I can't find who first said this, but a quote I've used - with success - is:
  "It is better to be approximately right than precisely wrong."
(Unfortunately I had *very* little success in persuading people to
replace figures like $10,563.27 with $10,563. And that was
deliberately limiting my intention knowing the likely resistance: what
I really wanted to get them to do was use $10,560 to make it clear
that the $10,563.27 was approximate, not exact.)

I don't know much about Numerical Analysis, but I do know enough to be
very wary about just increasing precision.