I encountered an expression which manual simplification is manageable, where the results are nicely in terms of generalized mean (power mean).
To avoid making this post too long, I created a post on Math StackExchange to display the kind of algebra I'm talking about.${}^1$
How can I formulate perhaps
ComplexityFunction
or adopt some good practice to use withFullSimplify
to arrive at an outcome that is "close enough"?
The math itself is rather standard calculus (2nd derivative), and I anticipate similar but more tedious calculations as one moves up the order (of derivatives). Therefore it would be nice to establish a reliable approach.
The codes below show what I'm trying to do.
(* quick and dirty *)
ClearAll[x, y, u, k, f, target, goal, myM, myAss];
myAss = x > y > 1 && k >= 1;
f = (Sqrt[1/(u + y)] - Sqrt[1/(u + x)]) Sqrt[2 (u + k)];
target = Divide[D[f, {u, 2}],
D[f, u]] /. {u -> 0} (* just take a look *)
target //FullSimplify[#, myAss]&
myM[p_] := If[p == 0, Sqrt[x y], (x^p + y^p)^(1/p)]
goal =
(-1/2 ) (1/k + 3(1/x + 1/y + 1/G + (M - G + (M G)/k)^(-1))) /.
{G -> myM @ 0, M -> myM[-1/2]};
goal - target //Simplify[#, myAss]&
(* verifies an equivalent form of the target expression *)
Having things in terms of generalized mean is not just for the aesthetics. Analytically it is helpful for estimation.
For example,${}^2$ here we have $\min = y < H < M < G < A < x = max$, in the order of generalized mean of powers $\{ -\infty,\, -1,\, \frac{-1}2,\, 0,\, 1,\, \infty \}$
I tried a bit manually forcing the change of variables like /.{Sqrt[x y] -> G}
and friends, but it's a total mess. There are many posts here regarding simplification and algebraic manipulation, and I'm still wading through.
I've had some experience of algebraic manipulation with MMA, and I know that one has to nudge it constantly to achieve a workable result. I guess what I'm asking is a guide for me to guide MMA for this particular type of task.
Footnote 1
The linked Math StackExchange post deals with only partially what the MMA code here contains. In manual simplification, it is trivial to first pull out the $1/k$, but not so much programmatically.
Footnote 2
Here $G$ is the geometric mean, $H$ is the harmonic mean, and $M$ is the mean with $\text{power}= \frac{-1}2$ (see the linked post). The arithmetic mean $A$ doesn't really appear in the final expression; included just for perspectives. The factors of averaging ($\frac1{\sqrt2}$ and $\frac12$) at various places are omitted here, since I'm just sketching an argument for the merit of such expressions.