Average Error: 0.0 → 0.0
Time: 36.7s
Precision: 64
Internal Precision: 576
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{e^{\log \left(1 - x\right) - \log_* (1 + x)}}\right)\]

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 add-exp-log0.0

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

  5. Applied add-exp-log0.0

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

  6. Applied div-exp0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{e^{\log \left(1 - x\right) - \log_* (1 + x)}}\right)\]

Runtime

Time bar (total: 36.7s)Debug logProfile

herbie shell --seed 2018254 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))