You are looking for partial evaluation. Perhaps, there is a shorter way, but here is one (rather subtle):

Clear[evalAtPattern] $evalAtPattern[expr_, p_] := expr /. pt : p :> With[{pp = pt}, pp /; True]; Clear[formula]; formula[x_List] := evalAtPattern[ Defer[(x[[1]] - x[[2]]) + (x[[3]] - x[[4]])], HoldPattern[x[[_]]]]; 

Now,

formula[{1, 2, 3, 4}] (* (1 - 2) + (3 - 4) *) 

And the output code can be executed. The solution combined injecting evaluated pieces inside held expressions via Trott - Strzebonski technique, and the use of Defer.

You are looking for partial evaluation. Perhaps, there is a shorter way, but here is one (rather subtle):

Clear[evalAtPattern] evalAtPattern[expr_, p_] := expr /. pt : p :> With[{pp = pt}, pp /; True]; Clear[formula]; formula[x_List] := evalAtPattern[ Defer[(x[[1]] - x[[2]]) + (x[[3]] - x[[4]])], HoldPattern[x[[_]]]]; 

Now,

formula[{1, 2, 3, 4}] (* (1 - 2) + (3 - 4) *) 

And the output code can be executed. The solution combined injecting evaluated pieces inside held expressions via Trott - Strzebonski technique, and the use of Defer.

You are looking for partial evaluation. Perhaps, there is a shorter way, but here is one (rather subtle):

Clear[evalAtPattern] evalAtPattern[expr_, p_] := expr /. pt : p :> With[{pp = pt}, pp /; True]; Clear[formula]; formula[x_List] := evalAtPattern[ Defer[(x[[1]] - x[[2]]) + (x[[3]] - x[[4]])], HoldPattern[x[[_]]]]; 

Now,

formula[{1, 2, 3, 4}] (* (1 - 2) + (3 - 4) *) 

And the output code can be executed. The solution combined injecting evaluated pieces inside held expressions via Trott - Strzebonski technique, and the use of Defer.

You are looking for partial evaluation. Perhaps, there is a shorter way, but here is one (rather subtle):

Clear[evalAtPattern] $evalAtPattern[expr_, p_] := expr /. pt : p :> With[{pp = pt}, pp /; True]; Clear[formula]; formula[x_List] := evalAtPattern[ Defer[(x[[1]] - x[[2]]) + (x[[3]] - x[[4]])], HoldPattern[x[[_]]]]; 

Now,

formula[{1, 2, 3, 4}] (* (1 - 2) + (3 - 4) *) 

And the output code can be executed. The solution combined injecting evaluated pieces inside held expressions via Trott - Strzebonski technique, and the use of Defer.