# Floating point numbers are real numbers

@article{Mascarenhas2016FloatingPN, title={Floating point numbers are real numbers}, author={Walter F. Mascarenhas}, journal={arXiv: Numerical Analysis}, year={2016} }

Floating point arithmetic allows us to use a finite machine, the digital computer, to reach conclusions about models based on continuous mathematics. In this article we work in the other direction, that is, we present examples in which continuous mathematics leads to sharp, simple and new results about the evaluation of sums, square roots and dot products in floating point arithmetic.

## 7 Citations

Error Bounds for Computer Arithmetics

- Mathematics, Computer Science2019 IEEE 26th Symposium on Computer Arithmetic (ARITH)
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This note summarizes recent progress in error bounds for compound operations performed in some computer arithmetic by identifying three types A, B, and C of weak sufficient assumptions implying new results and sharper error estimates.

Sharp estimates for perturbation errors in summations

- Computer ScienceMath. Comput.
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The main result is sharp for actual realizations of grids floating-point arithmetics are based on, for any feasible problem size, for IEEE 754 binary32 as well as binary64 format, there are examples satisfying the given bound with equality.

Fast and accurate normalization of vectors and quaternions

- Computer Science, MathematicsArXiv
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Fast and accurate ways to normalize two and three-dimensional vectors and quaternions and compute their length are presented.

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It is shown that if the tie-breaking rule is "to away" then the bound $3u$ is asymptotically optimal, and that asymptic optimal bounds are now $2.25u $ for base two and $2u$ for larger bases (such as base ten).

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This is an abstract of the PhD thesis Preconditioning techniques for singularly perturbed differential equations written by Thái Anh Nhan, under the supervision of Dr Niall Madden, at the School of…

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This thesis has analysed the potential of spectrally efficient optic al w reless transmission techniques and shown that OWC systems can greatly benefit from the application of MIMO schemes due to little differences between the multiple links.

Computing the exact sign of sums of products with floating point arithmetic

- Computer ScienceArXiv
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The algorithm is efficient and uses only of floating point arithmetic, which is much faster than exact arithmetic, and it is proved that the algorithm is correct and the efficient and tested C++ code for it is correct.

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