\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
Test:
math.log10 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 11.1 s
Input Error: 30.4
Output Error: 14.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\log \left(-re\right)}{\log 10} & \text{when } re \le -7.515261356325578 \cdot 10^{+104} \\ \sqrt[3]{{\left(\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}\right)}^3} & \text{when } re \le 4.3827392081136433 \cdot 10^{+102} \\ \frac{3}{\log 10} \cdot \log \left(\sqrt[3]{re}\right) & \text{otherwise} \end{cases}\)

    if re < -7.515261356325578e+104

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      51.8
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      51.8
    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 -7.515261356325578e+104 < re < 4.3827392081136433e+102

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      20.5
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      20.5
    3. Using strategy rm
      20.5
    4. Applied add-cbrt-cube to get
      \[\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\color{red}{\log 10}} \leadsto \frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\color{blue}{\sqrt[3]{{\left(\log 10\right)}^3}}}\]
      21.1
    5. Applied add-cbrt-cube to get
      \[\frac{\color{red}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}}{\sqrt[3]{{\left(\log 10\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}{\sqrt[3]{{\left(\log 10\right)}^3}}\]
      21.0
    6. Applied cbrt-undiv to get
      \[\color{red}{\frac{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}{\sqrt[3]{{\left(\log 10\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}{{\left(\log 10\right)}^3}}}\]
      20.6
    7. Applied simplify to get
      \[\sqrt[3]{\color{red}{\frac{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}{{\left(\log 10\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}\right)}^3}}\]
      20.5

    if 4.3827392081136433e+102 < re

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      51.0
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      51.0
    3. Using strategy rm
      51.0
    4. Applied add-cube-cbrt to get
      \[\frac{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10}\]
      51.0
    5. Using strategy rm
      51.0
    6. Applied pow3 to get
      \[\frac{\log \color{red}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{\log 10}\]
      51.0
    7. Applied log-pow to get
      \[\frac{\color{red}{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{\log 10} \leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}{\log 10}\]
      51.0
    8. Applied associate-/l* to get
      \[\color{red}{\frac{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}} \leadsto \color{blue}{\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}}\]
      51.0
    9. Applied taylor to get
      \[\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}} \leadsto \frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{re}\right)}}\]
      0.6
    10. Taylor expanded around inf to get
      \[\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\color{red}{re}}\right)}} \leadsto \frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\color{blue}{re}}\right)}}\]
      0.6
    11. Applied simplify to get
      \[\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{re}\right)}} \leadsto \frac{3}{\log 10} \cdot \log \left(\sqrt[3]{re}\right)\]
      0.7
    12. Applied final simplification
  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)))