\subsection{$\mathbf{\hat{~}}$}
\label{labpower}
\noindent Name: \textbf{$\mathbf{\hat{~}}$}\\
\phantom{aaa}power function\\[0.2cm]
\noindent Library names:\\
\verb|   sollya_obj_t sollya_lib_pow(sollya_obj_t, sollya_obj_t)|\\
\verb|   sollya_obj_t sollya_lib_build_function_pow(sollya_obj_t, sollya_obj_t)|\\
\verb|   #define SOLLYA_POW(x,y) sollya_lib_build_function_pow((x), (y))|\\[0.2cm]
\noindent Usage: 
\begin{center}
\emph{function1} \textbf{$\mathbf{\hat{~}}$} \emph{function2} : (\textsf{function}, \textsf{function}) $\rightarrow$ \textsf{function}\\
\emph{interval1} \textbf{$\mathbf{\hat{~}}$} \emph{interval2} : (\textsf{range}, \textsf{range}) $\rightarrow$ \textsf{range}\\
\emph{interval1} \textbf{$\mathbf{\hat{~}}$} \emph{constant} : (\textsf{range}, \textsf{constant}) $\rightarrow$ \textsf{range}\\
\emph{interval1} \textbf{$\mathbf{\hat{~}}$} \emph{constant} : (\textsf{constant}, \textsf{range}) $\rightarrow$ \textsf{range}\\
\end{center}
Parameters: 
\begin{itemize}
\item \emph{function1} and \emph{function2} represent functions
\item \emph{interval1} and \emph{interval2} represent intervals (ranges)
\item \emph{constant} represents a constant or constant expression
\end{itemize}
\noindent Description: \begin{itemize}

\item \textbf{$\mathbf{\hat{~}}$} represents the power (function) on reals. 
   The expression \emph{function1} \textbf{$\mathbf{\hat{~}}$} \emph{function2} stands for
   the function composed of the power function and the two
   functions \emph{function1} and \emph{function2}, where \emph{function1} is
   the base and \emph{function2} the exponent.
   If \emph{function2} is a constant integer, \textbf{$\mathbf{\hat{~}}$} is defined
   on negative values of \emph{function1}. Otherwise \textbf{$\mathbf{\hat{~}}$}
   is defined as $e^{y \cdot \ln x}$.

\item Note that whenever several \textbf{$\mathbf{\hat{~}}$} are composed, the priority goes
   to the last \textbf{$\mathbf{\hat{~}}$}. This corresponds to the natural way of
   thinking when a tower of powers is written on a paper.
   Thus, \verb|2^3^5| is read as $2^{3^5}$ and is interpreted as $2^{(3^5)}$.

\item \textbf{$\mathbf{\hat{~}}$} can be used for interval arithmetic on intervals
   (ranges). \textbf{$\mathbf{\hat{~}}$} will evaluate to an interval that safely
   encompasses all images of the power function with arguments
   varying in the given intervals. If the intervals given contain points
   where the power function is not defined, infinities and NaNs will be
   produced in the output interval.  Any combination of intervals with
   intervals or constants (resp. constant expressions) is
   supported. However, it is not possible to represent families of
   functions using an interval as one argument and a function (varying in
   the free variable) as the other one.
\end{itemize}
\noindent Example 1: 
\begin{center}\begin{minipage}{15cm}\begin{Verbatim}[frame=single]
> 5 ^ 2;
25
\end{Verbatim}
\end{minipage}\end{center}
\noindent Example 2: 
\begin{center}\begin{minipage}{15cm}\begin{Verbatim}[frame=single]
> x ^ 2;
x^2
\end{Verbatim}
\end{minipage}\end{center}
\noindent Example 3: 
\begin{center}\begin{minipage}{15cm}\begin{Verbatim}[frame=single]
> 3 ^ (-5);
4.1152263374485596707818930041152263374485596707819e-3
\end{Verbatim}
\end{minipage}\end{center}
\noindent Example 4: 
\begin{center}\begin{minipage}{15cm}\begin{Verbatim}[frame=single]
> (-3) ^ (-2.5);
NaN
\end{Verbatim}
\end{minipage}\end{center}
\noindent Example 5: 
\begin{center}\begin{minipage}{15cm}\begin{Verbatim}[frame=single]
> diff(sin(x) ^ exp(x));
sin(x)^exp(x) * ((cos(x) * exp(x)) / sin(x) + exp(x) * log(sin(x)))
\end{Verbatim}
\end{minipage}\end{center}
\noindent Example 6: 
\begin{center}\begin{minipage}{15cm}\begin{Verbatim}[frame=single]
> 2^3^5;
14134776518227074636666380005943348126619871175004951664972849610340958208
> (2^3)^5;
32768
> 2^(3^5);
14134776518227074636666380005943348126619871175004951664972849610340958208
\end{Verbatim}
\end{minipage}\end{center}
\noindent Example 7: 
\begin{center}\begin{minipage}{15cm}\begin{Verbatim}[frame=single]
> [1;2] ^ [3;4];
[1;16.000000000000000000000000000000000000000000000001]
> [1;2] ^ 17;
[1;131072]
> 13 ^ [-4;17];
[3.501277966457757081334687160813696999404782745702e-5;8650415919381337933]
\end{Verbatim}
\end{minipage}\end{center}
See also: \textbf{$+$} (\ref{labplus}), \textbf{$-$} (\ref{labminus}), \textbf{$*$} (\ref{labmult}), \textbf{/} (\ref{labdivide})
