How to plot Inverse of W-lambert function with pgfplots?

Question

This is my effort for plotting w-lambert and its inverse, but I get an error:

        \begin{figure}[t]
            \centering
            \begin{subfigure}[scale = 1]{0.4\textwidth}
    %           \includegraphics[width=\textwidth]{lambert_a.jpg}
                \begin{tikzpicture}
        \begin{axis}[
            samples=1001,
            enlarge x limits=false,
            axis lines=middle,
            xmin=-3.1,
            ymax=1.4,
            ytick={-1, 1},
            xtick={-3,-2,-1, 1, 2},
            yticklabels={{$-1$}, {$1$}},
            yticklabel pos=right,
            width=10cm,
            xlabel = \(x\),
            xlabel style={xshift=0.5cm, yshift=-0.2cm},
            every y tick label/.style={
                anchor=near yticklabel opposite,
                xshift=0.2em,
            },
            ylabel = \(y(x)\),
            ylabel style={xshift=-0.5cm, yshift=0.6cm},
            ]
            
    %       \addplot [red!80!black, domain=-4.2:-1, line width=0.3mm, dashed] (x * exp(x), x);
            
            \addplot [green!60!black, domain=-3:1.1, line width=0.5mm] (x, x * exp(x));
    %       
            \draw[dotted] (-1,-0.367879441171442321595523770161460867445811131031767834507836801) -- 
            (0, -0.367879441171442321595523770161460867445811131031767834507836801);
            \node[right] at (0, -0.4){\normalsize{$-1/e$}};
        \end{axis}
    \end{tikzpicture}
    
                \caption{
                    $y=x e^{x}$}
                \label{fig:first}
            \end{subfigure}
            \hfill
            \begin{subfigure}[scale =1]{0.4\textwidth}
    %           \includegraphics[width=\textwidth]{lambert_b.jpg}
                \begin{tikzpicture}[scale=1,thick]
                    \begin{axis}[
                        samples=1001,
                        enlarge y limits=false,
                        axis lines=middle,
                        xmin=-1.1,
                        ymax=1.4,
                        ytick={-4, -3, -2, -1, 1},
                        xtick={-1, 1, 2, 3},
                        yticklabels={{$-4$}, {$-3$}, {$-2$}, {$-1$}, {$1$}},
                        yticklabel pos=right,
                        width=10cm,
                        xlabel = \(x\),
                        xlabel style={xshift=0.5cm, yshift=-0.2cm},
                        every y tick label/.style={
                            anchor=near yticklabel opposite,
                            xshift=0.2em,
                        },
                        ylabel = \(W(x)\),
                        ylabel style={xshift=-0.5cm, yshift=0.6cm},
                        ]
                        
                        \addplot [red!80!black, domain=-4.2:-1, line width=0.3mm, dashed] (x * exp(x), x);
                        
                        \addplot [blue!80!black, domain=-1:1.1, line width=0.3mm] (x * exp(x), x);
                        
                        \draw[dotted] (-0.367879441171442321595523770161460867445811131031767834507836801,-1) -- (-0.367879441171442321595523770161460867445811131031767834507836801, 0);
                        \node[above] at (-0.4, 0){\tiny{$-1/e$}};
                    \end{axis}
                \end{tikzpicture}
                \caption{
                    $W(x),\quad x \geq \frac{-1}{e}$
                }
                \label{fig:second}
            \end{subfigure}
            \caption{
                $W(x)$
    and
    its inverse
            }
            \label{fig:figures}
        \end{figure}
    \end{definition}
``´

Answer 1

Now I see what you got stuck: The error Dimension too large may appear when trying to plot (with pgfplots that uses implicit scaling) the graph of the inverse of 2 branches of the Lambert W-function (that is, f(x)=x*exp(x)) on the same axes with the Lambert W-function.

The followings are some ways I propose.

1st way: using geometric transformation; plain TikZ; explicit scaling if we want; can be applied for any path.

First rotate a piece of graph (considering it as a TikZ's path) an angle 45 degrees counter-clockwise to the y-axe [rotate=45], then reflect to the y-axe [xscale=-1], and finally rotate back [rotate=-45]. Note that in TikZ, options on a path is acted from the right to the left, that is

[rotate=-45,xscale=-1,rotate=45]

Full code

\documentclass[tikz,border=5mm]{standalone}
\begin{document}
\begin{tikzpicture}[declare function={f(\x)=\x*exp(\x);}]
\def\pathA{plot[domain=-1:-3.5] ({f(\x)},\x)}
\def\pathB{plot[domain=-1:1.1] ({f(\x)},\x)}

% decorations
\draw[orange!50] (-3,-3)--(3,3);
\draw[->] (0,-3.5)--(0,3.5) node[below left]{$y$};
\draw[->] (-3.5,0)--(3.5,0) node[below right]{$x$};
\draw[dashed,orange,nodes={black}] 
(1,0) node[below]{$1$}--(1,{exp(1)}) 
(0,{exp(1)}) node[left]{$e$}--(1,{exp(1)})
(0,1) node[left]{$1$}--({exp(1)},1) 
({exp(1)},0) node[below]{$e$}--({exp(1)},1)
(0,-1) node[right]{$-1$}--({-1/exp(1)},-1) 
({-1/exp(1)},0) node[above,scale=.8]{$-\frac{1}{e}$}--({-1/exp(1)},-1)
(-1,0) node[above]{$-1$}--(-1,{-1/exp(1)}) 
(0,{-1/exp(1)}) node[right,scale=.8]{$-\frac{1}{e}$}--(-1,{-1/exp(1)})
;

