We analyze two fast and accurate algorithms recently presented by Borges for computing$x^{-1/2}$in binary floating-point arithmetic (assuming that efficient and correctly-rounded FMA and square root are available). The first algorithm is based on the Newton-Raphson iteration, and the second one uses an order-3 iteration. We give attainable relative-error bounds for these two algorithms, build counterexamples showing that in very rare cases they do not provide a correctly-rounded result, and characterize precisely when such failures happen in IEEE 754 binary32 and binary64 arithmetics. We then give a generic (i.e., precision-independent) algorithm that always returns a correctly-rounded result, and show how it can be simplified and made more efficient in the important cases of binary32 and binary64.

We develop fast and accurate algorithms for evaluating √x 2 +y 2 for two floating-point numbers x and y . Library functions that perform this computation are generally named hypot(x,y). We compare five approaches that we will develop in this article to the current resident library function that is delivered with Julia 1.1 and to the code that has been distributed with the C math library for decades. We will investigate the accuracy of our algorithms by simulation.

We examine the histogram method for estimating the parameters associated with a Markov random field. This method relies on the estimation of the local interaction sums from histogram data. We derive an estimator for these quantities that is optimal in a well-defined sense. Furthermore, we show that the final step of the histogram method, the solution of a least-squares problem, can be done substantially faster than one might expect if no equation culling is used. We also examine the use of weighted least-squares and see that this seems to lead to better estimates even with small amounts of data.

The complexity of computer graphics illumination models and the associated need to find ways of reducing evaluation time has led to the use of two methods for simplifying the spectral data needed for an exact solution. The first method, where spectral data is sampled at a number of discrete points, has been extensively investigated and bounds for the error are known. Unfortunately, the second method, where spectral data is replaced with tristimulus values (such as RGB values), is very little understood even though it is widely used. In this paper we examine the error incurred by the use of this method by investigating the problem of approximating the tristimulus coordinates of light reflected from a surface from those of the source and the surface. A variation on a well known and widely used approximation is presented. This variation used the XYZ primaries which have unique properties that yield straightforward analytic bounds for the approximation error. This analysis is important because it gives a sound mathematical footing to the widely used method of trichromatic approximation. The error bounds will give some insights into the factors that affect accuracy and will indicate why this method often works quite well in practice.