\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
Test:
math.log10 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 4.9 s
Input Error: 31.4
Output Error: 31.4
Log: ⚲
Profile: 🕒
\(\frac{3}{\frac{\frac{\log 10}{\frac{1}{3}}}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}}\)
  1. Started with
    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    31.4
  2. Applied simplify to get
    \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
    31.4
  3. Using strategy rm
    31.4
  4. Applied add-cube-cbrt to get
    \[\frac{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10}\]
    31.3
  5. Using strategy rm
    31.3
  6. Applied pow3 to get
    \[\frac{\log \color{red}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{\log 10}\]
    31.3
  7. Applied log-pow to get
    \[\frac{\color{red}{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{\log 10} \leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}{\log 10}\]
    31.4
  8. Applied associate-/l* to get
    \[\color{red}{\frac{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}} \leadsto \color{blue}{\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}}\]
    31.4
  9. Using strategy rm
    31.4
  10. Applied pow1/3 to get
    \[\frac{3}{\frac{\log 10}{\log \color{red}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}} \leadsto \frac{3}{\frac{\log 10}{\log \color{blue}{\left({\left(\sqrt{{re}^2 + im \cdot im}\right)}^{\frac{1}{3}}\right)}}}\]
    31.5
  11. Applied log-pow to get
    \[\frac{3}{\frac{\log 10}{\color{red}{\log \left({\left(\sqrt{{re}^2 + im \cdot im}\right)}^{\frac{1}{3}}\right)}}} \leadsto \frac{3}{\frac{\log 10}{\color{blue}{\frac{1}{3} \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right)}}}\]
    31.5
  12. Applied associate-/r* to get
    \[\frac{3}{\color{red}{\frac{\log 10}{\frac{1}{3} \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right)}}} \leadsto \frac{3}{\color{blue}{\frac{\frac{\log 10}{\frac{1}{3}}}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}}}\]
    31.4

Original test:


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