Can't get Mathematica to simplify an expression

Question

I am familiar with the use of Simplify, FullSimplify, Expand, Factor, Collect and Together for simplifying a long expression in Mathematica. But recently I have encountered an expression which Mathematica doesn't simplify even though it is completely clear to my eye that the expression can be written in a much shorter form.

This is the expression:

-2 Sqrt[1/(4 + (1. - 2. s + Sqrt[
  8 d^2 + (1. - 3. s) (1. - 1. s) + s^2])^2/(2 d^2))^2] - 
  1/4 Sqrt[(4 d^2 + (1. - 2. s + Sqrt[
  8 d^2 + (1. - 3. s) (1. - 1. s) + s^2])^2 - 
  4 d (1. - 2. s + Sqrt[
   8 d^2 + (1. - 3. s) (1. - 1. s) + s^2])^2)^2/(
 d^4 (4 + (1. - 2. s + Sqrt[
   8 d^2 + (1. - 3. s) (1. - 1. s) + s^2])^2/(2 d^2))^2)] + 
 1/4 Sqrt[(4 d^2 + (1. - 2. s + Sqrt[
  8 d^2 + (1. - 3. s) (1. - 1. s) + s^2])^2 + 
  4 d (1. - 2. s + Sqrt[
   8 d^2 + (1. - 3. s) (1. - 1. s) + s^2])^2)^2/(
  d^4 (4 + (1. - 2. s + Sqrt[
   8 d^2 + (1. - 3. s) (1. - 1. s) + s^2])^2/(2 d^2))^2)]

Question: How can we shorten this expression more and more with Mathematica functions. (All variables are reals)

Answer 1

A few things of note:

1. Like @m_goldberg suggested, it is wise to explicitly state d ∈ Reals, s ∈ Reals.
2. Because you introduced .'s after your numbers, you force mathematica to treat your integers as floats, which introduces rounding errors, removing these helps.
3. Simplification introduces absolutes variable epxressions, taking these to be positive or negative helps.

Positive:

FullSimplify[expr, 
 Assumptions -> {d ∈ Reals, 
   s ∈ 
    Reals, -6 d^2 + 
     16 d^3 + (1 + Sqrt[8 d^2 + (1 - 2 s)^2] - 2 s) (-1 + 2 s) - 
     4 d (1 + Sqrt[8 d^2 + (1 - 2 s)^2] - 2 s) (-1 + 2 s) > 0}]

$$\frac{1+\sqrt{8d^2+\left(1-2s\right)^2}-2s}{2\sqrt{8d^2+\left(1-2s\right)^2}}$$

Negative:

FullSimplify[expr, 
 Assumptions -> {d ∈ Reals, 
   s ∈ 
    Reals, -6 d^2 + 
     16 d^3 + (1 + Sqrt[8 d^2 + (1 - 2 s)^2] - 2 s) (-1 + 2 s) - 
     4 d (1 + Sqrt[8 d^2 + (1 - 2 s)^2] - 2 s) (-1 + 2 s) < 0}]

$$\begin{cases} -1 & \left(1+4d\right)\left(1+\sqrt{8d^{2}+\left(1-2s\right)}-2s\right)\left(-1+2s\right)\ge2d^{2}\left(3+8d\right)\\ \frac{2d\left(d-sd^{2}+2\left(1+\sqrt{8d^{2}+\left(1-2s\right)^{2}}\right)\right)\left(-1+2s\right)}{-8d^{2}+\left(1+\sqrt{8d^{2}+\left(1-2s\right)^{2}}-2s\right)\left(-1+2s\right)} & Else \end{cases}$$

Yielding an in total 3 cases environment.

Answer 2

Following from your expression expr:

exact = Rationalize[expr]

vars = Variables[expr]

Refine[exact, Element[vars, Reals]]

Simplify[%, Element[vars, Reals]]

(* substitution picked by me and copy-pasted from the last output *)
% /. Sqrt[8 d^2 + (1 - 2 s)^2] -> x1

(* another such substitution *)    
% /. (1 - 2 s + x1) -> x2

Answer 3

Just one approach, and not necessarily the kind of result you want:

expr  (* your expression *)

FullSimplify[Rationalize[%], Variables[%] ∈ Reals];
out = Experimental`OptimizeExpression[%];

new = Symbol /@ CharacterRange[63396, 63421];
old = DeleteDuplicates@Cases[out, s_Symbol /; Context[s] === "Compile`", {-1}]

Extract[out, {1, 2}, Defer] /.
  Cases[Flatten[{old, new}, {2}], {o_, n_} :> (o :> n)]