pgfplots gets "dimension too large"

Question

I am plotting a function, but always get"dimension too large" error. I have searched online and tried many ways without a success. Appreciate your suggestion.

\usetikzlibrary {spy}
\def\distance{1cm}%
\vspace{\distance}
\def\distance{2cm}%
\hspace{\distance}%
\begin{tikzpicture}
\begin{loglogaxis}[xlabel=$T$,ylabel=$\sigma$,xmin=1e-3, xmax=1e2, ymin=1e-1, ymax=1e18,restrict y to domain=1e-1:1e18]
\addplot [black,thick,domain=1e-3:1e2, y domain=1e-1:1e18,restrict y to domain=1e-1:1e18, samples=200]{x*(exp(4*10/x)-exp(3*10/x))};
\addplot[red,thick,domain=1e-3:1e2,y domain=1e-1:1e18,restrict y to domain=1e-1:1e18,  samples=400]{x*(exp(12*10/x)-exp(11*10/x))};
\end{loglogaxis}
\end{tikzpicture}

Answer 1

Here is exp(120000):

\documentclass{article}

\usepackage{xintexpr}

\xintDigits := 24;

\xintverbosetrue

\xintdeffloatvar e := `+`(rseq(1{;} (@<1e-24)?{abort}{@/i}, i = 1++));

\begin{document}

\xintthefloatexpr [16] e**120000\relax

\end{document}

This gives 2.176849428771918e52115. But the pgf manual says

The fpu provides a replacement set of math commands which can be installed in isolated placed to achieve large data ranges at reasonable accuracy. It provides at least the IEEE double precision data range, -10^324...+10^324

This suggests that the exponent 52115 is too big. There is a footnote in the pgf manual saying the exponent is a 32bit integer but it is not clear what that means. I don't know what is the exact maximal exponent, but for comparison the xfp is limited to 9999 as exponent.

---

Alternative: don't use loglogaxis.

Hopefully, I got the math right:

\documentclass{article}
\usepackage{pgfplots}
%\usetikzlibrary {spy}
%\usetikzlibrary{fpu}
\begin{document}
\begin{tikzpicture}
% \begin{loglogaxis}[xlabel=$T$,ylabel=$\sigma$,xmin=1e-3, xmax=1e2, ymin=1e-1, ymax=1e18,restrict y to domain=1e-1:1e18]
% \addplot [black,thick,domain=1e-3:1e2, y domain=1e-1:1e18,restrict y to domain=1e-1:1e18, samples=200]{x*(exp(4*10/x)-exp(3*10/x))};
% \addplot[red,thick,domain=1e-3:1e2,y domain=1e-1:1e18,restrict y to domain=1e-1:1e18,  samples=400]{x*(exp(12*10/x)-exp(11*10/x))};
% \end{loglogaxis}
% \begin{axis}[xlabel=$\log T$,ylabel=$\log \sigma$,xmin=-3, xmax=2, ymin=-1, ymax=18,restrict y to domain=-1:18]
% \addplot [black,thick,domain=-3:2, y domain=-1:18,restrict y to domain=-1:18,
% samples=200]{x + 40*exp(-x) + ln(1 - exp(-10/exp(x)))};
% \addplot[red,thick,domain=-3:2,y domain=-1:18,restrict y to domain=-1:18,
% samples=400]{x + 120*exp(-x) + ln(1 - exp(-10/exp(x)))};
% \end{axis}
\begin{axis}[xlabel=$\log T$,ylabel=$\log \sigma$,xmin=-3, xmax=2, ymin=5, ymax=50,restrict y to domain=5:50]
\addplot [black,thick,domain=-3:2, y domain=5:50,restrict y to domain=5:50,
samples=200]{x + 40*exp(-x) + ln(1 - exp(-10*exp(-x)))};
\addplot[red,thick,domain=-3:2,y domain=5:50,restrict y to domain=5:50,
samples=400]{x + 120*exp(-x) + ln(1 - exp(-10*exp(-x)))};
\end{axis}
\end{tikzpicture}
\end{document}

I needed to modify completely the (log y) domain to see something of the red curve.

Ah sorry I forgot a log(10) in the domain bounds. Will fix.

---

Here is with correct domain bounds after using ln. Apparently I could not use directly ln(10) in the specs for these, so I used coarse approximation.

\documentclass{article}
\usepackage{pgfplots}
%\usetikzlibrary {spy}
%\usetikzlibrary{fpu}
\begin{document}
\begin{tikzpicture}
\begin{axis}[xlabel=$\log T$,ylabel=$\log \sigma$,xmin=-6.9, xmax=4.6, ymin=-2.3, ymax=41.45,restrict y to domain=-2.3:41.45]
\addplot [black,thick,domain=-6.9:4.6, y domain=-2.3:41.45,restrict y to domain=-2.3:41.45,
samples=200]{x + 40*exp(-x) + ln(1 - exp(-10*exp(-x)))};
\addplot[red,thick,domain=-6.9:4.6, y domain=-2.3:41.45,restrict y to domain=-2.3:41.45,
samples=400]{x + 120*exp(-x) + ln(1 - exp(-10*exp(-x)))};
\end{axis}
\end{tikzpicture}
\end{document}

Of course \log in the labels is to refer to natural logarithm, not base 10 logarithm.

Answer 2

You try to compute too big values

I think that you are trying to compute a huge number with exp(12*10/x) when x=1e-3. Even for x=1e-2, I can't compute the output with Python and numpy. I suggest changing the x range (to 1:1e3 for example if it fits your needs) or computing the log values (if it's okay for your application) with an other tool (python and matplotlib2tikz for example) and plot the coordinates with pgf.

It seems that your maximum value on y is reached for the second function for x around 2.96, so I'd recommend changing the x range over the two solutions.