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\documentclass{article}
\usepackage{expl3}
\ExplSyntaxOn
\cs_new:Npn \hakviiaiii_tarai:nnn #1#2#3 {%
  \int_compare:nTF{#1 <= #2}
    {#2}
    {\hakviiaiii_tarai:nnn
      {\hakviiaiii_tarai:nnn{\int_eval:n{#1 - 1}}{#2}{#3}}
      {\hakviiaiii_tarai:nnn{\int_eval:n{#2 - 1}}{#3}{#1}}
      {\hakviiaiii_tarai:nnn{\int_eval:n{#3 - 1}}{#1}{#2}}}}
\cs_new_eq:NN \tarai \hakviiaiii_tarai:nnn
\ExplSyntaxOff
\begin{document}
  $\mathrm{Tarai}(14,12,10) = \tarai{14}{12}{10}$
\end{document}