I don't think the formula was built for speed Bessel's result was a landmark.How quickly does Eq 11 converge?
It is essentially a Fourier series and as mentioned it gives incorrect answers for certain values of the eccentricity (in Boost C++). So, in practice implementing series is probably NOT the best approach.
However, closed solutions are used to think about the qualitative properties of the solution such as distribution of zeroes, asymptotics. There is no rule to say that they should be used for numerical computation.
Classic iterative methods may be unpredictable in how long it takes to converge. Non-iterative methods (e.g. Fibonacci) converge in a fixed number of steps based on desired tolerance.You know a priori the number of function evaluations to achieve a given accuracy.
Ex. To give an idea, Brent minimisation converges in ~ 9 iterations of the Kepler equation to O(1.0e-15) accuracy.
I think Mathematica (and Maple?) support KE?