\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
Test:
math.log10 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 4.4 s
Input Error: 30.8
Output Error: 0.6
Log: ⚲
Profile: 🕒
\(\frac{\log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{\frac{\log 10}{3}}\)
  1. Started with
    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    30.8
  2. Applied simplify to get
    \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{im^2 + re^2}^*\right)}{\log 10}}\]
    0.6
  3. Using strategy rm
    0.6
  4. Applied add-cube-cbrt to get
    \[\frac{\log \color{red}{\left(\sqrt{im^2 + re^2}^*\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}^3\right)}}{\log 10}\]
    0.6
  5. Using strategy rm
    0.6
  6. Applied *-un-lft-identity to get
    \[\frac{\log \left({\left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}^3\right)}{\color{red}{\log 10}} \leadsto \frac{\log \left({\left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}^3\right)}{\color{blue}{1 \cdot \log 10}}\]
    0.6
  7. Applied pow3 to get
    \[\frac{\log \color{red}{\left({\left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}^3\right)}}{1 \cdot \log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}^{3}\right)}}{1 \cdot \log 10}\]
    0.6
  8. Applied log-pow to get
    \[\frac{\color{red}{\log \left({\left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}^{3}\right)}}{1 \cdot \log 10} \leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}}{1 \cdot \log 10}\]
    0.6
  9. Applied times-frac to get
    \[\color{red}{\frac{3 \cdot \log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{1 \cdot \log 10}} \leadsto \color{blue}{\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{\log 10}}\]
    0.6
  10. Applied taylor to get
    \[\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{\log 10} \leadsto \frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{\log 10}\]
    0.6
  11. Taylor expanded around 0 to get
    \[\frac{3}{1} \cdot \color{red}{\frac{\log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{\log 10}} \leadsto \frac{3}{1} \cdot \color{blue}{\frac{\log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{\log 10}}\]
    0.6
  12. Applied simplify to get
    \[\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{\log 10} \leadsto \frac{\log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{\frac{\log 10}{\frac{3}{1}}}\]
    0.6
  13. Applied final simplification
  14. Applied simplify to get
    \[\color{red}{\frac{\log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{\frac{\log 10}{\frac{3}{1}}}} \leadsto \color{blue}{\frac{\log \left(\sqrt[3]{\sqrt{im^2 + re^2}^*}\right)}{\frac{\log 10}{3}}}\]
    0.6

Original test:


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