Error: Arithmetic overflow

Question

Motivation

I'm trying to calculate different probabilities 'automatically', given only some variables such as the number of cards in a sample and the number of cards drawn.

Code

The following code consists of a combination of the code from "Reducing Fractions Automatically Using LaTeX3" and "Command for Multiplying Integers".

\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\cs_new:Nn \svend_gcd:nn
  {
    \int_compare:nNnTF {#2} = { 0 } {#1}
      { \svend_gcd:ff {#2} { \int_mod:nn {#1} {#2} } }
  }
\cs_generate_variant:Nn \svend_gcd:nn { ff }
\int_new:N \l__svend_tmp_int
\cs_new:Nn \svend_reduced:nn
  {
    \int_set:Nn \l__svend_tmp_int { \svend_gcd:nn {#1} {#2} }
    \frac { \svend_reduced_wrap:n { \int_eval:n { #1 / \l__svend_tmp_int } } }
          { \svend_reduced_wrap:n { \int_eval:n { #2 / \l__svend_tmp_int } } }
  }
\cs_new:Nn \svend_reduced_use_wrapper:N
  { \cs_set_eq:NN \svend_reduced_wrap:n #1 }
\svend_reduced_use_wrapper:N \use:n
\msg_new:nnn { svend } { malformed-fraction }
  {
    The~input~you~have~provided~is~malformed.~
    Please~provide~input~in~the~form~of~`p/q'.
  }
\NewDocumentCommand \ReducedFractionWrapper { m }
  { \svend_reduced_use_wrapper:N #1 }
\NewDocumentCommand \ReducedFraction { o > { \SplitList { / } } m }
  {
    \group_begin:
    \IfValueT{#1}{\ReducedFractionWrapper{#1}}
    \int_compare:nTF { \tl_count:n {#2} = 2 }
      { \svend_reduced:nn #2 }
      { \msg_error:nn { svend } { malformed-fraction } }
    \group_end:
  }
\NewExpandableDocumentCommand \produkt { m m }
  { \svend_multiply:nn {#1} {#2} }
\cs_new:Npn \svend_multiply:nn #1#2
  {
    \int_eval:n
      { 1 \int_step_function:nnN {#1} {#2} __svend_multiply:n }
  }
\cs_new:Npn __svend_multiply:n #1 { * #1 }
\ExplSyntaxOff
\usepackage{siunitx}
\begin{document}
\def\roed{5}
\def\sort{9}
\newcommand*\kortTotal{\fpeval{\roed+\sort}}
\def\udtraek{10}
\begin{equation}
  \ReducedFraction[\num]
    {(\produkt{\kortTotal-\udtraek+1}{\kortTotal}-\produkt{1}{\udtraek})
     /\produkt{\kortTotal-\udtraek+1}{\kortTotal}}
\end{equation}
\end{document}

Error

I get the error

! Arithmetic overflow.
<recently read> \__int_eval_end:
l.66 ...\produkt{\kortTotal-\udtraek+1}{\kortTotal}}

When I change \def\udtraek{10} to \def\udtraek{9}, it compiles just fine. The number is obviously too big (which is also indicated by Arithmetic overflow) but I don't know how to adjust the code so that I can use larger numbers.

Question

How do I change the code so that I can use larger input integers?

Answer 1

At present, your best option here is likely xint's \xintiieval, which can work with large integers and parse expressions. (There is some non-released expl3 code in this area, but it needs a few bits of finishing; bigintcalc doesn't do expressions.)

Taking that route and simplifying to highlight just the key points:

\documentclass{article}
\usepackage{xintexpr}
\makeatletter
\newcommand\svend@saved@gcd{}
\NewDocumentCommand\ReducedFraction{O{\@firstofone}mm}
  {%
    \edef\svend@saved@gcd{gcd(#2,#3)}%
    \expanded{%
      \noexpand\frac
        {%
          \noexpand#1%
            {\xintiieval{#2/\svend@saved@gcd}}%
        }%
        {%
          \noexpand#1%
            {\xintiieval{#3/\svend@saved@gcd}}%
        }%
     }% 
  }
\makeatother
\NewExpandableDocumentCommand\produkt{mm}
  {\xintiieval{pfactorial(#1,#2)}}
\usepackage{siunitx}
\begin{document}
\def\roed{5}
\def\sort{9}
\newcommand*\kortTotal{}
\edef\kortTotal{\xintiieval{\roed+\sort}}
\def\udtraek{9}
\begin{equation}
  \ReducedFraction[\num]
    {(\produkt{\kortTotal-\udtraek+1}{\kortTotal}-\produkt{1}{\udtraek})}
    {\produkt{\kortTotal-\udtraek+1}{\kortTotal}}
\end{equation}
\end{document}