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Derivation

1. Initial program 0.0

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]

2. Initial simplification0.0

   \[\leadsto \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \cdot 2\]
3. Using strategy rm

4. Applied add-cbrt-cube0.0

   \[\leadsto \tan^{-1} \left(\sqrt{\color{blue}{\sqrt[3]{\left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right) \cdot \frac{1 - x}{1 + x}}}}\right) \cdot 2\]
5. Using strategy rm

6. Applied add-cube-cbrt0.0

   \[\leadsto \tan^{-1} \left(\sqrt{\sqrt[3]{\color{blue}{\left(\left(\sqrt[3]{\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}} \cdot \sqrt[3]{\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}}\right) \cdot \sqrt[3]{\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}}\right)} \cdot \frac{1 - x}{1 + x}}}\right) \cdot 2\]

7. Final simplification0.0

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - x}{x + 1} \cdot \left(\left(\sqrt[3]{\frac{1 - x}{x + 1} \cdot \frac{1 - x}{x + 1}} \cdot \sqrt[3]{\frac{1 - x}{x + 1} \cdot \frac{1 - x}{x + 1}}\right) \cdot \sqrt[3]{\frac{1 - x}{x + 1} \cdot \frac{1 - x}{x + 1}}\right)}}\right)\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))

Details

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