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    $\begingroup$ Do the calculations and fail for 5cm? NEVER! The pride of the engineers is on stake! $\endgroup$
    – Ander Biguri
    Commented Mar 18, 2015 at 11:21
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    $\begingroup$ I can't find any NASA documentation, but Frank Borman explicitly says he recalls Apollo 8's final position error after lunar orbit insertion was "about a mile and a half from where we were supposed to be" - he quotes this in the Apollo 8 interviews and footage from "When We Left Earth". So this complements your answer nicely: actual positional errors were 4 to 5 orders of magnitude greater than GR's effect. $\endgroup$
    – Selene Routley
    Commented Mar 18, 2015 at 11:27
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    $\begingroup$ @WetSavannaAnimalakaRodVance - This error you mentioned alarmed NASA greatly. They formed multiple tiger teams to investigate it. It turned out that this ~2 km error was mostly a result of five large mascons (mass concentrations) on the near side of the Moon. After correcting for these mascons, Apollo 12 landed within ~160 meters of the intended target. $\endgroup$
    – David Hammen
    Commented Mar 18, 2015 at 17:32
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    $\begingroup$ Regarding those mascons, the "Bizarre Lunar Orbits" article at nasa.gov makes for some interesting reading. $\endgroup$
    – David Hammen
    Commented Mar 18, 2015 at 17:36
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    $\begingroup$ @DavidHammen That is interesting reading - I had no idea that the Moon was so lumpy. From the article "an astronaut in full spacesuit and life-support gear whose lunar weight was exactly 50 pounds at the edge of the mascon would weigh 50 pounds and 4 ounces when standing in the mascon's center" $\endgroup$
    – Selene Routley
    Commented Mar 18, 2015 at 22:48