When to evaluate a custom math function in a `tikzpicture` environment using `pgfplots`

Question

I want to draw a custom function using pgfplots, and add an annotation node:

\begin{tikzpicture}[
    >=stealth,
    evaluate={
        function f(\x) {
            return cos(2*\x);
        };
    }
    ]
    \begin{axis}[
        axis lines=middle,
        xtick=\empty,
        ytick=\empty,
        xmin=-200,
        xmax=200,
        ymin=-1.5,
        ymax=1.5
        ]
        \addplot[
            domain=-180:180,
            samples=361
            ]{f(x)};
        % \node[coordinate,pin=above right:{$\cos 2x$}] at (axis cs:45,{f(45)}) {};
        \node[coordinate,pin=above right:{$\cos 2x$}] at (axis cs:45,0) {};
    \end{axis}
\end{tikzpicture}

The result is:

My question is about the commented line above, which cannot be compiled correctly. The reason, IMHO, is probably related to the function evaluation process (say, function f above). I am not quite understand the mechanism of it, so is there any way to solve this problem? An extra explanation is quite appreciated!

BTW: Instead of evaluate, I have also tried declare function, which works fine in this scenario; yet the actual custom function involves a complicated expression (defined in a recursive manner), which might not be handled by declare function.

Answer 1

Your commented code does work provided you load the math library.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{math}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{tikzpicture}[
    >=stealth,
    evaluate={
        function f(\x) {
            return cos(2*\x);
        };
    }
    ]
    \begin{axis}[
        axis lines=middle,
        xtick=\empty,
        ytick=\empty,
        xmin=-200,
        xmax=200,
        ymin=-1.5,
        ymax=1.5
        ]
        \addplot[
            domain=-180:180,
            samples=361
            ]{f(x)};
        \node[coordinate,pin=above right:{$\cos 2x$}] at (axis cs:45,{f(45)}) {};
        %\node[coordinate,pin=above right:{$\cos 2x$}] at (axis cs:45,0) {};
    \end{axis}
\end{tikzpicture}
\end{document}

Note, however, that it might be nontrivial to marry math to the fpu library, which pgfplots uses in its plots, in full glory. That is, if you want to play with recursive functions that work with integers, you may need to locally switch off fpu.

As for the question in the comments: there is a problem with xelatex or lualatex running on a beamer document including this code and using the metropolis theme. A probably related problem is discussed here. Among other things, metropolis sets the version of pgfplots to the value 1.9. Here is a temporary workaround for that constellation. You need to choose a different variable for the function (which is of course just a placeholder, but this definitely should not be necessary). That is, use

evaluate={
    function f(\t) {
        return cos(2*\t);
    };
}

where \x got replaced by \t. Full code:

\documentclass{beamer}
\usetheme{metropolis} 
\usepackage{pgfplots}
\usetikzlibrary{math}
\begin{document}
\pgfplotsset{compat=1.17}

\begin{frame}[t,fragile]
\frametitle{}
\begin{tikzpicture}[
    >=stealth,
    evaluate={
        function f(\t) {
            return cos(2*\t);
        };
    }
    ]
    \begin{axis}[
        axis lines=middle,
        xtick=\empty,
        ytick=\empty,
        xmin=-200,
        xmax=200,
        ymin=-1.5,
        ymax=1.5
        ]
        \addplot[%variable=\t,
            domain=-180:180,
            samples=361
            ]{f(x)};
        \node[coordinate,pin=above right:{$\cos 2x$}] at (axis cs:45,{f(45)}) {};
        %\node[coordinate,pin=above right:{$\cos 2x$}] at (axis cs:45,0) {};
    \end{axis}
\end{tikzpicture}
\end{frame}

\end{document}