\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
Test:
math.log10 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 13.1 s
Input Error: 31.9
Output Error: 14.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\log \left(-re\right)}{\log 10} & \text{when } re \le -1.2232288122046868 \cdot 10^{+84} \\ \sqrt[3]{{\left(\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}\right)}^3} & \text{when } re \le 1.3480729395074825 \cdot 10^{+78} \\ \frac{\log re}{\log 10} & \text{otherwise} \end{cases}\)

    if re < -1.2232288122046868e+84

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      49.6
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      49.6
    3. Applied taylor to get
      \[\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10} \leadsto \frac{\log \left(-1 \cdot re\right)}{\log 10}\]
      0.6
    4. Taylor expanded around -inf to get
      \[\frac{\log \color{red}{\left(-1 \cdot re\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10}\]
      0.6
    5. Applied simplify to get
      \[\color{red}{\frac{\log \left(-1 \cdot re\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(-re\right)}{\log 10}}\]
      0.6

    if -1.2232288122046868e+84 < re < 1.3480729395074825e+78

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      22.1
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      22.1
    3. Using strategy rm
      22.1
    4. Applied add-cbrt-cube to get
      \[\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\color{red}{\log 10}} \leadsto \frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\color{blue}{\sqrt[3]{{\left(\log 10\right)}^3}}}\]
      22.7
    5. Applied add-cbrt-cube to get
      \[\frac{\color{red}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}}{\sqrt[3]{{\left(\log 10\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}{\sqrt[3]{{\left(\log 10\right)}^3}}\]
      22.6
    6. Applied cbrt-undiv to get
      \[\color{red}{\frac{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}{\sqrt[3]{{\left(\log 10\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}{{\left(\log 10\right)}^3}}}\]
      22.2
    7. Applied simplify to get
      \[\sqrt[3]{\color{red}{\frac{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}{{\left(\log 10\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}\right)}^3}}\]
      22.1

    if 1.3480729395074825e+78 < re

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      47.0
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      47.0
    3. Applied taylor to get
      \[\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10} \leadsto \frac{\log re}{\log 10}\]
      0.6
    4. Taylor expanded around inf to get
      \[\frac{\log \color{red}{re}}{\log 10} \leadsto \frac{\log \color{blue}{re}}{\log 10}\]
      0.6
  1. Removed slow pow expressions

Original test:


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