Average Error: 0.0 → 0.0
Time: 30.3s
Precision: 64
Internal Precision: 128
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\frac{\sqrt[3]{\sqrt{1 - x} \cdot \left(\sqrt{1 - x} \cdot \sqrt{1 - x}\right)}}{\sqrt{1 + x}}\right) \cdot 2\]

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 sqrt-div0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019068 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))