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MUM: flexible precise Monte Carlo algorithm for muon propagation through thick layers of matter

We present a new Monte Carlo muon propagation algorithm MUM (MUons+Medium) which possesses some advantages over analogous algorithms presently in use. The most important features of algorithm are described. Results on the test for accuracy of treatment the muon energy loss with MUM are presented and analyzed. It is evaluated to be of 0.002 or better, depending upon simulation parameters. Contributions of different simplifications which are applied at Monte Carlo muon transportation to the resulting error are considered and ranked. It is shown that when simulating muon propagation through medium it is quite enough to account only for fluctuations in radiative energy loss with fraction of energy lost being as large as 0.05 -- 0.1. Selected results obtained with MUM are given and compared with ones from other algorithms.