Average Error: 0.0 → 0.0
Time: 29.3s
Precision: 64
Internal Precision: 128
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{{x}^{3} + 1} \cdot \left(x \cdot x - \left(x - 1\right)\right)}\right)\]

Error

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Results

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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. Final simplification0.0

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

Runtime

Time bar (total: 29.3s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))