Let a complete intersection $d$-fold singularity be cut out by $k$ polynomials $\{f_1,\ldots,f_k\}$ in $\mathbb{C}^{k+d}$. Is there a relatively simple algorithm to check whether this is analytically equivalent to a $d$-fold node (ordinary double point) given by $x_1^2 + \ldots + x_{d+1}^2 = 0 ~~\text{in}~\mathbb{C}^{d+1}$?
I think I do have a rather clunky algorithm which works, but would like something more efficient if possible.