Bug introduced in 7.0 or earlier and fixed in 10.2.0
I found an unexpected behavior (that I think of as a bug) in evaluation of the equality operator applied to mathematical functions with exact arguments (i.e. not containing any approximate floating-point numbers):
AppellF1[1/2, 1/2, 1/2, 3/2, 1/2 + I √3 / 2, 1/2 - I √3 / 2] == EllipticK[3/4] / 2 (* False *)
This result is wrong. In fact, these two expressions are equal, so the ideal correct result would be
True. I understand that Mathematica may not be able to immediately prove this equality, so it could return it unevaluated.
I guess that the wrong answer
False is a result of round-off errors occurred when Mathematica tried to evaluate both operands numerically. But I believe that numerical approximations should only be used to decide equality if precision is enough to establish provably non-zero difference.
Here are some excerpts from Mathematica help confirming this point:
When given precise numbers, the Wolfram Language does not convert them to an approximate representation, but gives a precise result.
Equalcannot decide whether two numeric expressions are equal it returns unchanged.
I reported this issue to email@example.com (CASE:3353933), but received a response saying this is not a bug. Update: we followed up on this issue and agreed that it's indeed a bug.
So I would like to know your opinion, whether this behavior is indeed by design?