if im < -9.791989538518476e+123

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   54.8
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   54.8
3. Using strategy rm
   54.8
4. Applied *-un-lft-identity to get
   \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\log base \cdot \log base} \leadsto \frac{\color{blue}{1 \cdot \left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}}{\log base \cdot \log base}\]
   54.8
5. Applied times-frac to get
   \[\color{red}{\frac{1 \cdot \left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{1}{\log base} \cdot \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base}}\]
   54.8
6. Applied simplify to get
   \[\frac{1}{\log base} \cdot \color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base}} \leadsto \frac{1}{\log base} \cdot \color{blue}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}\]
   54.8
7. Applied taylor to get
   \[\frac{1}{\log base} \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right) \leadsto \frac{1}{\log base} \cdot \log \left(-1 \cdot im\right)\]
   0.4
8. Taylor expanded around -inf to get
   \[\frac{1}{\log base} \cdot \log \color{red}{\left(-1 \cdot im\right)} \leadsto \frac{1}{\log base} \cdot \log \color{blue}{\left(-1 \cdot im\right)}\]
   0.4
9. Applied simplify to get
   \[\color{red}{\frac{1}{\log base} \cdot \log \left(-1 \cdot im\right)} \leadsto \color{blue}{\frac{\log \left(-im\right)}{\log base}}\]
   0.3

if -9.791989538518476e+123 < im < 5.11104370095671e-307 or 2.9099428310613175e-275 < im < 1.3135365250902935e+68

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   20.7
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   20.7
3. Using strategy rm
   20.7
4. Applied div-inv to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right) \cdot \frac{1}{\log base \cdot \log base}}\]
   20.7

if 5.11104370095671e-307 < im < 2.9099428310613175e-275

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   33.8
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   33.8
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \left(-1 \cdot re\right) + 0}{\log base \cdot \log base}\]
   0.5
4. Taylor expanded around -inf to get
   \[\frac{\log base \cdot \log \color{red}{\left(-1 \cdot re\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{\left(-1 \cdot re\right)} + 0}{\log base \cdot \log base}\]
   0.5
5. Applied simplify to get
   \[\color{red}{\frac{\log base \cdot \log \left(-1 \cdot re\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{\log \left(-re\right)}{\log base}}\]
   0.4

if 1.3135365250902935e+68 < im

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   46.9
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   46.9
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log im + 0}{\log base \cdot \log base}\]
   0.5
4. Taylor expanded around 0 to get
   \[\frac{\log base \cdot \log \color{red}{im} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{im} + 0}{\log base \cdot \log base}\]
   0.5
5. Applied simplify to get
   \[\color{red}{\frac{\log base \cdot \log im + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{\log im}{\log base}}\]
   0.4