A note on Dekker’s FastTwoSum algorithm

  title={A note on Dekker’s FastTwoSum algorithm},
  author={Marko Lange and Shin'ichi Oishi},
  journal={Numerische Mathematik},
  • M. Lange, S. Oishi
  • Published 1 June 2020
  • Mathematics, 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… 
Computing the exact sign of sums of products with floating point arithmetic
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.


Ultimately Fast Accurate Summation
  • S. Rump
  • Computer Science, Mathematics
    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
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
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
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
Abstract. Exact computer arithmetic has a variety of uses, including the robust implementation of geometric algorithms. This article has three purposes. The first is to offer fast software-level
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
Given a vector pi of ∞oating-point numbers with exact sum s, we present a new algorithm with the following property: Either the result is a faithful rounding of s, or otherwise the result has a
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
  • J. Shewchuk
  • Mathematics, Computer Science
    Discret. Comput. Geom.
  • 1997
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.