\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
Test:
arccos
Bits:
128 bits
Bits error versus x
Time: 8.6 s
Input Error: 0.0
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1 - x}{1 + x}\right)}^3}}\right)\)
  1. Started with
    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
    0.0
  2. Using strategy rm
    0.0
  3. Applied add-cbrt-cube to get
    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{\color{red}{1 + x}}}\right) \leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^3}}}}\right)\]
    0.0
  4. Applied add-cbrt-cube to get
    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{\color{red}{1 - x}}{\sqrt[3]{{\left(1 + x\right)}^3}}}\right) \leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\color{blue}{\sqrt[3]{{\left(1 - x\right)}^3}}}{\sqrt[3]{{\left(1 + x\right)}^3}}}\right)\]
    0.0
  5. Applied cbrt-undiv to get
    \[2 \cdot \tan^{-1} \left(\sqrt{\color{red}{\frac{\sqrt[3]{{\left(1 - x\right)}^3}}{\sqrt[3]{{\left(1 + x\right)}^3}}}}\right) \leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\sqrt[3]{\frac{{\left(1 - x\right)}^3}{{\left(1 + x\right)}^3}}}}\right)\]
    0.0
  6. Applied simplify to get
    \[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\color{red}{\frac{{\left(1 - x\right)}^3}{{\left(1 + x\right)}^3}}}}\right) \leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\color{blue}{{\left(\frac{1 - x}{1 + x}\right)}^3}}}\right)\]
    0.0

Original test:


(lambda ((x default))
  #:name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))