This is actually a centuries-old technique that was largely
killed by computers, but could possibly revive or still exist in
some technical niche. It is even known to have been used by ancient Greek mathematicians and physicists.
The question asks why one would prefer to tabulate the results of a
function, in a table indexed by its parameters, so as to replace
computation by table lookup.
Of course, the minimum to expect is that table lookup is cheaper than
computation of the result, but that is often the case with properly
The answer given by Luke Mathieson describes the best known case of
function tabulation, which is memoisation, i.e. simply the
preservation in a table of results that had to be computed previously,
in case they are needed again.
Raphael argues that a systematic precomputing for a small domain can bring more
efficiency to computations when the values are needed often, though,
as he proposes lazy computation of the table, the difference with
the previous answer is not too clear.
In a comment, I also suggested filling the table in advance, even with
values not yet needed, when there is free/cheap computer time
available for it.
But all this seems somehow restricted to a single program, which
limits the usefulness of the effort.
However, the problem should probably be considered in a more general
context, and has been in the past, before computers existed.
Complex calculations are an ancient problem in mathematics, sciences and
engineering, and for a long time it was done by hand. These computations were used
for all kinds of purposes, including astronomical (and astrological)
predictions (including discovery of planets), computing tides,
cartography and triangulation, compounded interest rates. In
particular, logarithms were used to replace multiplication by
addition, which implied using exponentiation on the results.
All these computations made use of hard to compute functions, such as
trigonometric functions, logarithm, and others. So it became a
business (and a very tedious job) to create precise tables for these
functions, that were printed as books and sold to engineers and all
people who needed them to conduct calculations. These were functions
on the reals, and the tables were designed to reach a given precision,
with some improvements using interpolation techniques. The tables were
also designed so that they could be used reversibly: the same table
could be used for logarithm and for exponential. These books were
extremely valuable tools that one would keep around all the time. And
they survived until the 1980's, when the microcomputers and especially
sophisticated hand-held calculators became available, i.e. not much
more than twenty to thirty years ago.
Using efficiently these tabulated functions was part of standard
engineering curricula in universities and engineering schools.
Another way of tabulating functions was using a graphical form, called
nomogram, nomograph, chart or abaque. They were tabulating a variety
of complex functions used in exact sciences, either computed or obtained
The formerly ubiquitous slide rule of engineers was yet another way of
It could be that there are still useful functions that are too costly
to compute with good precision, even with a standard computer. Then it
can make sense to have them computed with powerful machines, then
tabulated and made accessible, either on some memory device, or
through the Internet. But I would not be enough in that kind of
scientific work to know, and my search on the Internet was not fruitful.