Radical Denest: an ancient difficult task in symbolic computation

Posted 1 year ago
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Posted 1 year ago
 Very good! Is there another similar function that does similar things when variables (rather than just numbers) are involved?For example, there is the function denestSqrt[e_, domain_, x_] := Replace[y /. Solve[Simplify[Reduce[Reduce[y == e && domain, x], y, Reals], domain], y], {{r_} :> r, _ -> e}] written by @CarlWoll (denestSqrt).Here is an example: denestSqrt[Sqrt[(1 - 2 x) (1 - x - 2 x^2) (2 - x + 2 Sqrt[1 - x - 2 x^2])], 1/3 < x <= 1/8 (2 + Sqrt[2]), x] (* 1 - x - 2 x^2 + Sqrt[-(1 + x) (-1 + 2 x)^3] *) 
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Posted 1 year ago
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Posted 6 days ago
 Looks fantastic, this is just what I need for recent work. However, the current algorithm is unable to find some denesting result even when one truly exists, e.g. ones related to non-algebraic numbers and transcendental functions, like this below: $$\frac{1}{2} \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}}$$From my test today, the function posted is unable to find the denesting result of the compound radicals above, but there truly exists one: $$\frac{1}{2} \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}}=\sin \left(\frac{\pi }{96}\right)$$So I hope you can improve your function, making it able to solve compound radicals related to non-algebraic numbers and transcendental functions.Despite its imperfection, I am still very impressive about Corey Ziegler and Bill Gosper, Daniel Lichtblau, Swastik Banerjee and their team's excellent work. This is no doubt a good start. Please keep going, making this function more and more powerful!
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