Average Error: 0.0 → 0.0
Time: 37.5s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\frac{1}{\sqrt{1 + x}} \cdot \sqrt{1 - x}\right) \cdot 2\]

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 div-inv0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))