Here is a related question, but not quite what I'm after.
I would like to define a function f[x_] that has the distributive attribute, such that:
f[a+b+c]
f[a]+f[b]+f[c]
Of course I could do
f[Plus[x_,y___]]:=f[x]+f[Plus[y]] but I suspect that this would be much slower than an actual Attribute. Unfortunately, I cannot find Distributive in the list of allowed attributes.
How should one define such a property to make sure that it performs as efficiently as possible?
f[x__Plus] := Plus @@ (f /@ (List @@ x)) This is also about the same speed as:
f[x__Plus] := Map[f, x] Which is also about the same speed as:
f[x__Plus] := Distribute[Unevaluated[f[x]]] You can test the efficiency of this like so:
var = Sum[a[i], {i, 1, 100000}]; AbsoluteTiming[f[var];] I am reasonably sure there isn't an attribute based solution, and I believe that most of the built-in functions that aren't coded in C rely on pattern matching to manage the distributive property. The trick here is to avoid relying on pattern matching more often than necessary: this splits out every summand in a single replacement.
Distribute? e.g.Distribute[f[a + b + c]]$\endgroup$