Extremely Large Telescopes (ELTs) are very challenging with respect to their adaptive optics (AO) requirements. Their diameters and the specifications required by the astronomical science for which they are being designed imply a huge increment in the number of degrees of freedom in the deformable mirrors. Faster algorithms are needed to implement the real-time reconstruction and control in AO at the required speed. We present the results of a study of the AO correction performance of three different algorithms applied to the case of a $42\text{-}\mathrm{m}$ ELT: one considered as a reference, the matrix-vector multiply (MVM) algorithm; and two considered fast, the fractal iterative method (FrIM) and the Fourier transform reconstructor (FTR). The MVM and the FrIM both provide a maximum a posteriori estimation, while the FTR provides a least-squares one. The algorithms are tested on the European Southern Observatory (ESO) end-to-end simulator, OCTOPUS. The performance is compared using a natural guide star single-conjugate adaptive optics configuration. The results demonstrate that the methods have similar performance in a large variety of simulated conditions. However, with respect to system misregistrations, the fast algorithms demonstrate an interesting robustness.