Gal's accurate tables is a method devised by Shmuel Gal to provide accurate values of special functions using a lookup table and interpolation. It is a fast and efficient method for generating values of functions like the exponential or the trigonometric functions to within last-bit accuracy for almost all argument values without using extended precision arithmetic.
The main idea in Gal's accurate tables is not to use tables of equally spaced argument values in which the rounding error prevents obtaining last-bit accuracy. In order to achieve a small error the following idea is used: Perturb the original, equally spaced, argument values in such a way that the function value will be very close to numbers that can be exactly represented by the computer (much closer than the usual double-precision representation). Thus each table entry has a perturbed argument value and an associated function value. The function value for a given argument is interpolated using these more accurate end values from the table. This method enables controlling the error introduced by the computer representation of real numbers and extends the accuracy.
The problem of generating function values which are accurate to the last bit is known as the table-maker's dilemma.