if re < -1.4799672839841313e+124

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   54.2
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
   54.2
3. Using strategy rm
   54.2
4. Applied add-cube-cbrt to get
   \[\frac{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10}\]
   54.2
5. Using strategy rm
   54.2
6. Applied *-un-lft-identity to get
   \[\frac{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}{\color{red}{\log 10}} \leadsto \frac{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}{\color{blue}{1 \cdot \log 10}}\]
   54.2
7. Applied pow3 to get
   \[\frac{\log \color{red}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{1 \cdot \log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{1 \cdot \log 10}\]
   54.2
8. Applied log-pow to get
   \[\frac{\color{red}{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{1 \cdot \log 10} \leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}{1 \cdot \log 10}\]
   54.2
9. Applied times-frac to get
   \[\color{red}{\frac{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{1 \cdot \log 10}} \leadsto \color{blue}{\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}}\]
   54.2
10. Using strategy rm
    54.2
11. Applied add-cbrt-cube to get
    \[\frac{3}{1} \cdot \color{red}{\frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}} \leadsto \frac{3}{1} \cdot \color{blue}{\sqrt[3]{{\left(\frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}\right)}^3}}\]
    54.2
12. Applied add-cbrt-cube to get
    \[\color{red}{\frac{3}{1}} \cdot \sqrt[3]{{\left(\frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}\right)}^3} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{3}{1}\right)}^3}} \cdot \sqrt[3]{{\left(\frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}\right)}^3}\]
    54.2
13. Applied cbrt-unprod to get
    \[\color{red}{\sqrt[3]{{\left(\frac{3}{1}\right)}^3} \cdot \sqrt[3]{{\left(\frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}\right)}^3}} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{3}{1}\right)}^3 \cdot {\left(\frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}\right)}^3}}\]
    54.2
14. Applied taylor to get
    \[\sqrt[3]{{\left(\frac{3}{1}\right)}^3 \cdot {\left(\frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{3}{1}\right)}^3 \cdot {\left(\frac{\log \left(\sqrt[3]{-1 \cdot re}\right)}{\log 10}\right)}^3}\]
    0.7
15. Taylor expanded around -inf to get
    \[\sqrt[3]{{\left(\frac{3}{1}\right)}^3 \cdot {\left(\frac{\log \left(\sqrt[3]{\color{red}{-1 \cdot re}}\right)}{\log 10}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{3}{1}\right)}^3 \cdot {\left(\frac{\log \left(\sqrt[3]{\color{blue}{-1 \cdot re}}\right)}{\log 10}\right)}^3}\]
    0.7
16. Applied simplify to get
    \[\sqrt[3]{{\left(\frac{3}{1}\right)}^3 \cdot {\left(\frac{\log \left(\sqrt[3]{-1 \cdot re}\right)}{\log 10}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{\log \left(\sqrt[3]{-re}\right)}{\log 10}\right)}^3 \cdot \frac{{3}^3}{1}}\]
    0.7

---

17. Applied final simplification
18. Applied simplify to get
    \[\color{red}{\sqrt[3]{{\left(\frac{\log \left(\sqrt[3]{-re}\right)}{\log 10}\right)}^3 \cdot \frac{{3}^3}{1}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(\log \left(\sqrt[3]{-re}\right)\right)}^3}{{\left(\frac{\log 10}{3}\right)}^3}}}\]
    0.7

if -1.4799672839841313e+124 < re < 3.751610550051807e+91

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   21.0
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
   21.0
3. Using strategy rm
   21.0
4. Applied add-cube-cbrt to get
   \[\frac{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10}\]
   21.0
5. Using strategy rm
   21.0
6. Applied pow3 to get
   \[\frac{\log \color{red}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{\log 10}\]
   21.0
7. Applied log-pow to get
   \[\frac{\color{red}{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{\log 10} \leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}{\log 10}\]
   21.0
8. Applied associate-/l* to get
   \[\color{red}{\frac{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}} \leadsto \color{blue}{\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}}\]
   21.0

if 3.751610550051807e+91 < re

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   50.1
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
   50.1
3. Using strategy rm
   50.1
4. Applied add-cube-cbrt to get
   \[\frac{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10}\]
   50.1
5. Using strategy rm
   50.1
6. Applied *-un-lft-identity to get
   \[\frac{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}{\color{red}{\log 10}} \leadsto \frac{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}{\color{blue}{1 \cdot \log 10}}\]
   50.1
7. Applied pow3 to get
   \[\frac{\log \color{red}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{1 \cdot \log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{1 \cdot \log 10}\]
   50.1
8. Applied log-pow to get
   \[\frac{\color{red}{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{1 \cdot \log 10} \leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}{1 \cdot \log 10}\]
   50.1
9. Applied times-frac to get
   \[\color{red}{\frac{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{1 \cdot \log 10}} \leadsto \color{blue}{\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}}\]
   50.1
10. Applied taylor to get
    \[\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10} \leadsto \frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{re}\right)}{\log 10}\]
    0.6
11. Taylor expanded around inf to get
    \[\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\color{red}{re}}\right)}{\log 10} \leadsto \frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\color{blue}{re}}\right)}{\log 10}\]
    0.6
12. Applied simplify to get
    \[\color{red}{\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{re}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt[3]{re}\right)}{\frac{\log 10}{3}}}\]
    0.6