Arithmetic Operations Beyond Floating Point Number Precision

TL;DR

This paper introduces an undergraduate project on arbitrary precision arithmetic using Fortran, demonstrating how to perform calculations beyond standard floating point limits, with applications in scientific computing and digital signal processing.

Contribution

It presents a practical implementation of arbitrary precision arithmetic in Fortran, illustrating how to handle numbers exceeding machine precision for scientific and engineering applications.

Findings

Demonstrates calculations beyond floating point limits using arrays in Fortran

Shows applications in digital signal processing hardware and firmware

Highlights importance for high-precision scientific computing

Abstract

In basic computational physics classes, students often raise the question of how to compute a number that exceeds the numerical limit of the machine. While technique of avoiding overflow/underflow has practical application in the electrical and electronics engineering industries, it is not commonly utilized in scientific computing, because scientific notation is adequate in most cases. We present an undergraduate project that deals with such calculations beyond a machine's numerical limit, known as arbitrary precision arithmetic. The assignment asks students to investigate the approach of calculating the exact value of a large number beyond the floating point number precision, using the basic scientific programming language Fortran. The basic concept is to utilize arrays to decompose the number and allocate finite memory. Examples of the successive multiplication of even number and the multiplication and division of two overflowing floats are presented. The multiple precision scheme has been applied to hardware and firmware design for digital signal processing (DSP) systems, and is gaining importance to scientific computing. Such basic arithmetic operations can be integrated to solve advanced mathematical problems to almost arbitrarily-high precision that is limited by the memory of the host machine.