Hyperbolic arc cosine fails on (-1) :: Complex r.

When allowing for complex results, the hyperbolic arc cosine is continuously defined on all ℝ.

In the (x < (-1)) real ray of the complex plane, acosh equals \z -> i * pi + acosh(abs z), which works fine for almost all arguments. Thus, acosh (-1) should equal i * pi; however due to the implementation as

acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))

where the denominator in the root becomes zero at z = -1, this comes out as NaN :+ NaN.

Could be fixed trivially by adding a special case

acosh ((-1):+0) = 0:+pi

to the instance (RealFloat a) => Floating (Complex a) in Data.Complex.

Trac metadata

Trac field	Value
Version	7.6.3
Type	Bug
TypeOfFailure	OtherFailure
Priority	low
Resolution	Unresolved
Component	libraries/base
Test case	acosh(-1) :: Complex Double
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