How can I add third point in plot?

I seems to work when you replace /2 by *0.5 -- for whatever reason.

% used PGFPlots v1.16
\documentclass[border=5pt]{standalone}
\usepackage{xparse}
\usepackage{pgfplots}
    \usetikzlibrary{matrix}
    \pgfplotsset{
        point legend/.style={},
        point 1/.style={anchor=south},
        point 2/.style={anchor=south},
        point 3/.style={anchor=south},                  % <-- added
        azetinaplot/.style={
            width=10cm,
            height=8cm,
            axis lines=middle,
            xlabel=$x$,
            ylabel=$y$,
            enlarge y limits,
            clip=false,
        },
    }
    \NewDocumentCommand{\derivative}{O{f} m m m m O{rel axis cs:1,1}}{
        \tikzset{declare function={#1(\x)=#2;}}
        %
        \addplot [thick, red, latex-latex] {#1(x)}
            node [anchor=west] {#3}
        ;
        \addplot [black, mark=*] coordinates {
            (#4,{#1(#4)})
            (#5,{#1(#5)})
        }
            node [pos=0,/pgfplots/point 1] {$P_1$}
            node [pos=1,/pgfplots/point 2] {$P_2$}
        ;
        % we need another `\addplot' for the third point to avoid drawing a line
        % also to point 2
        \addplot [black, mark=*, forget plot] coordinates {
            ({0.5*(#5+#4)},{#1(0.5*(#5+#4))})           % <-- repaced `/2' by `*0.5'
        }
            node [pos=0.5,/pgfplots/point 3] {$P_3$}    % <-- added
        ;
        \pgfplotsextra{
            \pgfmathsetmacro\first{#1(#4)}
            \pgfmathsetmacro\second{#1(#5)}
            \pgfmathsetmacro\xdiff{#5-#4}
            \pgfmathsetmacro\ydiff{#1(#5)-#1(#4)}
            %
            \draw (axis cs:#4,\first) -| (axis cs:#5,\second);
            \draw [|-|,yshift=-2ex] (axis cs:#4,\first)
                -- node [inner sep=1pt,fill=white] {\pgfmathprintnumber{\xdiff}}
                    (axis cs:#5,\first);
            \draw [|-|,xshift=+2ex] (axis cs:#5,\first)
                -- node [inner sep=1pt,fill=white] {\pgfmathprintnumber{\ydiff}}
                    (axis cs:#5,\second);
            %
            \matrix at (#6) [matrix of math nodes,/pgfplots/point legend] {
                P_1=(#4\,,\,\pgfmathparse{#1(#4)}\pgfmathprintnumber{\pgfmathresult})\\
                P_2=(#5\,,\,\pgfmathparse{#1(#5)}\pgfmathprintnumber{\pgfmathresult})\\
            };
        }
    }
\begin{document}
\begin{tikzpicture}
    \begin{axis}[
        azetinaplot,
        domain=-20:370, samples=100,
        xmin=-20, xmax=370,
        point 1/.append style={anchor=south east},
        point 2/.append style={anchor=east},
    ]
        \derivative{sin(\x)}
            {$f(x)=\sin(x)$}
            {290}{340}
            [axis cs:150,-1] % position for the legend

        % above we used the default function name f
        % here we use the first optional argument to give the function the name g instead
        \derivative[g]{cos(\x)+2}
            {$f(x)=\cos(x)$}
            {200}{300}
            [axis cs:170,2.5] % position for the legend

    \end{axis}
\end{tikzpicture}
\end{document}