Polar plot with small values and coordinate transformations leads to a "Dimension too large error"

Question

EDIT : As it seems to be a bug in pgfplots an issue has been opened: https://github.com/pgf-tikz/pgfplots/issues/360

While plotting polar axes, I faced an issue when transforming small quandtities with the y coord trafo/y coord inv trafo options.

My plots are basically sine functions show in the polar domain. As an example take the following cosine cos(3\x). This function has positive and negative values and is plotted as expected in an axis environment.

Now if one wants to plot this very same function inside a polaraxis environment, here is what comes out :

Basically, negative values are handled as a 180° phase shift, instead of being simply shown as a negative value at the defined angular coordinate.

This behavior can be circumvented with some well-chosen y coord trafo/y coord inv trafo. In this very case,

y coord trafo/.code=\pgfmathparse{#1+2},
y coord inv trafo/.code=\pgfmathparse{#1-2},

works just fine.

However, if some small values have to be handled, say around 1e-6 amplitude instead of 1 as in the above examples, the computation fails due to precision errors. These problems should be solvable through the fpu unit, but I can't figure the correct sequence. I tried e.g

y coord trafo/.code={\pgfkeys{/pgf/fpu=true}\pgfmathparse{#1+2e-6}\pgfkeys{/pgf/fpu=false}},
y coord inv trafo/.code={\pgfkeys{/pgf/fpu=true}\pgfmathparse{#1-2e-6}\pgfkeys{/pgf/fpu=false}},

but this leads to illegal unit of measure errors.

Here is the MWE to reproduce the figures and the problem once the commented part is reactivated.

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.17}
\usepgfplotslibrary{polar}
\usepackage{amsmath}
\begin{document}
\begin{tikzpicture}
    \begin{axis}[%
            xmin=0,
            xmax=360,
            ymin=-2,
            ymax=2,
            domain = 0:360,
        ]
        \addplot+ ({\x},{cos(3*\x)});
    \end{axis}
\end{tikzpicture}
\begin{tikzpicture}
    \begin{polaraxis}[%
            xmin=0,
            xmax=360,
            ymin=-2,
            ymax=2,
            domain = 0:360,
        ]
        \addplot+ ({\x},{cos(3*\x)});
    \end{polaraxis}
\end{tikzpicture}

\begin{tikzpicture}
    \begin{polaraxis}[%
            xmin=0,
            xmax=360,
            ymin=-2,
            ymax=2,
            y coord trafo/.code=\pgfmathparse{#1+2},
            y coord inv trafo/.code=\pgfmathparse{#1-2},
            domain = 0:360,
        ]
        \addplot+ ({\x},{cos(3*\x)});
    \end{polaraxis}
\end{tikzpicture}

% Not working
% \begin{tikzpicture}
%     \begin{polaraxis}[%
%             xmin=0,
%             xmax=360,
%             ymin=-2e-6,
%             ymax=2e-6,
%             y coord trafo/.code={\pgfmathparse{#1+2e-6}},
%             y coord inv trafo/.code={\pgfmathparse{#1-2e-6}},
%             domain = 0:360,
%         ]
%         \addplot+ ({\x},{1e-6*cos(3*\x)});
%     \end{polaraxis}
% \end{tikzpicture}

\end{document}

Answer 1

Your idea of using fpu for the coordinate transformations is good. You just need to add /pgf/fpu/output format=fixed.

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.17}
\usepgfplotslibrary{polar}
\newcommand{\pgfmathparseFPU}[1]{\begingroup%
  \pgfkeys{/pgf/fpu,/pgf/fpu/output format=fixed}%
  \pgfmathparse{#1}%
  \pgfmathsmuggle\pgfmathresult\endgroup}
\usepackage{amsmath}
\begin{document}
\begin{tikzpicture}
    \begin{axis}[%
            xmin=0,
            xmax=360,
            ymin=-2,
            ymax=2,
            domain = 0:360,
        ]
        \addplot+ ({\x},{cos(3*\x)});
    \end{axis}
\end{tikzpicture}
\begin{tikzpicture}
    \begin{polaraxis}[%
            xmin=0,
            xmax=360,
            ymin=-2,
            ymax=2,
            domain = 0:360,
        ]
        \addplot+ ({\x},{cos(3*\x)});
    \end{polaraxis}
\end{tikzpicture}

\begin{tikzpicture}
    \begin{polaraxis}[%
            xmin=0,
            xmax=360,
            ymin=-2,
            ymax=2,
            y coord trafo/.code=\pgfmathparse{#1+2},
            y coord inv trafo/.code=\pgfmathparse{#1-2},
            domain = 0:360,
        ]
        \addplot+ ({\x},{cos(3*\x)});
    \end{polaraxis}
\end{tikzpicture}

% Now working
\begin{tikzpicture}
    \begin{polaraxis}[%
            xmin=0,
            xmax=360,
            ymin=-2e-6,
            ymax=2e-6,
            y coord trafo/.code={\pgfmathparseFPU{#1+2e-6}},
            y coord inv trafo/.code={\pgfmathparseFPU{#1-2e-6}},
            domain = 0:360,
        ]
        \addplot+ ({\x},{1e-6*cos(3*\x)});
    \end{polaraxis}
\end{tikzpicture}

\end{document}