Are there any calculations or results that have similar answers and when compared numerically look the same, but in actual fact after so much precision, the answers diverge from each other?
An example I could think of would be a fractional approximation for $\pi$ such as $\frac{245850922}{78256779}$. If one was unaware of $\pi$'s irrationality then they might think (through numerical "proof") that they were the same thing. Obviously this is a trivial and silly example so I am looking for examples that are more major and may even have been of a surprise to some.