Related papers: Floating point numbers are real numbers

Many algorithms, especially in the field of computational geometry, are based on the premise that arithmetic operations are performed exactly. Real machines are based on inexact floating-point arithmetic. Various number types have been…

We guide the reader on a journey through mathematical modeling and numerical analysis, emphasizing the crucial interplay of both disciplines. Targeting undergraduate students with basic knowledge in dynamical systems and numerical methods…

Analog computers can be revived as a feasible technology platform for low precision, energy efficient and fast computing. We justify this statement by measuring the performance of a modern analog computer and comparing it with that of…

We provide tools to help automate the error analysis of algorithms that evaluate simple functions over the floating-point numbers. The aim is to obtain tight relative error bounds for these algorithms, expressed as a function of the unit…

Numerical approximate computation can solve large and complex problems fast. It has the advantage of high efficiency. However it only gives approximate results, whereas we need exact results in many fields. There is a gap between…

Since the early twentieth century, it has been understood that mathematical definitions and proofs can be represented in formal systems systems with precise grammars and rules of use. Building on such foundations, computational proof…

Finite-precision floating point arithmetic unavoidably introduces rounding errors which are traditionally bounded using a worst-case analysis. However, worst-case analysis might be overly conservative because worst-case errors can be…

We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the precision. We present a systematic recurrence for a martingale…

Probabilistic model checking computes probabilities and expected values related to designated behaviours of interest in Markov models. As a formal verification approach, it is applied to critical systems; thus we trust that probabilistic…

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses…

If the non-zero finite floating-point numbers are interpreted as point intervals, then the effect of rounding can be interpreted as computing one of the bounds of the result according to interval arithmetic. We give an interval…

I present two new methods for exactly summing a set of floating-point numbers, and then correctly rounding to the nearest floating-point number. Higher accuracy than simple summation (rounding after each addition) is important in many…

We extend the Stainless deductive verifier with floating-point support, providing the first automated verification support for floating-point numbers for a subset of Scala that includes polymorphism, recursion and higher-order functions. We…

This paper aims to introduce high school students to the intriguing world of continued fractions, a mathematical concept that provides a unique representation of numbers. The study focuses on the exploration and development of the…

The recent hardware trend towards reduced precision computing has reignited the interest in numerical techniques that can be used to enhance the accuracy of floating point operations beyond what is natively supported for basic arithmetic…

Modern programmable digital signal processing relies on floating-point numbers for their ease of use. Fixed-point number formats have the potential to save resources and improve execution time, but realising this potential burdens the…

Continuous models used in physics and other areas of mathematics applications become discrete when they are computerized, e.g., utilized for computations. Besides, computers are controlling processes in discrete spaces, such as films and…

In recent years, half precision floating-point arithmetic has gained wide support in hardware and software stack thanks to the advance of artificial intelligence and machine learning applications. Operating at half precision can…

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.…

Programs with floating-point computations are often derived from mathematical models or designed with the semantics of the real numbers in mind. However, for a given input, the computed path with floating-point numbers may differ from the…