New Results on the Distance between a Segment and Z². Application to the Exact Rounding

Abstract

This paper presents extensions to Lefèvre's algorithm that computes a lower bound on the distance between a segment and a regular grid Z 2. This algorithm and, in particular, the extensions are useful in the search for worst cases for the exact rounding of unary elementary functions or base-conversion functions. The proof that is presented here is simpler and less technical than the original proof. This paper also gives benchmark results with various optimization parameters, explanations of these results, and an application to base conversion.

GPU-Accelerated Generation of Correctly Rounded Elementary Functions

“…However increasing the number of intervals per thread do not improve performances since the occupancy of each kernel is already high enough. We show in Table 4 performance results of the HRcase search over the binade [1, 2[ as it corresponds to the general case according to [19]. First, we remark that Lefèvre and the regular HR-case searches take advantage of the two-way SMT on the multi-core tests as we have a parallel speedup higher than the number of cores.…”

“…In this section we describe two algorithms for HRcase search: Lefèvre HR-case search originally described in [19] and our new and more regular HR-case search. Both algorithms make use of Boolean tests which rely on the properties described in Sect.…”

“…In [19], Lefèvre presented an algorithm to search for (p, ǫ ′ ) HR-cases of a polynomial P i (x). This algorithm relies on a Boolean test on Q i (x) (the truncation of P i (x) to degree one) which computes a lower bound of the set {b − a · x mod 1 | x < # p D i } and returns Success if the inequality (1) does not hold, Failure otherwise.…”

“…In practice, we can make use of different division implementations. We can apply a subtractive division, a division instruction, or combine both in an hybrid approach as presented and analysed in [8], [19].…”

Numeration and discrete dynamical systems

“…As an example, a fast algorithm for computing a lower bound on the distance between a straight line and the points of a regular grid is given in [65,64], which is used to find worst cases when trying to round correctly the elementary functions in floatingpoint arithmetic; this is the so-called Table Maker's Dilemma [65], see also [77]. This algorithm is closely related to the algorithm (based on Ostrowski's map T 0 ) which is presented in [11], and which provides a greedy representation algorithm for double-base number systems.…”