# A note on Dekker’s FastTwoSum algorithm

@article{Lange2020ANO,
title={A note on Dekker’s FastTwoSum algorithm},
author={Marko Lange and Shin'ichi Oishi},
journal={Numerische Mathematik},
year={2020},
volume={145},
pages={383-403}
}
• Published 1 June 2020
• Computer Science
• Numerische Mathematik
More than 45 years ago, Dekker proved that it is possible to evaluate the exact error of a floating-point sum with only two additional floating-point operations, provided certain conditions are met. Today the respective algorithm for transforming a sum into its floating-point approximation and the corresponding error is widely referred to as $${{\,\mathrm{FastTwoSum}\,}}$$ FastTwoSum . Besides some assumptions on the floating-point system itself—all of which are satisfied by any binary IEEE…
1 Citations
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## References

SHOWING 1-10 OF 18 REFERENCES
Ultimately Fast Accurate Summation
• S. Rump
• Computer Science
SIAM J. Sci. Comput.
• 2009
Two new algorithms to compute a faithful rounding of the sum of floating-point numbers and the other for a result “as if” computed in $K$-fold precision, which are the fastest known in terms of flops.
Accurate Floating-Point Summation Part I: Faithful Rounding
• Computer Science
SIAM J. Sci. Comput.
• 2008
This paper presents an algorithm for calculating a faithful rounding of a vector of floating-point numbers, which adapts to the condition number of the sum, and proves certain constants used in the algorithm to be optimal.
Fast and Accurate Floating Point Summation with Application to Computational Geometry
• Computer Science
Numerical Algorithms
• 2004
The results show that in the absence of massive cancellation (the most common case) the cost of guaranteed accuracy is about 30–40% more than the straightforward summation, and the accurate summation algorithm improves the existing algorithm by a factor of two on a nearly coplanar set of points.
Algorithm 908
• Computer Science
ACM Trans. Math. Softw.
• 2010
A novel, online algorithm for exact summation of a stream of floating-point numbers that is the fastest, most accurate, and most memory efficient among known algorithms.
Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates
This article proposes a technique for adaptive precision arithmetic that can often speed software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values when they are used to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound.
Minima of Functions of Several Variables with Inequalities as Side Conditions
The problem of determining necessary conditions and sufficient conditions for a relative minimum of a function $$f({x_1},{x_2},....,{x_n})$$ in the class of points $$x = ({x_1},{x_2},....,{x_n})$$
Pracniques: further remarks on reducing truncation errors
The AND and NOT operatimts are transf0rlned to multiplication and subtraction operations as described in (1) and (3).
Fast high precision summation
• Computer Science
• 2010
Given a vector pi of ∞oating-point numbers with exact sum s, a new algorithm is presented that is fast in terms of measured computing time because it allows good instruction-level parallelism and more accurate and faster than competitors such as XBLAS.
Floating-point computation
(a) Write a function in a programming language of your choice that takes a (32-bit IEEE format) float and returns a float with the property that: given zero, infinity or a positive normalised
Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates
This article offers fast software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values and proposes a technique for adaptive precision arithmetic that can often speed these algorithms when they are used to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound.