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fullFactor[f_, x_] := Roots[f == 0, x] /. Equal -> ((#1 - #2) &) /. Or -> Times fullFactor[x^5 - 1, x] (* (-1 + x) ((-1)^(1/5) + x) (-(-1)^(2/5) + x) ((-1)^(3/5) + x) (-(-1)^(4/5) + x) *)

A function from the article that cormullion linked is shorter and faster than what I proposed below. Transcribed in terse style: uf[m_, 1] := {{}} uf[1, n_] := {{}} uf[m_, n_?PrimeQ] := If[m < n, {}, {{n}}] uf[m_, n_] := uf[m, n] = Join @@ Table[Prepend[#, d] & /@ uf[d, n/d], {d, Select[Rest@Divisors@n, # <= m &]}] uf[n_] := uf[n, n] ...

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]= ...

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); ...

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 ...