Before revision:  We demonstrate that geometric computing in the Siegel-Klein disk allows one (i) to bypass the time-costly recentering operations to the origin (Siegel translations) required at each iteration of the SEB algorithm in the Siegel-Poincar\'e disk model, and (ii) to approximate numerically fast the Siegel distance with guaranteed lower and upper bounds  .
After revision:  We demonstrate that geometric computing in the Siegel-Klein disk allows one (i) to bypass the time-costly recentering operations to the origin (Siegel translations) required at each iteration of the SEB algorithm in the Siegel-Poincar\'e disk model, and (ii) to approximate numerically fast the Siegel distance with guaranteed lower and upper bounds derived from nested Hilbert geometries .