IEEE Standard 754 Floating Point Numbers
Steve
Hollasch
/ Last update 2005-Feb-24
IEEE Standard 754 floating point is the most common representation today for real numbers
on computers, including Intel-based PC's, Macintoshes, and most Unix platforms. This article
gives a brief overview of IEEE floating point and its representation. Discussion of arithmetic
implementation may be found in the book mentioned at the bottom of this article.
What Are Floating Point Numbers?
There are several ways to represent real numbers on computers. Fixed point places a radix
point somewhere in the middle of the digits, and is equivalent to using integers that represent
portions of some unit. For example, one might represent 1/100ths of a unit; if you have four
decimal digits, you could represent 10.82, or 00.01. Another approach is to use rationals, and
represent every number as the ratio of two integers.
Floating-point representation - the most common solution - basically represents reals in
scientific notation. Scientific notation represents numbers as a base number and an exponent.
For example, 123.456 could be represented as 1.23456 × 10
2
. In hexadecimal, the number
123.abc might be represented as 1.23abc × 16
2
.
Floating-point solves a number of representation problems. Fixed-point has a fixed
window of representation, which limits it from representing very large or very small numbers.
Also, fixed-point is prone to a loss of precision when two large numbers are divided.
Floating-point, on the other hand, employs a sort of "sliding window" of precision
appropriate to the scale of the number. This allows it to represent numbers from
1,000,000,000,000 to 0.0000000000000001 with ease.
Storage Layout
IEEE floating point numbers have three basic components: the sign, the exponent, and the
mantissa. The mantissa is composed of the
fraction
and an implicit leading digit (explained
below). The exponent base (2) is implicit and need not be stored.
The following figure shows the layout for single (32-bit) and double (64-bit) precision
Page 1 of 8
IEEE Standard 754 Floating-Point
3/6/2009
mhtml:http://www.hlam.ece.ufl.edu/EEL4712/Labs/IEEEStandard754Floating-Point.mht

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