`Contrary to the current description, it should hold y == tan(atan(y)) for ALL y.`

However, x == atan(tan(x)) only holds for those x which are in the range of atan, which are those x with -pi/2 < x < pi/2.

Of course, those equalities are limited by precision. On my machine

tan(atan(1000)) returns 1000.0000000001.

atan(tan(0)) returns 0 (correct).

atan(tan(M_PI)) returns -1.2246467991474E-16 instead of 0.