\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
Test:
arccos
Bits:
128 bits
Bits error versus x
Time: 7.5 s
Input Error: 0.1
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(2 \cdot \tan^{-1} \left(\sqrt{e^{\log \left(\frac{1 - x}{1 + x}\right)}}\right)\)
  1. Started with
    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
    0.1
  2. Using strategy rm
    0.1
  3. Applied add-exp-log to get
    \[2 \cdot \tan^{-1} \left(\sqrt{\color{red}{\frac{1 - x}{1 + x}}}\right) \leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{e^{\log \left(\frac{1 - x}{1 + x}\right)}}}\right)\]
    0.1
  4. Removed slow pow expressions

Original test:


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