HP-UX Floating-Point Guide

22 Chapter 1
Introduction
Overview of Floating-Point Principles
Overview of Floating-Point Principles
In the context of computer programming, the term floating-point refers
to the ways in which modern computer systems represent real numbers
and perform real arithmetic. Computers use special representations for
floating-point numbers. They also have special rules for performing
floating-point arithmetic that differ from the rules for performing integer
arithmetic. Usually, a computer has special hardware for performing
floating-point calculations at a higher speed than would be possible
using the computer’s integer-oriented hardware.
In all modern computer systems, representations of real numbers are
inherently inexact. There are an infinite number of real numbers, and a
digital computer can represent only a finite subset of them.
When you write a program that attempts to generate an unrepresentable
value, the computer approximates the value by choosing a representable
value close to the one you attempted to generate. Data that is input into
a computer in floating-point format is almost always approximate, and
the calculations performed by the computer are usually approximations
of the intended mathematical operations; therefore, the results you
receive from a mathematical computation are also usually
approximations.
The approximate nature of floating-point arithmetic has several
important ramifications:
• Results from floating-point calculations are almost never exactly
equal to the corresponding mathematical value.
• Results from a particular calculation may vary slightly from one
computer system to another, and all may be valid. However, when the
computer systems conform to the same standard, the amount of
variation is drastically reduced.
• Incorrect results are not necessarily caused by programming errors in
the traditional sense. Correcting the problems may require an
understanding of the floating-point approximation techniques used
by the computer system executing the program.
The types of incorrect results and unexpected errors that floating-point
applications sometimes generate can be very difficult to interpret if you
do not understand how your computer performs floating-point