% 2 banches of the Lambert W-function
\draw[red,densely dashed,thick] \pathA node[above left]{$W_{-1}(x)$};
\draw[blue,densely dashed,thick] \pathB node[above]{$W_0(x)$};

% 2 banches of the inverse of the Lambert W-function
% It is exactly the graph of f(\x)=\x*exp(\x)
\draw[red,thick,rotate=-45,xscale=-1,rotate=45] \pathA +(-.5,0) node[right]{$y=x e^x$};
\draw[blue,thick,rotate=-45,xscale=-1,rotate=45] \pathB node[right]{$y=x e^x$};

% turing points
\fill (-1,{-1/exp(1)}) circle(1.5pt);
\fill ({-1/exp(1)},-1) circle(1.5pt);
\end{tikzpicture}
\end{document}

2nd way: just using plot of plain TikZ: plot ({f(\x)},\x) or plot (\x,{f(\x)}); explicit scaling if we want; gives the same figure.

\documentclass[tikz,border=5mm]{standalone}
\begin{document}
\begin{tikzpicture}[declare function={f(\x)=\x*exp(\x);}]
% decorations
\draw[orange!50] (-3,-3)--(3,3);
\draw[->] (0,-3.5)--(0,3.5) node[below left]{$y$};
\draw[->] (-3.5,0)--(3.5,0) node[below right]{$x$};
\draw[dashed,orange,nodes={black}] 
(1,0) node[below]{$1$}--(1,{exp(1)}) 
(0,{exp(1)}) node[left]{$e$}--(1,{exp(1)})
(0,1) node[left]{$1$}--({exp(1)},1) 
({exp(1)},0) node[below]{$e$}--({exp(1)},1)
(0,-1) node[right]{$-1$}--({-1/exp(1)},-1) 
({-1/exp(1)},0) node[above,scale=.8]{$-\frac{1}{e}$}--({-1/exp(1)},-1)
(-1,0) node[above]{$-1$}--(-1,{-1/exp(1)}) 
(0,{-1/exp(1)}) node[right,scale=.8]{$-\frac{1}{e}$}--(-1,{-1/exp(1)})
;

% 2 banches of the Lambert W-function
\draw[red,densely dashed,thick] plot[domain=-1:-3.5] ({f(\x)},\x) node[above left]{$W_{-1}(x)$};
\draw[blue,densely dashed,thick] plot[domain=-1:1.1] ({f(\x)},\x) node[above]{$W_0(x)$};

% 2 banches of the inverse of the Lambert W-function
% It is exactly the graph of f(\x)=\x*exp(\x)
\draw[red,thick] plot[domain=-1:-3.5] (\x,{f(\x)}) +(0,-.5) node[right]{$y=x e^x$};
\draw[blue,thick] plot[domain=-1:1.1] (\x,{f(\x)}) node[right]{$y=x e^x$};

% turing points
\fill (-1,{-1/exp(1)}) circle(1.5pt);
\fill ({-1/exp(1)},-1) circle(1.5pt);
\end{tikzpicture}
\end{document}

3nd way: is a better way with plain Asymptote (also see this answer); we can use either explicit scaling (unitsize(1cm,5mm);) or implicit scaling (size(8cm);), or both, as we wish.

unitsize(1cm);
// size(8cm);    // for auto (implicit) scaling
import graph;
import gsl;      // for Lambert W-functions
real f(real x){return x*exp(x);}
pen penlegend=orange;
pen penLeft=red+.8pt;
pen penRight=blue+.8pt;
pen mydashed=linetype(new real[]{2,2});

real a=3.5;
draw((-3,-3)--(3,3),penlegend+white);
draw(Label("$y$",EndPoint,align=2SW),(0,-a)--(0,a),Arrow(TeXHead));
draw(Label("$x$",EndPoint,align=2SE),(-a,0)--(a,0),Arrow(TeXHead));

pair A=(1,f(1)), B=(-1,f(-1));
draw((A.x,0)--A^^(0,A.y)--A^^(B.x,0)--B^^(0,B.y)--B,mydashed+penlegend);
label("$1$",(A.x,0),S);  label("$e$",(0,A.y),W);
label("$-1$",(B.x,0),N); label(scale(.8)*"$-\frac{1}{e}$",(0,B.y),E);

pair Aw=(f(1),1), Bw=(f(-1),-1);
draw((Aw.x,0)--Aw^^(0,Aw.y)--Aw^^(Bw.x,0)--Bw^^(0,Bw.y)--Bw,mydashed+penlegend);
label("$e$",(Aw.x,0),S);  label("$1$",(0,Aw.y),W);
label("$-1$",(0,Bw.y),E); label(scale(.8)*"$-\frac{1}{e}$",(Bw.x,0),N);

path fLeft=graph(f,-a,-1);
path fRight=graph(f,-1,1.1);
path invfRight=graph(W0,f(-1),a);
path invfLeft=graph(Wm1,f(-1),f(-a));

draw(Label("$y=x e^x$",EndPoint,align=E),fRight,penRight);
draw(Label("$y=x e^x$",BeginPoint,align=3SE),fLeft,penLeft);
draw(Label("$W_{-1}(x)$",EndPoint,align=NW),invfLeft,penLeft+mydashed);
draw(Label("$W_0(x)$",EndPoint,align=NW),invfRight,penRight+mydashed);

dot(B^^Bw);