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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 div-sub0.0

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

6. Applied frac-sub0.0

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

7. Applied sqrt-div0.0

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

8. Simplified0.0

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

9. Final simplification0.0

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

Runtime

Time bar (total: 29.8s)Debug logProfile

herbie shell --seed 2018354 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))