How accurate is the elliptic integral solution of simple pendulum? Wikipedia indicates the exact solution to the simple pendulum period oscillation can be written using elliptic integrals.
Can anyone tell me if experiments have been carried out (some links with tables) to show what is the accordance with this formula?
 A: The "high school solution" of the pendulum is obtained replacing $\sin(\theta) \approx \theta$ for small $\theta$. If you avoid this simplification by solving the full dynamics, you obtain an exact solution that captures the non-isochronicity of the pendulum for large angles.
The main missing ingredient in is solution is now the dissipative component, like viscous friction from the air or the friction in the bolt. These couple to the pendulum mass altering the frequency to some extent and becoming the main source of uncertainty. However, as they depend on the specific setup (and note that they may be well negligible), it is impossible to give a "universal table".
What one could do is to include the dissipative terms (at least by switching to numerical integration). At that point the model gets probably enough degrees of freedom (gravity, length, mass and various friction coefficients) to be fitted to the data from a real pendulum obtaining agreement beyond the precision of whatever instrument.
