>perl -MMath::Complex -le "print exp(1) ** (pi * i);"
-1+1.22460635382238e-16i
[download]

Copying Rob's one-liner into my system gives yet another answer:

-1+1.22464679914735e-16i

Using perl -v, my perl is "

This is perl 5, version 32, subversion 1 (v5.32.1) built for x86_64-linux-gnu-thread-multi

". Possibly different compilers or options used to build the executables. As somebody who has dealt with floating point issues, 10**(-16) in a operation return like this is frequently noise. I'd not expect perl to return exactly zero to sin(pi) and I don't get it, but 1.22464679914735e-16.

That's hinting at something, but I'm not quite sure what.

Tried (and hopefully succeeded) in fixing the formatting, so my reply isn't in my sig.

---

Information about American English usage here and here. Floating point issues? Please read this before posting. — emc

Copying Rob's one-liner into my system gives yet another answer:
-1+1.22464679914735e-16i

That's the same result as the OP got.
The difference is that perl's print function rounded the final 2 digits away, whereas Raku's didn't.
Now that I'm back home, I can see the same with perl on Ubuntu-20.04.
The figure that I reported came from Windows, so there's a discrepancy in the underlying C libraries.

C:\>perl -le "printf '%a',  1.2246467991473532e-16;
0x1.1a62633145c07p-53
C:\>perl -le "printf '%a',  1.2246063538223773e-16;
0x1.1a6p-53
[download]

The latter (which is what windows is reporting) is suspiciously truncated.

With the MPC library, the given inputs produce a result of -1+2.8954204500590832e-16.

Cheers,
Rob

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