One other aspect should be mentioned. When the program was transferred to El Segundo from Santa Monica, I naturally duplicated some runs. The printout was eight-decimal places, I believe. For a number of steps the new and old tab sheets would check exactly. But then after a while there would be a gradual drift; first the eighth place would not check, then the seventh, and finally down to the sixth or fifth, so that I could not be certain I met my goal of five place accuracy. I talked and worked with the computer staff on the problem for about two weeks and no explanation was found. The discrepancy could not be tolerated, so in desperation I sat down at a Marchant and, using exactly the same mathematics and roundoff procedures, set out to duplicate a run by hand. After about a day and a half of steady calculating I found my first disagreement. This gave some guidance as to where to look for the trouble. The deck of cards containing the program for this problem was about 0.9-in. thick. In order to reduce the frequency of reloading the deck in the hopper, eleven duplicate decks were made, making a dozen in all. What we found when we traced through all the decks was that one card had been misplaced so that eleven times instructions were correct but were wrong the twelfth time. The erroneous card determined the value of one of the last terms in the Taylor series extrapolation so the error was small. This experience has made me extremely cautious about trusting the output of a large scale computer on a complicated problem, because there are so many possibilities for error.
I was not sure of the propagation and growth of roundoff errors, so after finding all the solutions, I checked by rerunning them again with two more terms in the Taylor series and longer steps. Everything checked. I hope this chronicle gives you the flavor of automatic computers in the early days.Tuncer Cebeci in Legacy of a Gentle Genius: The Life of AMO Smith