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I'm trying to coax Mathematica into rewriting $ x\sqrt{1-y^2/x^2} = \text{sgn}(x)\sqrt{x^2-y^2} $ for $ x $ and $ y $ real. I have tried all combinations of FullSimplify and Refine with the appropriate assumptions. Mathematica refuses to cooperate.

With the additional assumption $ x>0 $, I can get Mathematica to simplify to $ \sqrt{x^2-y^2} $, but I don't want to assume this. I want the general result with the $ \text{sgn}(x) $.

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    $\begingroup$ Sometimes, it is actually hard to tell if one form is simpler than another. But if *Simplify etc. refuse to carry out further transformations, that means, Mathematica determines, according to its own built-in principles, that the "simplest" form has already been reached. $\endgroup$ – Αλέξανδρος Ζεγγ Nov 18 '18 at 14:13
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For this particular expression you can use

ComplexExpand[Re[x Sqrt[1 - y^2/x^2]], {x, y}] // 
  FullSimplify[#, 
    FunctionDomain[x Sqrt[1 - y^2/x^2], {x, y}, Reals]] & // ExpandAll

(*Sqrt[x^2 - y^2] Sign[x]*)

You can also use TargetFunctions with Sign or RealSign as an option to ComplexExpand for more complicated expressions.

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It is not clear, why do you need to make such a replacement by Mathematica, when it is much faster to straightforwardly type the desired expression.

However, if you are insisting, try this

expr = x*Sqrt[1 - y^2/x^2];
expr1 = expr /. x -> Sign[z]*Abs[z]

(*  Abs[z] Sqrt[1 - y^2/(Abs[z]^2 Sign[z]^2)] Sign[z]  *)

and then

(expr1 // Simplify[#, z ∈ Reals && z != 0] &) /. z -> x

(*  Sqrt[x^2 - y^2] Sign[x]  *)

Done. Have fun!

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  • $\begingroup$ I just gave a simplified example for the purposes of asking the question. I don't need to simplify this expression in particular. I need to be able to give Mathematica an expression of unknown form and allow it to perform the simplification using $\text{sgn}$, if it's possible. $\endgroup$ – Donjon Oct 22 '18 at 8:14
  • $\begingroup$ So, I gave you an example. There are, however, several methods. One chooses the best one according to the expression. $\endgroup$ – Alexei Boulbitch Oct 23 '18 at 18:29

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