Mathematica Module versus With or Block - Guideline, rule of thumb for usage?

Question

Leonid wrote in chapter iv of his book : "... Module, Block and With. These constructs are explained in detail in Mathematica Book and Mathematica Help, so I will say just a few words about them here. ..."

From what I have read ( been able to find ) I am still in the dark. For packaged functions I ( simply ) use Module, because it works and I know the construct. It may not be the best choice though. It is not entirely clear to me ( from the documentation ) when, where or why to use With ( or Block ).

Question. Is there a rule of thumb / guideline on when to use Module, With or Block ( for functions in packages )? Are there limitations compared to Module? The docs say that With is faster. I want to be able to defend my =choice= for Module ( or another construct ).

Answer 1

A more practical difference between Block and Module can be seen here:

Module[{x}, x]
Block[{x}, x]
(*
-> x$1979
   x
*)

So if you wish to return eg x, you can use Block. For instance,

Plot[D[Sin[x], x], {x, 0, 10}]

does not work; to make it work, one could use

Plot[Block[{x}, D[Sin[x], x]], {x, 0, 10}]

(of course this is not ideal, it is simply an example).

Another use is something like Block[{$RecursionLimit = 1000},...], which temporarily changes $RecursionLimit (Module would not have worked as it renames $RecursionLimit).

One can also use Block to block evaluation of something, eg

Block[{Sin}, Sin[.5]] // Trace
(*
-> {Block[{Sin},Sin[0.5]],Sin[0.5],0.479426}
*)

ie, it returns Sin[0.5] which is only evaluated after the Block has finished executing. This is because Sin inside the Block is just a symbol, rather than the sine function. You could even do something like

Block[{Sin = Cos[#/4] &}, Sin[Pi]]
(*
-> 1/Sqrt[2]
*)

(use Trace to see how it works). So you can use Block to locally redefine built-in functions, too:

Block[{Plus = Times}, 3 + 2]
(*
-> 6
*)