evaluation - Replacement inside held expression

I wish to make a replacement inside a held expression:

f[x_Real] := x^2;
Hold[{2., 3.}] /. n_Real :> f[n]

The desired output is Hold[{4., 9.}], but I get Hold[{f[2.], f[3.]}] instead. What is the best way to make such a replacement without evaluation of the held expression?

Answer

Generally, you want the Trott-Strzebonski in-place evaluation technique:

f[x_Real]:=x^2;
Hold[{Hold[2.],Hold[3.]}]/.n_Real:>With[{eval = f[n]},eval/;True]

(* Hold[{Hold[4.],Hold[9.]}] *)

It will inject the evaluated r.h.s. into an arbitrarily deep location in the held expression, where the expression was found that matched the rule pattern. This is in contrast with Evaluate, which is only effective on the first level inside Hold (won't work in the example above). Note that you may evaluate some things and not evaluate others:

g[x_] := x^3;
Hold[{Hold[2.], Hold[3.]}] /. n_Real :> With[{eval = f[n]}, g[eval] /; True]


(* Hold[{Hold[g[4.]], Hold[g[9.]]}] *)

The basic idea is to exploit the semantics of rules with local variables shared between the body of With and the condition, but within the context of local rules. Since the condition is True, it forced the eval variable to be evaluated inside the declaration part of With, while the code inside the Condition , here the body of With (g[eval]), is treated then as normally the r.h.s. of RuleDelayed is. It is important that With is used, since it can inject into unevaluated expressions. Module and Block also have the shared variable semantics, but wouldn't work here: while their declaration part would evaluate, they would not be able to communicate that result to their body that remains unevaluated (more precisely, only the part of the body that is inside Condition will remain unevaluated - see below). The body of With above was not evaluated either, however With injects the evaluated part ( eval here) into it - this is why the g function above remained unevaluated when the rule applied. This can be further illustrated by the following:

Hold[{Hold[2.],Hold[3.]}]/.n_Real:>Module[{eval=f[n]},
   With[{eval = eval},g[eval]/;True]]

(* Hold[{Hold[g[4.]],Hold[g[9.]]}] *)

Note b.t.w. that only the part of code inside With that is inside Condition is considered a part of the "composite rule" and therefore not evaluated. So,

Hold[{Hold[2.],Hold[3.]}]/.n_Real:>Module[{eval = f[n]},

    With[{eval = eval},Print[eval];g[eval]/;True]]

(* print: 4. *)
(* print: 9. *)
(* Hold[{Hold[g[4.]],Hold[g[9.]]}] *)

But

Hold[{Hold[2.],Hold[3.]}]/.n_Real:>Module[{eval = f[n]},
   With[{eval = eval},(Print[eval];g[eval])/;True]]


(* Hold[{Hold[Print[4.];g[4.]],Hold[Print[9.];g[9.]]}] *)

This should further clarify this mechanism.