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Given sufficient familiarity with the basics of CPUs, it's possible to study and understand documentation specific to ARM or another architecture. Looking at the output of a C compiler really helps to build familiarity and experience.
Once we’ve defined arccosh and square root carefully, we can see that the expressions Mathematica returns for sinh(arccosh(x)) and tanh(arccosh(x)) are correct for all complex inputs, while the simpler expressions in the table above implicitly assume we’re working with values of x for which arccosh(x) is real.