When should I use Apply (or Function) and when @@ (or &)?

Question

This is a rather general question, which I fail to answer myself. I guess it is mainly due to my insufficient knowledge of the precise terms.

If I understand correctly, the following are equivalent:

f = Function[u, 3 + u]
f = Function[3 + #]
f = (3+#) &
f[x_]:=3+x

In all cases f[a] would yield 3+a. Similarly, f[a] is equivalent to f@@{a}.

I have mainly two questions:

1. What is the right term to call the shorter version, i.e. the version where one uses @@, #, & etc.?
2. When should I prefer the one method and not the other? It seems like many answers given around MA.SE uses the "shorter version". What are the advantages and disadvantages...

Bonus question: Note my first question. I hope I could get more fishing rods and less fish. In other words, what is the official term for these two notions? Where are can I find documentation of these differences? The examples I presented in the this questions are merely examples, and I try to understand where can I rigorously study these notions and similar ones.

Answer 1

The brief forms /@, and @@@ are not exactly equivalent to the full forms of these operations, and also f[x] is not exactly equivalent to either of f[#]&[x] or f @@ {x}.

I will mention two aspects of the problem, but there likely are more (of course, in addition to the precedence - related aspect, already extensively described in the linked past discussions).

Evaluation

Consider the following example:

Hold[Print[1]]

which evaluates trivially to itself. However, the other forms you mentioned:

Hold @@ {Print[1]}

or

Hold[#] & [Print[1]]

will lead to evaluation leaks (Print will be evaluated).

One can probably come up with some contrived examples of other differences of a similar kind, but generally it is enough to say that these forms define different evaluation routes, so with enough effort, we can generate all kinds of differences. It is another matter that for most practical cases, such differences either are not there or don't matter.

Heads option

The main point here is that the brief forms like f @@@ expr and f @@ expr are parsed as Apply[f,expr,{1}] and Apply[f,expr], respectively, and do not allow one to explicitly pass the Heads option. Therefore, they will always use the global value for this option. And, should anyone change that option globally, there may return different results (from what one would normally expect) - just as the equivalent literal forms I mentioned above do.

You can do the following experiment (make sure you don't have any unsaved work):

SetOptions[Apply, Heads -> True] 

Then (taking this example from the help page for Apply):

f @@@ p[x][q[y]]

(* f[y][f[y]] *)

but the explicit literal form allows to pass the option explicitly:

Apply[f, p[x][q[y]], {1}, Heads -> False]

(* p[x][f[y]] *)

A similar situation is with Map and @@. I have described this in more detail here. Basically, you can't specify the Heads option when you use the short forms.

Note that resetting Heads option globally, as I did here, is strongly discouraged and can lead to disastrous consequences.

SetOptions[Apply, Heads -> False] 

Answer 2

If you do a performance test (say, using a million random numbers), you'll find that f = (3+#) & is the fastest. f[x_]:=3+x is significantly slower.

One way to think of @@ (i. e. Apply) is as "replace Head". Because that's exactly what it does. It removes the Head and puts whatever other Head you have provided. Very practical, but not always run-time efficient. Just compare Plus @@ somelist with Total, or even Tr @ somelist.

Also, if you look at Apply in http://reference.wolfram.com/legacy/flash/ you can see quite visually that Apply means "replace Head".

Answer 3

These are all functionally equivalent forms of a Pure Function, sometimes called an anonymous function.

body[#] &

I use the operator form body[#]& when there is only one argument, but you can define a more complicated body using #[[2]] (Part) and #2 (Slot) used with List and Sequence arguments respectively.

Function[body[#]]

The full form Function[body[#]] is useful when you are having problems with precedence including other functions into your body. Compare

f@g[#] &@3 // Trace
f@Function[g[#]]@3 // Trace

Usually I use Pure Functions to apply several operations sequentially. I might even string together Built-In functions, user defined functions, and Pure Functions in a one-liner. But if I need to compute several side-calculations using different parts of several input variables, and then combine their results, I might go for

Function[vars, body[vars]]

where body includes a 'Module'. The only time I would do that is if I needed a complex operation which I only need once. Otherwise I would just make a user defined function through setting its DownValues

f[vars_]:=body[vars]

Whether I use Apply vs Prefix depends on whether I have a single List of arguments or several arguments in a Sequence.

f@Sequence[a, b] == f[a, b]
f @@ List[a, b] == f[a, b]
f@List[a, b] == f[{a, b}]

Answer 4

In answer to your first question, you can refer to the various special forms of input depending on where and how it is used in the expression. Mathematics and Mathematica both use forms such as prefix, infix and postfix form. You can read more here: Special ways to Input Expressions

For completeness, I'll show several obvious examples:

Prefix form

• f(x): Here f is the prefix operator

Infix form

• a + b: Here + is the infix operator because it sits between its arguments

Postfix form

• 5!: Here ! comes after the 5 on which it is operating.

It is probably worth mentioning that, regardless of the input form used, Mathematica translates all input into a standard prefix form: Head[argument1, argument2, ...]