Given a custom function f, what is the difference between passing arguments like
f[x_][y_][w_]
and
f[x_,y_,w_]
or other such variations among these? It is often that I see functions using both/some notations, but I can't quite understand the underlying difference.
In addition to @Marco's comment on currying, see "operator forms", for instance V10's Operator Forms - what are they good for? and in the docs.
Note in f[x][y][w], any Attributes that f has apply only to f[x]; in f[x, y, w], the Attributes of f apply to f[x, y, z]. For instance HoldAll. If f has the attribute HoldAll, then in evaluating f[x][y][w], the argument x is passed to f unevaluated; but y and w are evaluated and their values are passed to f. In evaluating f[x, y, w], all arguments would be passed unevaluated.
If no attributes, the evaluation sequence is in the two cases shown by TracePrint:
TracePrint[f[x][y][w]]
f[x][y][w] <- Expression to be evaluated
f[x][y] <- The head of f[x][y][w] (to be eval.)
f[x] <- The head of f[x][y] (to be eval.)
f <- The head of f[x] (evaluated)
x <- The arg of f[x] (evaluated)
y <- The arg of f[x][y] (evaluated)
w <- The arg of f[x][y][w] (evaluated)
f[x][y][w] <- The result of f[x][y][w] (evaluated)
TracePrint[f[x, y, z]]
f[x,y,z] <- Expression to be evaluated
f <- The head of f[x,y,w] (evaluated)
x <- The 1st arg of f[x,y,w] (evaluated)
y <- The 2nd arg of f[x,y,w] (evaluated)
z <- The 3rd arg of f[x,y,w] (evaluated)
f[x, y, z] <- The result of f[x][y][w] (evaluated)
Note that in the curried/operator form f[x] or f[x][y] could evaluate to something else:
ClearAll[ff, gg];
ff[a_][b_] := gg[a + b];
TracePrint[ff[x][y][w]]
(* Out[]= <exercise for the reader> *)
