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Derivation

1. Initial program 0.0

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

3. Applied flip3--0.0

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

4. Applied associate-/l/0.0

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

5. Simplified0.0

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

7. Applied flip-+0.0

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

8. Applied flip-+0.0

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

9. Applied frac-times0.0

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

10. Simplified0.0

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

11. Simplified0.0

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

12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019007 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))

Details

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