzburg},
  abstract = {Affine arithmetic is one of the extensions of interval arithmetic that aim at counteracting the variable dependency problem. With affine arithmetic, defined in [5] by Stolfi and Figueiredo, variables are represented as ane combination of symbolic noises. It differs from the generalized interval arithmetic, defined by Hansen in [1], where variables are represented as affine combination of intervals. Non-affne operations are realized through the introduction of a new noise, that accounts for nonlinear terms. Variants of ane arithmetic have been proposed, they aim at limiting the number of noise symbols. Let us mention [4] by Messine and [6] by Vu, Sam-Haroud and Faltings to quote only a few. The focus here is on the implementation of affine arithmetic using floating-point arithmetic, specified in [2]. With floating-point arithmetic, an issue is to handle roundoff errors and to incorporate them in the final result, so as to satisfy the inclusion property, which is the fundamental property of interval arithmetic. In [4], [5] and [6], roundoff errors are accounted for in a manner that implies frequent switches of the rounding mode; this incurs a severe time penalty. Implementations of these variants are available in YalAA, developed by Kiel [2].  We propose an implementation that uses one dedicated noise symbol for accumulated roundoff errors. For accuracy purposes, the roundoff error $\epsilon$ of each arithmetic operation is computed exactly via EFT (Error Free Transforms). For efficiency purposes, the rounding mode is never switched. Instead, a brute-force bound on the roundoff error $\varepsilon$ incurred by the accumulation of the $\epsilon$s mentioned above is used. Experimental results are presented. The proposed implementation is one of the most accurate and its execution time is significantly reduced; it can be up to 50{\%} faster than other implementations. Furthermore, the use of a FMA (Fused Multiply-and-Add) reduces the cost of the EFT and the overall performance is even better}
}