I'd not expect perl to return exactly zero to sin(pi) and I don't get it, but 1.22464679914735e-16

The reliance on the sine and cosine trig functions can be seen at https://www.math.toronto.edu/mathnet/questionCorner/complexexp.html.
For the case under consideration in this thread, the imaginary component should be sin(3.1415926535897931 * log(2.7182818284590451))
log(2.7182818284590451) is calculated to be exactly 1 and, as already noted, sin(3.1415926535897931) is calculated to be 1.2246467991473532e-16.

I don't know how the MPC library is coming up with 2.8954204500590832e-16, but I'm about to ask them about that, and I'll update this post when I get the answer.

UPDATE: I've changed my mind - the value returned for these inputs is very sensitive to minute changes in the evaluation of those inputs. For example:

sisyphus@sisyphus5-desktop:~$ perl -MMath::Complex -E "say sin(pi * 1)
+;"
1.22464679914735e-16
sisyphus@sisyphus5-desktop:~$ perl -MMath::Complex -E "say sin(pi * 0.
+99999999999999994);"
5.66553889764798e-16
[download]

I don't see much point in investigating further when such a minute alteration to the arguments can make such a large relative difference, especially when we consider that the absolute difference is so minute.

Cheers,
Rob