Problems with plotting planar functions with Tikz

Question

I tried to plot this graph using tikz and pgfplots. The code compiled just fine, but I got a peculiar result that I can't explain. My code.

\documentclass{article}

\usepackage{amsmath}
\usepackage{tikz}
\usepackage{pgfplots}

\begin{document}

\begin{tikzpicture}
\begin{axis}[xlabel=$x$,ylabel=$y$,
xmin=-6,xmax=6,ymin=-6,ymax=6, axis lines=center, axis equal]

\addplot[domain=-6:6,color=blue,samples=400]({sin(\x^2+\y^2)},{cos(\x*\y r)});

\end{axis}
\end{tikzpicture}

\end{document}

This is the riddle I was presented with.

The shape my code produced (a line, strangely enough) perplexes me. I truly have no idea what I did wrong. Is there any internal process at fault? Can you please modify my code in such a way that the desired plot will actually show up? I'd be grateful if you (kind users who already mastered tikz and pgfplots) would help me. Thank you in advance.

Answer 1

I have seen this graph when I saw an introductory article on the small application GrafEq. The ideal behind is to compute the LHS and RHS using interval arithmetic. We cannot (at least it is pretty difficult to) do interval arithmetic in TeX. But we have the following for free

\documentclass[tikz]{standalone}

\pgfdeclarefunctionalshading{interesting-1}{\pgfpoint{-25bp}{-25bp}}{\pgfpoint{25bp}{25bp}}{}{
    3 mul exch 3 mul exch % x y
    2 copy                % x y x y
    dup mul exch dup mul  % x y y² x²
    add sin               % x y sin(x²+y²)
    3 1 roll              % sin(x²+y²) x y
    mul cos               % sin(x²+y²) cos(xy)
    le                    % < or not <
    {0 0 0}{1 1 1}ifelse  % black or white
}
\pgfdeclarefunctionalshading{interesting-2}{\pgfpoint{-25bp}{-25bp}}{\pgfpoint{25bp}{25bp}}{}{
    3 mul exch 3 mul exch % x y
    2 copy                % x y x y
    dup mul exch dup mul  % x y y² x²
    add sin               % x y sin(x²+y²)
    3 1 roll              % sin(x²+y²) x y
    mul cos               % sin(x²+y²) cos(xy)
    sub dup mul           % (L-R)²
    .01 le                % <.01 or not <.01
    {0 0 0}{1 1 1}ifelse  % black or white
}

\begin{document}
    \tikz\path[shading=interesting-1](-10,-10)rectangle(10,10);
    \tikz\path[shading=interesting-2](-10,-10)rectangle(10,10);
\end{document}

OR GNUPLOT

\documentclass[tikz]{standalone}
    \usepackage{pgfplots}
\begin{document}
    \begin{tikzpicture}
        \begin{axis}[view={0}{90}]
            \addplot3[domain=-40:40,samples=100,contour gnuplot={levels={0},labels=false}]
                {sin(x^2+y^2)-cos(x*y)};
        \end{axis}
    \end{tikzpicture}
\end{document}

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