Definition:Floating-Point Representation/Computer Implementation

Floating-Point Representation: Computer Implementation

$x = f \times \beta^e$

is subject to the following considerations.

Both the mantissa $f$ and the exponent $e$ have a limited range:

$f$ is a number with $t$ digits base $\beta$

$e$ lies in an interval $\closedint L U$.

In such a system there is a finite set of numbers which can be so implemented, and $x$ can be written as:

$\ds x$

$=$

$\ds \pm \sqbrk {0.f_1 f_2 \ldots f_t} \times \beta^e$

$\ds $

$=$

$\ds \pm \paren {\dfrac {f_1} \beta + \dfrac {f_2} {\beta^2} + \cdots + \dfrac {f_t} {\beta^t} } \times \beta^e$

where each digit $f_i$ satisfies $0 \le f_i \le \beta - 1$.

On most computers, $\beta = 2$.

$x = f \times \beta^e$

such that:

$x = \pm \sqbrk {0.f_1 f_2 \ldots f_t} \times \beta^e$

This representation is said to be normalized if and only if $f_1 \ne 0$.