# Best way to apply a list of functions to a list of values?

This question is closely related to questions 83720, 17460, and 11298.

How would you write the operator

F = Through[#1[#2]] &


in the prettiest, fastest, or most poetic way? It just looks clumsy the way I've written it.

Also, can you think of any pitfalls? Edge cases where this function does not behave regularly?

What F does is to apply a list of functions to a list of values:

F[{Sin, Cos, Tan}, {x, y}]
(*    {{Sin[x], Sin[y]}, {Cos[x], Cos[y]}, {Tan[x], Tan[y]}}    *)


### benchmarks of solutions

Here's a benchmark of all solutions, sorted from fastest to slowest.

(* random functions and random data *)
a = Table[Function[x, Evaluate[x + RandomReal[]]], {10^4}];
b = RandomReal[{0, 1}, 10^4];


original post: very fast

F = Through[#1[#2]] &;
c0 = F[a, b]; // AbsoluteTiming // First
(*    0.506063    *)


contracted syntax: a little bit slower

F = Through@*Construct;
c1 = F[a, b]; // AbsoluteTiming // First
(*    0.85128    *)


dataset query: still a bit slower

F = Query[#1][#2] &;
c2 = F[a, b]; // AbsoluteTiming // First
(*    1.45958    *)


explicit outer-product construction: terribly slow

F = Outer[Construct, ##] &;
c3 = F[a, b]; // AbsoluteTiming // First
(*    83.0245    *)


check that all solutions agree on the result:

c0 == c1 == c2 == c3
(*    True    *)

• There is one pitfall you have to be aware of, which is that you're relying on your functions to be Listable in the examples you're giving (Cos, Sin and Tan). For a list of non-listable functions you'd need F = Through[(Map /@ #1)[#2]] &. – Sjoerd Smit Jun 10 at 13:20
• Thanks @SjoerdSmit! Yes that's a very important one. – Roman Jun 10 at 13:26

To streamline the syntax of F = Through[#1[#2]]& we can do

F = Through@*Construct


Alternatively, an explicit outer product:

F = Outer[Construct, ##] &

• What is the purpose or function of * here? – CA Trevillian Jun 10 at 13:16
• @CATrevillian operator composition – Roman Jun 10 at 13:22
• sweet!! That’s awesome as all get-out! Many thanks. Side note: does any resource exist of just these shorthand inputs? They’re my favourites – CA Trevillian Jun 10 at 13:31
• @CATrevillian Here's the list of all operator input forms sorted by precedence. – Roman Jun 10 at 13:40
• +1, ro-man, +1 :) – CA Trevillian Jun 10 at 13:42