Related papers: On the precision attainable with various floating-…
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…
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…
An improvement on precision of recursive function simulation in IEEE floating point standard is presented. It is shown that the average of rounding towards negative infinite and rounding towards positive infinite yields a better result than…
Recently we introduced a class of number representations denoted RN-representations, allowing an un-biased rounding-to-nearest to take place by a simple truncation. In this paper we briefly review the binary fixed-point representation in an…
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…
Scientific computing applications, such as computational fluid dynamics and climate modeling, typically rely on 64-bit double-precision floating-point operations, which are extremely costly in terms of computation, memory, and energy. While…
Computer programs may go wrong due to exceptional behaviors, out-of-bound array accesses, or simply coding errors. Thus, they cannot be blindly trusted. Scientific computing programs make no exception in that respect, and even bring…
Presented here are algorithms for converting between (decimal) scientific-notation and (binary) IEEE-754 double-precision floating-point numbers. By employing a rounding integer quotient operation these algorithms are much simpler than…
Using exact computer arithmetic, it is possible to determine the (exact) solution of a numerical model without rounding error. For such purposes, a corresponding system of equations should be exactly defined, either directly or by…
We introduce new rounding methods to improve the accuracy of finite precision quantum arithmetic. These quantum rounding methods are applicable when multiple samples are being taken from a quantum program. We show how to use multiple…
If several independent algorithms for a computer-calculated quantity exist, then one can expect their results (which differ because of numerical errors) to follow approximately Gaussian distribution. The mean of this distribution,…
When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…
With disks and networks providing gigabytes per second, parsing decimal numbers from strings becomes a bottleneck. We consider the problem of parsing decimal numbers to the nearest binary floating-point value. The general problem requires…
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 consider the error due to a single bit-flip in a floating point number. We assume IEEE 754 double precision arithmetic, which encodes binary floating point numbers in a 64-bit word. We assume that the bit-flip happens randomly so it has…
The error function of real argument can be uniformly approximated to a given accuracy by a single closed-form expression for the whole variable range either in terms of addition, multiplication, division, and square root operations only, or…
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…
Large-scale numerical computations make increasing use of low-precision (LP) floating point formats and mixed precision arithmetic, which can be enhanced by the technique of stochastic rounding (SR), that is, rounding an intermediate…
The use of low-precision fixed-point arithmetic along with stochastic rounding has been proposed as a promising alternative to the commonly used 32-bit floating point arithmetic to enhance training neural networks training in terms of…
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…