Partial evaluation

In computing, partial evaluation is a technique for several different types of program optimization by specialization. The most straightforward application is to produce new programs that run faster than the originals while being guaranteed to behave in the same way.
A computer program "prog" is seen as a mapping of input data into output data:
where formula_2, the "static data", is the part of the input data known at compile time.
The partial evaluator transforms formula_3 into formula_4 by precomputing all static input at compile time. formula_5 is called the "residual program" and should run more efficiently than the original program. The act of partial evaluation is said to "residualize" formula_6 to formula_5.
Futamura projections.
A particularly interesting example of the use of partial evaluation, first described in the 1970s by Yoshihiko Futamura, is when "prog" is an interpreter for a programming language.
If "I"static is source code designed to run inside that interpreter, then partial evaluation of the interpreter with respect to this data/program produces "prog"*, a version of the interpreter that only runs that source code, is written in the implementation language of the interpreter, does not require the source code to be resupplied, and runs faster than the original combination of the interpreter and the source. In this case "prog"* is effectively a compiled version of "I"static.
This technique is known as the first Futamura projection, of which there are three:
Furthermore, applying the tool in #3 to itself yields the tool itself, so it is a Quine.
They were first described by Futamura in 1983.