In a paragraph written by a person emphasizing that rigour is not everything in mathematics (I wish I had written down the details), it was stated that the moon landings would have been impossible without the use of nonrigorous algorithms (I believe he was referring to the calculations of the trajectories of the spacecraft). What was the nonrigorous algorithm that was used? And why were rigorous algorithms insufficient? Was it because they were too slow on 60's hardware or was it because rigorous methods could not provide the answers required? And if such computations were to be done today, could they be done rigorously?
The existence of the Arenstorf Orbits was discovered in 1963 on the basis of numerical computations. The Arenstorf orbits appear as periodic solutions to the equations for the plane restricted three body problem (see the original paper by R. Arenstorf):
When $\mu=0$, the solutions describe the classical Keplerian motion. There are two types of the closed orbits which can be obtained by perturbation methods from $\mu=0$. The closed orbits of the first kind were discovered by Poincaré. They are close to circular Keplerian orbits. The closed orbits of the second kind (i.e. the Arenstorf 8-shaped orbits) are close to elliptic Keplerian orbits of arbitrary eccentricity.
The estimates showing that the Arenstorf orbits can pass very near the Moon and the Earth probably would not have been found without extensive numerical computations.