Average Error: 0.0 → 0.0
Time: 19.7s
Precision: 64
Internal Precision: 320
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\frac{1 - x}{1 - x \cdot x} \cdot \left(1 - x\right)}\right) \cdot 2\]

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 flip-+0.0

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

  5. Applied associate-/r/0.0

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

  6. Final simplification0.0

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

Runtime

Time bar (total: 19.7s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))