\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
Test:
math.log10 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 3.1 s
Input Error: 14.6
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\sqrt[3]{{\left(\frac{\log \left(\sqrt{im^2 + re^2}^*\right)}{\log 10}\right)}^3}\)
  1. Started with
    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    14.6
  2. Applied simplify to get
    \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{im^2 + re^2}^*\right)}{\log 10}}\]
    0.3
  3. Using strategy rm
    0.3
  4. Applied add-cbrt-cube to get
    \[\frac{\log \left(\sqrt{im^2 + re^2}^*\right)}{\color{red}{\log 10}} \leadsto \frac{\log \left(\sqrt{im^2 + re^2}^*\right)}{\color{blue}{\sqrt[3]{{\left(\log 10\right)}^3}}}\]
    0.3
  5. Applied add-cbrt-cube to get
    \[\frac{\color{red}{\log \left(\sqrt{im^2 + re^2}^*\right)}}{\sqrt[3]{{\left(\log 10\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\log \left(\sqrt{im^2 + re^2}^*\right)\right)}^3}}}{\sqrt[3]{{\left(\log 10\right)}^3}}\]
    0.3
  6. Applied cbrt-undiv to get
    \[\color{red}{\frac{\sqrt[3]{{\left(\log \left(\sqrt{im^2 + re^2}^*\right)\right)}^3}}{\sqrt[3]{{\left(\log 10\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(\log \left(\sqrt{im^2 + re^2}^*\right)\right)}^3}{{\left(\log 10\right)}^3}}}\]
    0.3
  7. Applied simplify to get
    \[\sqrt[3]{\color{red}{\frac{{\left(\log \left(\sqrt{im^2 + re^2}^*\right)\right)}^3}{{\left(\log 10\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{im^2 + re^2}^*\right)}{\log 10}\right)}^3}}\]
    0.3

Original test:


(lambda ((re default) (im default))
  #:name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))