Common subexpression from two expressions

Question

I am working with some unpleasantly tedious polynomials, which need to be manipulated in various ways (integrate with respect to some variable, differentiate with respect to another). Since these will eventually be part of numeric routines, I'd like to pull out common subexpressions that I can reuse, e.g.

(G u^2 (6 p (2 h + p) - 8 (h + p) u + 3 u^2))/(12 h^2)
(G (3 h + 3 p - 2 u) u^2)/(3 h^2)

should at least note that both have a common factor of

(G u^2)/(3 h^2)

Is there a convenient way to instruct Mathematica to look for this sort of computational-expense-reduction in pairs of expressions? Ideally it would notice even in the case where it's not just a factor multiplied by both, e.g. if I added + 1 to the second equation, I'd still like it to find that common subexpression.

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Just for clarification, this is what I do by hand:

e1 = (G u^2)/(12 h^2) * (6 p (2 h + p) - 8 (h + p) u + 3 u^2))
e2 = (G u^2)/( 3 h^2) * (3 h + 3 p - 2 u)

A = (G u^2)/(12 h^2)
e1 = A * (6 p (2 h + p) - 8 (h + p) u + 3 u^2))
e2 = A * (12 (h + p) - 6 u)

B = u^2
C = h + p
A = (G B)/(12 h^2)
e1 = A * (6 p (2 h + p) - 8 C u + 3 B)
e2 = A * (12 C - 6 u)

to go from 6 additions, 20 multiplications, 2 divisions, and 2 assignments, to 5 additions, 14 multiplications, 1 division, and 5 assignments (the extra three of which are temporary so can be in registers and cost essentially nothing).

Answer 1

The engine behind this inside Compile is a well-hidden function called OptimizeExpression. it has two levels, 1 and 2. Setting to 2 makes it work harder to find CSEs.

e1 = (G u^2 (6 p (2 h + p) - 8 (h + p) u + 3 u^2))/(12 h^2);
e2 = (G (3 h + 3 p - 2 u) u^2)/(3 h^2);

Experimental`OptimizeExpression[{e1, e2}, 
 OptimizationLevel -> 2]

(* Out[40]= Experimental`OptimizedExpression[
 Block[{Compile`$7, Compile`$9, Compile`$10}, Compile`$7 = h^2; 
  Compile`$9 = 1/Compile`$7; 
      Compile`$10 = 
   u^2; {1/12 G Compile`$9 Compile`$10 (6 p (2 h + p) - 8 (h + p) u + 
          3 Compile`$10), 
   1/3 G Compile`$9 (3 h + 3 p - 2 u) Compile`$10}]] *)

Answer 2

Level may provide a means of getting started. Level breaks down your polynomials into their constituent parts, with various "levels" of complexity. As @Rex Kerr noted in a comment, Level 1 happens to be interesting in the present example.

a=(G u^2 (6 p (2 h+p)-8 (h+p) u+3 u^2))/(12 h^2);
b=(G (3 h+3 p-2 u) u^2)/(3 h^2);
Intersection[Level[a,1],Level[b,1]]

Answer 3

Since you are going to work with numeric functions, Compile will optimize your functions along those lines. If you define :

f = Compile[{{g, _Real}, {u, _Real}, {h, _Real}, {p, _Real}},
 {(g u^2 (6 p (2 h + p) - 8 (h + p) u + 3 u^2))/(12 h^2), 
  (g (3 h + 3 p - 2 u) u^2)/(3 h^2)}
    ]

you can already see some of it in the output. To get an even more in depth look try :

Needs["CompiledFunctionTools`"]
CompilePrint[f]