Computing Integer Powers in Floating-Point Arithmetic

We introduce two algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a fused multiply-add (fma) instruction is available. We show that our log-time algorithm always produce faithfully-rounded results, discuss the possibility of getting correctly rounded results, and show that results correctly rounded in double precision can be obtained if extended-precision is available with the possibility to round into double precision (with a single rounding).Comment: Laboratoire LIP : CNRS/ENS Lyon/INRIA/Universit\'e Lyon

Numerical simulation of the stabilisation of the steady state operation of an alternator feeding an infinite busbar by a circuit field coupled model.

In this paper, the authors present a study of the stabilisation of the steady state operation of an alternator by means of a time stepped 2D finite element field circuit coupled model. The results of the study show that it can be very difficult to stabilise the global quantities of the alternator. Ripples of numerical origin are observed on these quantities. The studies presented in this paper are intended toexplain these difficulties and show that these ripples are due to the famous backward Euler algorithm commonly used in this kind of model

Some notes on the possible under/overflow of the most common elementary functions

The purpose of this short note is not to describe when underflow or overflow must be signalled (it is quite clear that the rules are the same as for the basic arithmetic operations). We just want to show that for some of the most common functions and floating-point formats, in many cases, we can know in advance that the results will always lie in the range of the numbers that are representable by normal floating-point numbers, so that in these cases there is no need to worry about underflow or overflow. Note that when it is not the case, an implementation is still possible using a run-time test

Pleistocene hominins as a resource for carnivores. A c. 500,000-year-old human femur bearing tooth-marks in North Africa (Thomas Quarry I, Morocco)

In many Middle Pleistocene sites, the co-occurrence of hominins with carnivores, who both contributed to faunal accumulations, suggests competition for resources as well as for living spaces. Despite this, there is very little evidence of direct interaction between them to-date. Recently, a human femoral diaphysis has been recognized in South-West of Casablanca (Morocco), in the locality called Thomas Quarry I. This site is famous for its Middle Pleistocene fossil hominins considered representatives of Homo rhodesiensis. The bone was discovered in Unit 4 of the Grotte à Hominidés (GH), dated to c. 500 ky and was associated with Acheulean artefacts and a rich mammalian fauna. Anatomically, it fits well within the group of known early Middle Pleistocene Homo, but its chief point of interest is that the diaphyseal ends display numerous tooth marks showing that it had been consumed shortly after death by a large carnivore, probably a hyena. This bone represents the first evidence of consumption of human remains by carnivores in the cave. Whether predated or scavenged, this chewed femur indicates that humans were a resource for carnivores, underlining their close relationships during the Middle Pleistocene in Atlantic Morocco

On-The-Fly Range Reduction

In several cases, the input argument of an elementary function evaluation is given bit-serially, most significant bit first. We suggest a solution for performing the first step of the evaluation (namely, the range reduction) on the fly: the computation is overlapped with the reception of the input bits. This algorithm can be used for the trigonometric functions sin, cos, tan as well as for the exponential function.Il arrive que l’oprande dont on doit calculer une fonction élémentaire soit disponible chiffre après chiffre, en série, en commençant par les poids forts. Nous proposons une solution permettant d’effectuer la première phase de l’évaluation(la réduction d’argument)au vol: le calcul et la réception des chiffres d’entré se recouvrent. Cet algorithme peut être utilisé pour les fonctions trigonométriques sin, cos, tan ainsi que pour l'exponentiell

On the maximum relative error when computing integer powers by iterated multiplications in floating-point arithmetic

International audienceWe improve the usual relative error bound for the computation of x^n through iterated multiplications by x in binary floating-point arithmetic. The obtained error bound is only slightly better than the usual one, but it is simpler. We also discuss the more general problem of computing the product of n terms