Average Error: 0.0 → 0.0
Time: 33.9s
Precision: 64
Internal Precision: 128
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\log_* (1 + (e^{\tan^{-1} \left(\sqrt{\log \left(e^{\frac{1 - x}{1 + x}}\right)}\right)} - 1)^*) \cdot 2\]

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. Using strategy rm

  3. Applied add-log-exp0.0

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

  5. Applied log1p-expm1-u0.0

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

  6. Final simplification0.0

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

Runtime

Time bar (total: 33.9s)Debug logProfile

herbie shell --seed 2018349 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))