Related papers: Accuracy of Mathematical Functions in Julia
We present a detailed study of roundoff errors in probabilistic floating-point computations. We derive closed-form expressions for the distribution of roundoff errors associated with a random variable, and we prove that roundoff errors are…
Recent renewed interest in optimizing and analyzing floating-point programs has lead to a diverse array of new tools for numerical programs. These tools are often complementary, each focusing on a distinct aspect of numerical programming.…
Alternative expressions for calculating the oblate spheroidal radial functions of both kinds R1ml and R2ml are shown to provide accurate values over very large parameter ranges using 64 bit arithmetic, even where the traditional expressions…
Errors in floating-point programs can lead to severe consequences, particularly in critical domains such as military, aerospace, and financial systems, making their repair a crucial research problem. In practice, some errors can be fixed…
Due to the limited number of bits in floating-point or fixed-point arithmetic, rounding is a necessary step in many computations. Although rounding methods can be tailored for different applications, round-off errors are generally…
Floating-point computations are quickly finding their way in the design of safety- and mission-critical systems, despite the fact that designing floating-point algorithms is significantly more difficult than designing integer algorithms.…
Verification of C++ programs has seen considerable progress in several areas, but not for programs that use these languages' mathematical libraries. The reason is that all libraries in widespread use come with no guarantees about the…
The vast use of computers on scientific numerical computation makes the awareness of the limited precision that these machines are able to provide us an essential matter. A limited and insufficient precision allied to the truncation and…
Our RLibm project generates a single implementation for an elementary function that produces correctly rounded results for multiple rounding modes and representations with up to 32-bits. They are appealing for developing fast reference…
In this work, approximate eight-bit floating-point operations performed using simple integer operations is discussed. For two-bit mantissa formats, faithful rounding can always be obtained for the considered operations. For all operations,…
The Mathieu functions are used to solve analytically some problems in elliptical cylinder coordinates. A computational toolbox was implemented in Matlab. Since the notation and normalization for Mathieu functions vary in the literature, we…
Modern computer architectures support low-precision arithmetic, which present opportunities for the adoption of mixed-precision algorithms to achieve high computational throughput and reduce energy consumption. As a growing number of…
This article introduces new acceleration methods for fixed-point iterations. Extrapolations are computed using two or three mappings alternately and a new type of step length is proposed with good properties for nonlinear applications. The…
IIn computational geometry, the construction of essential primitives like convex hulls, Voronoi diagrams and Delaunay triangulations require the evaluation of the signs of determinants, which are sums of products. The same signs are needed…
Many algorithms feature an iterative loop that converges to the result of interest. The numerical operations in such algorithms are generally implemented using finite-precision arithmetic, either fixed- or floating-point, most of which…
We introduce data structures and algorithms to count numerical inaccuracies arising from usage of floating numbers described in IEEE 754. Here we describe how to estimate precision for some collection of functions most commonly used for…
This paper provides a bound on the number of numeric operations (fixed or floating point) that can safely be performed before accuracy is lost. This work has important implications for control systems with safety-critical software, as these…
Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were…
We introduce the Scheduling.jl Julia package, which is intended for collaboratively conducting scheduling research and for sharing implementations of algorithms. It provides the fundamental building blocks for implementing scheduling…
In various applications, computers are required to compute approximations to univariate elementary and special functions such as $\exp$ and $\arctan$ to modest accuracy. This paper proposes a new heuristic for automating the design of such…