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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 flip3-+0.0

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

5. Applied associate-/r/0.0

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

6. Simplified0.0

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

8. Applied associate-*l/0.0

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

9. Applied sqrt-div0.0

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

10. Final simplification0.0

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

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2018263 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))