When I was using cmath.atanh, I found that this function generates wrong output.

For example:

>>> import cmath
>>> cmath.atanh(-1.89)
(-0.5888951591901462+1.5707963267948966j)

However, the correct output should be:
(-0.5888951591901462-1.5707963267948966j)

Clearly, the sign of the imaginary part is wrong.

Both are correct, since atanh requires a branch cut.

> Both are correct
Well, strictly speaking only the output with positive imaginary part is correct here: the recommendations of C99 Annex G (which Python's cmath module follows) use the sign of the zero imaginary part to determine which 'side' of the branch cut the input lies on.  In this case, -1.89 is interpreted as complex(-1.89, 0.0), so the imaginary part is a positive zero, and the sign of the imaginary part of the result matches that for complex(-1.89, small_and_positive).

So all four of the following are correct:

>>> cmath.atanh(complex(-1.89, 0.0))
(-0.5888951591901462+1.5707963267948966j)
>>> cmath.atanh(complex(-1.89, -0.0))
(-0.5888951591901462-1.5707963267948966j)
>>> cmath.atanh(complex(1.89, 0.0))
(0.5888951591901462+1.5707963267948966j)
>>> cmath.atanh(complex(1.89, -0.0))
(0.5888951591901462-1.5707963267948966j)