A radix \(\beta\) floating point number \(x\) is of the form \(m\beta{e}\), where \(|m|<\beta\) is called the significand and \(e\) is the exponent.
A floating point number is characterized by four integers:
- A radix (base) \(\beta\ge2\)
- A precision \(p\ge2\)
- \(e_{\textit{min}}\): The smallest possible exponent
- \(e_{\textit{max}}\): The largest possible exponent
In the IEEE 754, \(e_{\textit{min}}=1-e_{\textit{max}}\)
A finite floating point number \(x\) is such that \(x=M\beta^{e-p+1}\) where:
- \(|M|\le\beta^{p}-1\). It is called the integral significand.
- Note that this just means that \(|M|\) can be represented exactly in base \(\beta\) within \(p\) characters.
- \(e_{\textit{min}}\le e\le e_{\textit{max}}\)
The tuple \((M,e)\) need not be unique. The set of equivalent \((M,e)\) is called a cohort.
The number \(\beta^{e-p+1}\) is called the quantum of \(x\).
There is another representation:
\begin{equation*}
x=(-1)^{s}m\beta^{e}
\end{equation*}
- \(m=|M|\beta^{1-p}\) and is called the normal significand (or just the significand). It has one digit before the radix point.
Let \(x\) be any number. The infinitely precise significand of \(x\) in \(\beta\) is the number:
\begin{equation*}
\frac{x}{\beta^{\lfloor\log_{\beta}|x|\rfloor}}
\end{equation*}