I'm a little rusty on the details, but IIUC the core of PyPy is an "abstract interpreter" for a restricted subset of Python known as RPython. Abstract interpretation is essentially a fold (in the functional programming sense) of the IR for a language, where you can parameterize the particular operation applied to each node. When the operation specified is "evaluate", you get partial evaluation, as described in the Wikipedia article.
The interesting part of PyPy happens when the RPython program to be evaluated is itself a Python interpreter, which it is. In this case, 'prog' is the Python interpreter written in RPython, and Istatic is your Python program. The first Futamura projection will give you a JIT; you specialize the RPython interpreter with your particular program, and it evaluates everything statically known at compile time (namely, your program) and produces an executable.
The second Futamura projection will give you a JIT compiler. Remember, since RPython is a subset of Python, an interpreter for Python written in RPython could be interpreted by itself. Moreover, it could be compiled by itself by the process described in the paragraph above. When you compile your JIT through the same process, you get a program that means the same thing (i.e. it will translate a Python program into an executable) but runs faster. The PyPy JIT that everybody's excited about for running Python scripts faster is this executable.
The third Futamura projection involves running PyPy on the toolchain that generated PyPy. Remember, this whole specializer machinery is all written in RPython, which can be interpreted by itself. So when your run the specializer machinery through itself, you get - an optimized toolset for building JIT compilers. That's what made programming language theorists excited about PyPy long before the Python community was. It's not just "faster Python", but it's a toolset for building a JIT out of anything you can build an interpreter for. Write an interpreter in RPython - for anything, Ruby, PHP, Arc, whatever - and run it through the PyPy toolchain, and it will give you a JIT compiler.
I'm a little rusty on the details, but IIUC the core of PyPy is an "abstract interpreter" for a restricted subset of Python known as RPython. Abstract interpretation is essentially a fold (in the functional programming sense) of the IR for a language, where you can parameterize the particular operation applied to each node. When the operation specified is "evaluate", you get partial evaluation, as described in the Wikipedia article.
The interesting part of PyPy happens when the RPython program to be evaluated is itself a Python interpreter, which it is. In this case, 'prog' is the Python interpreter written in RPython, and Istatic is your Python program. The first Futamura projection will give you a JIT; you specialize the RPython interpreter with your particular program, and it evaluates everything statically known at compile time (namely, your program) and produces an executable.
The second Futamura projection will give you a JIT compiler. Remember, since RPython is a subset of Python, an interpreter for Python written in RPython could be interpreted by itself. Moreover, it could be compiled by itself by the process described in the paragraph above. When you compile your JIT through the same process, you get a program that means the same thing (i.e. it will translate a Python program into an executable) but runs faster. The PyPy JIT that everybody's excited about for running Python scripts faster is this executable.
The third Futamura projection involves running PyPy on the toolchain that generated PyPy. Remember, this whole specializer machinery is all written in RPython, which can be interpreted by itself. So when your run the specializer machinery through itself, you get - an optimized toolset for building JIT compilers. That's what made programming language theorists excited about PyPy long before the Python community was. It's not just "faster Python", but it's a toolset for building a JIT out of anything you can build an interpreter for. Write an interpreter in RPython - for anything, Ruby, PHP, Arc, whatever - and run it through the PyPy toolchain, and it will give you a JIT compiler.