Average Error: 0.0 → 0.0
Time: 39.7s
Precision: 64
Internal Precision: 128
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - x}}{1 + x} \cdot \left(\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}\right)}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied times-frac0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019090 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))