Numerical Fallback

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\documentclass{article}
\usepackage[elliptic]{qed}

\begin{document}

\section*{Numerical fallback}

The Legendre relation is a relation among the elliptic integrals of the first
($\ellipk$) and second ($\ellipe$) kinds, given by
%
\begin{qed}
    \ellipe(k^2) \ellipk(1-k^2) + \ellipe(1 - k^2) \ellipk(k^2)
    - \ellipk(k^2) \ellipk(1 - k^2) = \frac{\pi}{2}
\end{qed}
%
Unfortunately, \textsf{sympy} doesn't know how to automatically simplify it,
so \QED cannot analytically determine its validity. Nevertheless,
when we typeset it, we still got a passing badge. That's because \QED falls
back to numerical evaluation automatically.

\end{document}

Building the PDF

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pdflatex numerical_fallback.tex
qed
pdflatex numerical_fallback.tex

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tectonic numerical_fallback.tex --keep-intermediates
qed
tectonic numerical_fallback.tex

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