Applied combinatorics
The accuracy of floating point summation
SIAM Journal on Scientific Computing
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
SIAM Journal on Scientific Computing
Semantics of roundoff error propagation in finite precision calculations
Higher-Order and Symbolic Computation
Accurate Floating-Point Summation Part I: Faithful Rounding
SIAM Journal on Scientific Computing
Enhancing the implementation of mathematical formulas for fixed-point and floating-point arithmetics
Formal Methods in System Design
Ultimately Fast Accurate Summation
SIAM Journal on Scientific Computing
Optimizing correctly-rounded reciprocal square roots for embedded VLIW cores
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
A new abstract domain for the representation of mathematically equivalent expressions
SAS'12 Proceedings of the 19th international conference on Static Analysis
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In this article, we focus on numerical algorithms for which, in practice, parallelism and accuracy do not cohabit well. In order to increase parallelism, expressions are reparsed, implicitly using mathematical laws like associativity, and this reduces the accuracy. Our approach consists in focusing on summation algorithms and in performing an exhaustive study: we generate all the algorithms equivalent to the original one and compatible with our relaxed time constraint. Next we compute the worst errors which may arise during their evaluation, for several relevant sets of data. Our main conclusion is that relaxing very slightly the time constraints by choosing algorithms whose critical paths are a bit longer than the optimal makes it possible to strongly optimize the accuracy. We extend these results to the case of bounded parallelism and to accurate sum algorithms that use compensation techniques.