Average Error: 0.0 → 0.0
Time: 29.1s
Precision: 64
Internal Precision: 320
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{(e^{\log_* (1 + \frac{1 - x}{1 + x})} - 1)^*}\right) \cdot 2\]

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. Using strategy rm

  3. Applied expm1-log1p-u0.0

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

  4. Final simplification0.0

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

Runtime

Time bar (total: 29.1s)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))