Average Error: 0.0 → 0.0
Time: 48.4s
Precision: 64
Internal Precision: 128
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\left(\left(\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}\right) \cdot \left(\frac{1 - x}{x + 1} \cdot \frac{\sqrt[3]{1 - x}}{x + 1}\right)\right) \cdot \frac{1 - x}{x + 1}}}\right)\]

Error

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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-cbrt-cube0.0

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

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

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

  7. Applied add-cube-cbrt0.0

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

  8. Applied times-frac0.0

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

  9. Applied associate-*l*0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Runtime

Time bar (total: 48.4s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))