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

↓

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

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Derivation

Initial program 0.0
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
Using strategy rm
Applied add-sqr-sqrt0.0
\[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}}\right)\]
Applied add-cube-cbrt0.0
\[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\color{blue}{\left(\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}\right) \cdot \sqrt[3]{1 - x}}}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}\right)\]
Applied times-frac0.0
\[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}}{\sqrt{1 + x}} \cdot \frac{\sqrt[3]{1 - x}}{\sqrt{1 + x}}}}\right)\]

herbie shell --seed 2018208 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))