Editing XOR swap algorithm (section)

==Variations==
The underlying principle of the XOR swap algorithm can be applied to any operation meeting criteria L1 through L4 above. Replacing XOR by addition and subtraction gives various slightly different, but largely equivalent, formulations. For example:<ref>{{cite book |last1=Warren |first1=Henry S. |title=Hacker's delight |date=2003 |publisher=Addison-Wesley |location=Boston |isbn=0201914654 |page=39}}</ref>

<syntaxhighlight lang="c">
void AddSwap( unsigned int* x, unsigned int* y )
{
    *x = *x + *y;
    *y = *x - *y;
    *x = *x - *y;
}
</syntaxhighlight>

Unlike the XOR swap, this variation requires that the underlying processor or programming language uses a method such as [[modular arithmetic]] or [[bignum]]s to guarantee that the computation of <code>X + Y</code> cannot cause an error due to [[integer overflow]]. Therefore, it is seen even more rarely in practice than the XOR swap.

However, the implementation of <code>AddSwap</code> above in the C programming language always works even in case of integer overflow, since, according to the C standard, addition and subtraction of unsigned integers follow the rules of [[modular arithmetic]], i. e. are done in the [[cyclic group]] <math>\mathbb Z/2^s\mathbb Z</math> where <math>s</math> is the number of bits of <code>unsigned int</code>. Indeed, the correctness of the algorithm follows from the fact that the formulas <math>(x + y) - y = x</math> and <math>(x + y) - ((x + y) - y) = y</math> hold in any [[abelian group]]. This generalizes the proof for the XOR swap algorithm: XOR is both the addition and subtraction in the abelian group <math>(\mathbb Z/2\mathbb Z)^{s}</math> (which is the [[direct sum]] of ''s'' copies of <math>\mathbb Z/2\mathbb Z</math>).

This doesn't hold when dealing with the <code>signed int</code> type (the default for <code>int</code>). Signed integer overflow is an undefined behavior in C and thus modular arithmetic is not guaranteed by the standard, which may lead to incorrect results.

The sequence of operations in <code>AddSwap</code> can be expressed via matrix multiplication as:
 
:<math>
\begin{pmatrix}1 & -1\\0 & 1\end{pmatrix}
\begin{pmatrix}1 & 0\\1 & -1\end{pmatrix}
\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}
=
\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}
</math>