Tikz thinks I'm dividing by 0 when plotting Planck's Law

Question

I'm trying to plot Planck's law using tikz. This works fine for the most part, but tikz thinks I'm dividing by 0 if I try to plot certain areas of the law. Here's a minimally working example of what I'm trying to do :

\documentclass[8pt,xcolor=dvipsnames,compress]{beamer}
\usepackage{tikz}
\begin{document}
\begin{frame}
  \begin{tikzpicture}
    \draw[->] (0,0) -- (4.2,0) node[right] {$\lambda$};
    \draw[->] (0,0) -- (0,4.2) node[above] {Flux};
    \draw[scale=3,domain=0.16:1,smooth,variable=\x,blue] plot ({\x},{((1/(exp(1/\x)-1)*(1/(\x^5)))/20});
    \end{tikzpicture}
  \end{frame}
\end{document}

Now this works, but notice that my domain start at 0.16. I'd like it to start earlier, at least around 0.1, but if I start my domain at 0.1 I get a division by 0 error :

! Package PGF Math Error: You've asked me to divide `1' by `0.0', but I cannot divide any number by `0.0' (in '{((1/(exp(1/0.1)-1)*(1/(0.1^5)))/20}').

Any ideas as to what's causing this ?

Answer 1

You need to make sure that the fpu library is used so that arithmetic is does with floating point numbers. The easiest way to do this is to use pgfplots ( built on top of tikz), which will do this by default.

In pgfplots the default style for axes are boxed, so some tweaking is needed to produce the style of plot you are after. But the actual plotting command for the function is simple.

\documentclass[8pt,xcolor=dvipsnames,compress]{beamer}
\usepackage{pgfplots}
\pgfplotsset{compat=1.12}

\begin{document}

\begin{frame}
  \begin{tikzpicture}
    \begin{axis}[xlabel={\( \lambda \)},ylabel={Flux},
      axis x line=bottom,axis y line=left,
      every axis x label/.append style={at={(1,0)},right},
      every axis y label/.append style={at={(0,1)},above,rotate=-90}]
      \addplot[blue,domain=0.01:1,samples=200]
      {1/((exp(1/x)-1)*x^5*20)};
    \end{axis}
    \end{tikzpicture}
  \end{frame}

\end{document}

Answer 2

You have x⁵ in the denominator with 0.1 as smallest value. The precision of TeX's arithmetic via dimen registers in unit pt is only 2^-16 (= 1 sp). Therefore, 0.1⁵ becomes 0.00001 and 0.00001 pt is smaller than 1 sp and the result is truncated to zero.

TikZ supports better arithmetics, e.g., via library fpu, see the other answer of Andrew Swann. This example shows the use of library fixedpointarithmetic, which relies on package fp. The calculations take more time, but the result is more precise and the lower limit can be decreased to 0.025 before arithmetic overflow. I have also increased the number of samples to get a better curve shape.

\documentclass[8pt,xcolor=dvipsnames,compress]{beamer}
\usepackage{tikz}
\usepackage{fp}
\usetikzlibrary{fixedpointarithmetic}
\begin{document}
\begin{frame}
  \begin{tikzpicture}
    \draw[->] (0,0) -- (4.2,0) node[right] {$\lambda$};
    \draw[->] (0,0) -- (0,4.2) node[above] {Flux};
    \draw[
      samples=100,
      fixed point arithmetic,
      scale=3,domain=0.025:1,smooth,variable=\x,blue] plot
    ({\x},{((1/(exp(1/\x)-1)*(1/(\x^5)))/20});
    \end{tikzpicture}
  \end{frame}
\end{document}