\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
Test:
math.log/2 on complex, imaginary part
Bits:
128 bits
Bits error versus re
Bits error versus im
Bits error versus base
Time: 5.4 s
Input Error: 31.2
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\frac{\tan^{-1}_* \frac{im}{re}}{\log base} - 0\)
  1. Started with
    \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
    31.2
  2. Applied simplify to get
    \[\color{red}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} - 0}{\log base}}\]
    0.3
  3. Using strategy rm
    0.3
  4. Applied flip3-- to get
    \[\frac{\color{red}{\tan^{-1}_* \frac{im}{re} - 0}}{\log base} \leadsto \frac{\color{blue}{\frac{{\left(\tan^{-1}_* \frac{im}{re}\right)}^{3} - {0}^{3}}{{\left(\tan^{-1}_* \frac{im}{re}\right)}^2 + \left({0}^2 + \tan^{-1}_* \frac{im}{re} \cdot 0\right)}}}{\log base}\]
    11.1
  5. Applied associate-/l/ to get
    \[\color{red}{\frac{\frac{{\left(\tan^{-1}_* \frac{im}{re}\right)}^{3} - {0}^{3}}{{\left(\tan^{-1}_* \frac{im}{re}\right)}^2 + \left({0}^2 + \tan^{-1}_* \frac{im}{re} \cdot 0\right)}}{\log base}} \leadsto \color{blue}{\frac{{\left(\tan^{-1}_* \frac{im}{re}\right)}^{3} - {0}^{3}}{\log base \cdot \left({\left(\tan^{-1}_* \frac{im}{re}\right)}^2 + \left({0}^2 + \tan^{-1}_* \frac{im}{re} \cdot 0\right)\right)}}\]
    11.2
  6. Applied simplify to get
    \[\frac{{\left(\tan^{-1}_* \frac{im}{re}\right)}^{3} - {0}^{3}}{\color{red}{\log base \cdot \left({\left(\tan^{-1}_* \frac{im}{re}\right)}^2 + \left({0}^2 + \tan^{-1}_* \frac{im}{re} \cdot 0\right)\right)}} \leadsto \frac{{\left(\tan^{-1}_* \frac{im}{re}\right)}^{3} - {0}^{3}}{\color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \log base}}\]
    11.2
  7. Applied taylor to get
    \[\frac{{\left(\tan^{-1}_* \frac{im}{re}\right)}^{3} - {0}^{3}}{\left(\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \log base} \leadsto \frac{{\left(\tan^{-1}_* \frac{im}{re}\right)}^{3} - {0}^{3}}{\left(\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \log base}\]
    11.2
  8. Taylor expanded around 0 to get
    \[\frac{\color{red}{{\left(\tan^{-1}_* \frac{im}{re}\right)}^{3}} - {0}^{3}}{\left(\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \log base} \leadsto \frac{\color{blue}{{\left(\tan^{-1}_* \frac{im}{re}\right)}^{3}} - {0}^{3}}{\left(\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \log base}\]
    11.2
  9. Applied simplify to get
    \[\frac{{\left(\tan^{-1}_* \frac{im}{re}\right)}^{3} - {0}^{3}}{\left(\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \log base} \leadsto \frac{{\left(\tan^{-1}_* \frac{im}{re}\right)}^3 - 0}{\tan^{-1}_* \frac{im}{re} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)}\]
    11.2
  10. Applied final simplification
  11. Applied simplify to get
    \[\color{red}{\frac{{\left(\tan^{-1}_* \frac{im}{re}\right)}^3 - 0}{\tan^{-1}_* \frac{im}{re} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)}} \leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base} - 0}\]
    0.3

Original test:


(lambda ((re default) (im default) (base default))
  #:name "math.log/2 on complex, imaginary part"
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0)) (+ (* (log base) (log base)) (* 0 0))))