Average Error: 0.0 → 0.0
Time: 32.6s
Precision: 64
Internal Precision: 128
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{\sqrt{1 + x}} \cdot \frac{1}{\sqrt{1 + x}}}\right)\]

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 add-sqr-sqrt0.0

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

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

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

  5. Applied times-frac0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019053 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))