Interactive Educational Modules in
Scientific Computing

Floating-Point Systems

This module graphically illustrates the finite, discrete nature of floating-point number systems. A floating-point number system is characterized by four integer parameters: the base β, precision p, lower exponent limit L, and upper exponent limit U. The total number of normalized floating-point numbers in such a system is 2 ( β − 1) β ^p−1 (U − L + 1) + 1. The smallest positive normalized floating-point number, called the underflow level, is given by UFL = β^L, and the largest floating-point number, called the overflow level, is given by OFL = β^U+1 (1 − β ^{− p}).

The user selects values for the base, precision, and lower and upper limits for the exponent range. The machine numbers in the resulting floating-point number system are indicated by tick marks on the real number line. The largest and smallest positive machine numbers, as well as the total number of machine numbers, are also printed. Clicking the mouse anywhere on the number line highlights in red the corresponding rounded value, and both the selected value and rounded result are printed below.

Reference: Michael T. Heath, Scientific Computing, An Introductory Survey, 2nd edition, McGraw-Hill, New York, 2002. See Section 1.3, especially Example 1.9 and Figure 1.3.

Developers: Nicholas Exner and Michael Heath