mirror of
https://github.com/janishutz/eth-summaries.git
synced 2026-03-14 23:10:03 +01:00
47 lines
2.2 KiB
TeX
47 lines
2.2 KiB
TeX
\subsubsection{Floating Point Representation}
|
|
Floating point numbers instead use the representation:
|
|
$$
|
|
a = \underbrace{(-1)^s}_\text{Sign} \cdot \underbrace{M}_\text{Mantissa} \cdot \underbrace{2^E}_\text{Exponent}
|
|
$$
|
|
|
|
Single precision and Double precision floating point numbers store the $3$ parameters in separate bit fields $s, e, m$:
|
|
|
|
\begin{center}
|
|
Single Precision:
|
|
\begin{tabular}{|c|c|c|}
|
|
\hline
|
|
$31$: Sign & $30-23$: Exponent & $22-0$: Mantissa \\
|
|
\hline
|
|
\end{tabular} \\
|
|
Bias: $127$, Exponent range: $[-126, 127]$
|
|
\end{center}
|
|
\begin{center}
|
|
Double Precision:
|
|
\begin{tabular}{|c|c|c|}
|
|
\hline
|
|
$63$: Sign & $62-52$: Exponent & $51-0$: Mantissa \\
|
|
\hline
|
|
\end{tabular}\\
|
|
Bias: $1023$, Exponent range: $[-1022, 1023]$
|
|
\end{center}
|
|
|
|
Most of the extra precision in $64$b floating point numbers is associated to the mantissa. Note how double precision is necessary to represent all $32$b signed Integers, and not all $64$b signed Integers can be represented in either format.
|
|
|
|
\newpage
|
|
|
|
The way these bitfields are interpretd \textit{differs} based on the exponent field $e$:
|
|
|
|
\begin{enumerate}
|
|
\item \textbf{Normalized Values}: Exponent bit field $e$ is neither all $1$s nor all $0$s.\\
|
|
In this case, $E$ is read in \textit{biased} form: $E = e - b$. The bias is $b=2^{k-1}-1$, where $k$ is the amount of bits reserved for $e$. This produces the exponent ranges $E \in [-(b-1), b]$.\\
|
|
The mantissa field $m$ is interpreted as $M = 0.m_{n-1}\ldots m_1 m_0 + 1$, where $n$ is the amount of bits reserved for $m$
|
|
\item \textbf{Denormalized Values}: Exponent bit field $e$ is all $0$s.\\
|
|
In this case, $E$ is read in \textit{biased} form $E = 1 - b$. (Instead of $E = e - b$)\\
|
|
The mantissa field $m$ is interpreted as $M = 0.m_{n-1}\ldots m_1 m_0$ (without adding $1$)
|
|
\item \textbf{Special Values}: Exponent bit field $e$ is all $1$s.\\
|
|
$m = 0$ represents infinitiy, which is signed using $s$.\\
|
|
$m \neq 0$ is \verb|NaN|, regardless of what is in $m$ or $s$.
|
|
\end{enumerate}
|
|
|
|
\content{Why is the Bias chosen this way?} It allows smooth transitions between normalized and denormalized values.
|