Mathematica doesn't want simplify the square root of the square even under assumptions

Question

Suppose a simple function

f[x_,y_] = Sqrt[(x-y)^2]+Sqrt[(y-x)^2]

Knowing that $x>y>0$, without any commands like Simplify one can obtain (x-y) by using the Sqrt[(x-y)^2]-> (x-y) and Sqrt[(x-y)^2]-> (x-y):

f[s2, m2] /. {Sqrt[(x-y)^2] -> x-y, 
  Sqrt[(y-x)^2] -> x-y}

This might be used for the case of complicated functions involving many terms and other, when the Simplify command takes so much time. I tried to use it for one complicated function defined dynamically, but there was no result.

What is the reason and how to construct formal rule replacing the expressions like $\sqrt{(x-y)^2}$?

P.S. PowerExpand cripples the expression for the function...

Answer 1

You can target the Sqrt[..] expressions for simplification like this:

Sqrt[(x - y)^2] + Sqrt[(y - x)^2] /. 
 rt : (Sqrt[_] | 1/Sqrt[_]) :> Simplify[rt, x > y]
(*  2 x - 2 y  *)

Perhaps that is what you're after.

I've often done such targeted simplification, when the size of the main expression leads to a combinatorial explosion of rabbit trails for Simplify to follow. You can also use ExcludedForms to prevent Simplify from touching a subexpression, in case you want a somewhat complementary process (thanks to Carl Woll for pointing out this option). You can combine these in a sequence of steps to slowly get an expression down to a manageable size.