The main idea in Gal's accurate tables is a different tabulation for the special function being computed.
Gal's idea is to not precompute equally spaced values, but rather to perturb the points x so that both x and f(x) are very nearly exactly representable in the chosen numeric format.
If this approximation is more than ±1/1000 of a bit away from exactly midway between two representable values (which happens 99.8% of the time), then the correctly rounded result is clear.
Combined with an extended-precision fallback algorithm, this can compute the correctly rounded result in very reasonable average time.
The problem of generating function values which are accurate to the last bit is known as the table-maker's dilemma.