if im < -1.3639623149762622e+140

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   58.0
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   58.0
3. Using strategy rm
   58.0
4. Applied add-cbrt-cube to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}\right)}^3}}\]
   58.1
5. Applied simplify to get
   \[\sqrt[3]{\color{red}{{\left(\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}\right)}^3}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}\right)}^3}}\]
   58.1
6. Applied taylor to get
   \[\sqrt[3]{{\left(\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{\log \left(-1 \cdot im\right)}{\log base}\right)}^3}\]
   0.7
7. Taylor expanded around -inf to get
   \[\sqrt[3]{{\left(\frac{\log \color{red}{\left(-1 \cdot im\right)}}{\log base}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{\log \color{blue}{\left(-1 \cdot im\right)}}{\log base}\right)}^3}\]
   0.7
8. Applied simplify to get
   \[\sqrt[3]{{\left(\frac{\log \left(-1 \cdot im\right)}{\log base}\right)}^3} \leadsto \frac{\log \left(-im\right)}{\log base}\]
   0.4

---

9. Applied final simplification

if -1.3639623149762622e+140 < im < 2.56976209479494e+109

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   19.9
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   19.9
3. Using strategy rm
   19.9
4. Applied *-un-lft-identity to get
   \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\log base \cdot \log base} \leadsto \frac{\color{blue}{1 \cdot \left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}}{\log base \cdot \log base}\]
   19.9
5. Applied times-frac to get
   \[\color{red}{\frac{1 \cdot \left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{1}{\log base} \cdot \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base}}\]
   19.9
6. Applied simplify to get
   \[\frac{1}{\log base} \cdot \color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base}} \leadsto \frac{1}{\log base} \cdot \color{blue}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}\]
   19.9
7. Applied simplify to get
   \[\frac{1}{\log base} \cdot \log \color{red}{\left(\sqrt{im \cdot im + re \cdot re}\right)} \leadsto \frac{1}{\log base} \cdot \log \color{blue}{\left(\sqrt{{im}^2 + re \cdot re}\right)}\]
   19.9

if 2.56976209479494e+109 < im

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   50.7
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   50.7
3. Using strategy rm
   50.7
4. Applied div-inv to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right) \cdot \frac{1}{\log base \cdot \log base}}\]
   50.7
5. Applied taylor to get
   \[\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right) \cdot \frac{1}{\log base \cdot \log base} \leadsto \left(\log base \cdot \log im + 0\right) \cdot \frac{1}{\log base \cdot \log base}\]
   0.5
6. Taylor expanded around 0 to get
   \[\left(\log base \cdot \log \color{red}{im} + 0\right) \cdot \frac{1}{\log base \cdot \log base} \leadsto \left(\log base \cdot \log \color{blue}{im} + 0\right) \cdot \frac{1}{\log base \cdot \log base}\]
   0.5
7. Applied simplify to get
   \[\left(\log base \cdot \log im + 0\right) \cdot \frac{1}{\log base \cdot \log base} \leadsto \frac{\log im \cdot \log base}{\log base \cdot \log base}\]
   0.5

---

8. Applied final simplification
9. Applied simplify to get
   \[\color{red}{\frac{\log im \cdot \log base}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{\log im}{\log base}}\]
   0.4