The GIsqrt function computes a Gaussian integer approximation to the square root of x. The approximation is exact for perfect squares; the distance between x and its approximation is less than or equal to $\frac{\sqrt{2}}{2}$ otherwise.

$\mathrm{with}\left(\mathrm{GaussInt}\right)\:$

$\mathrm{GIsqrt}\left(-5+12\mathrm{I}\right)$

$2+3\mathrm{I}$

$\mathrm{GIsqrt}\left(7+9\mathrm{I}\right)$

$3+\mathrm{I}$