\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
Test:
math.log10 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 5.8 s
Input Error: 30.9
Output Error: 31.0
Log: ⚲
Profile: 🕒
\(\frac{1}{\sqrt[3]{\frac{{\left(\log 10\right)}^3}{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}\)
  1. Started with
    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    30.9
  2. Applied simplify to get
    \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
    30.9
  3. Using strategy rm
    30.9
  4. Applied clear-num to get
    \[\color{red}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}}}\]
    30.9
  5. Using strategy rm
    30.9
  6. Applied add-cbrt-cube to get
    \[\frac{1}{\frac{\log 10}{\color{red}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}}} \leadsto \frac{1}{\frac{\log 10}{\color{blue}{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}}\]
    31.0
  7. Applied add-cbrt-cube to get
    \[\frac{1}{\frac{\color{red}{\log 10}}{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}} \leadsto \frac{1}{\frac{\color{blue}{\sqrt[3]{{\left(\log 10\right)}^3}}}{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}\]
    31.3
  8. Applied cbrt-undiv to get
    \[\frac{1}{\color{red}{\frac{\sqrt[3]{{\left(\log 10\right)}^3}}{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}} \leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{{\left(\log 10\right)}^3}{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}}\]
    31.0

Original test:


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