A note on Dekker’s FastTwoSum algorithm

Abstract

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 $$754$$ 754 standard conform arithmetic, the main practical limitation of $${{\,\mathrm{FastTwoSum}\,}}$$ FastTwoSum is that the summands have to be ordered according to their exponents. In most preceding applications of $${{\,\mathrm{FastTwoSum}\,}}$$ FastTwoSum , however, a more stringent condition is used, namely that the summands have to be sorted according to their absolute value. In remembrance of Dekker’s work, this note reminds the original assumptions for an error-free transformation via$${{\,\mathrm{FastTwoSum}\,}}$$ FastTwoSum . Moreover, we generalize the conditions for arbitrary bases and discuss a possible modification of the $${{\,\mathrm{FastTwoSum}\,}}$$ FastTwoSum algorithm to extend its applicability even further. Subsequently, a range of programs exploiting the wider applicability is presented. This comprises the OnlineExactSum algorithm by Zhu and Hayes, an error-free transformation from a product of three floating-point numbers to a sum of the same number of addends, and an algorithm for accurate summation proposed by Demmel and Hida.

Floating-point arithmetic

“…If the floating-point exponents and of and are such that ≥ , and if the first operation does not overflow, then is the error of the floating-point addition RN( + ), i.e. + = + , as shown by Knuth (1998) and later formally proved by Daumas, Rideau and Théry (2001); see also the work by Lange and Oishi (2020) for further extensions and applications. If we make the stronger assumption | | ≥ | |, the proof becomes a straightforward consequence of the Sterbenz theorem (Theorem 2.5).…”

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

“…Lemma 1 is about the exact computation of the difference − of floating point numbers. There are versions of this lemma since the late 1960s [2,7,8], but we prove it here for the case in which we round upwards for completeness, and because the details are not obvious (as stated, Lemma 1 is false if we round downwards for instance.) In essence, it states that we can represent − exactly as the difference of two floating point numbers and , with the additional feature that is much smaller than .…”

“…The inexactness of floating point arithmetic makes it hard to compute this sign exactly in some cases. There are already several references about the computation sums of floating point numbers [3,4,5,6,7,8,10,12,13,15,16,17], with applications in computations geometry. The idea of decomposing products in sums is also as old as [2,11], and the fourth chapter of [14] presents an algorithm for obtaining the signs of such sums.…”