# Sorting a list of pairs under a given function

If I have a list

p={{15/16,1/4},{3/4,13/64},{3/4,1/4},{3/16,1/64},{3/16,1/16},{0,1/64}}


I apply a function f[x_,y_]:=2 x-5 y; on it by command 4^3 f@@@p so i got

 {40,31,16,19,4,-5}


I want to sort p by using f. I try this command SortBy[p,f] but it gives

{{0,1/64},{3/16,1/64},{3/16,1/16},{3/4,13/64},{3/4,1/4},{15/16,1/4}}


, but i need actually

{{0,1/64},{3/16,1/16},{3/4,1/4},{3/16,1/64},{3/4,13/64},{15/16,1/4}}


You need to use

SortBy[list, Apply[f]]


By the way:

SortBy[list, f] sorts the elements of list in the order defined by applying f to each of them.

Is a little bit misleading because apply has a special meaning in Mathematica. Apply f at arg (f @@ arg) does something different than f @ arg. And the latter is what SortBy really does.

That is the problem of collision of common language with name choices made for basic Mathematica functions. To be consistent within Wolfram Language, but maybe less clear for a beginner, it should be:

SortBy[list, f] sorts the elements of list in the order defined by f composed with each of them.

As Compose is what that really is: Is there a name for #1@#2&?

Just to amplify point:

f[x_, y_] := 2 x - 5 y;
g[{x_, y_}] := 2 x - 5 y


So (1. as Kub 2.redeine)

SortBy[p, f @@ # &]
SortBy[p, g]