if im < -7.25960525145735e+115

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   53.0
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   53.0
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{{\left(\log base\right)}^2}\]
   53.0
4. Taylor expanded around 0 to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{{\left(\log base\right)}^2}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{{\left(\log base\right)}^2}}\]
   53.0
5. Applied simplify to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{{\left(\log base\right)}^2}} \leadsto \color{blue}{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\]
   53.0
6. Applied taylor to get
   \[\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base} \leadsto \frac{\log \left(-1 \cdot im\right)}{\log base}\]
   0.4
7. Taylor expanded around -inf to get
   \[\frac{\log \color{red}{\left(-1 \cdot im\right)}}{\log base} \leadsto \frac{\log \color{blue}{\left(-1 \cdot im\right)}}{\log base}\]
   0.4
8. Applied simplify to get
   \[\color{red}{\frac{\log \left(-1 \cdot im\right)}{\log base}} \leadsto \color{blue}{\frac{\log \left(-im\right)}{\log base}}\]
   0.4

if -7.25960525145735e+115 < im < -6.407630827637074e-218 or 2.3918651035698753e-213 < im < 1.3041855436404e+134

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   18.2
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   18.2
3. Using strategy rm
   18.2
4. Applied *-un-lft-identity to get
   \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\log base \cdot \log base} \leadsto \frac{\color{blue}{1 \cdot \left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}}{\log base \cdot \log base}\]
   18.2
5. Applied times-frac to get
   \[\color{red}{\frac{1 \cdot \left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{1}{\log base} \cdot \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base}}\]
   18.2
6. Applied simplify to get
   \[\frac{1}{\log base} \cdot \color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base}} \leadsto \frac{1}{\log base} \cdot \color{blue}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}\]
   18.2
7. Applied simplify to get
   \[\frac{1}{\log base} \cdot \log \color{red}{\left(\sqrt{im \cdot im + re \cdot re}\right)} \leadsto \frac{1}{\log base} \cdot \log \color{blue}{\left(\sqrt{{im}^2 + re \cdot re}\right)}\]
   18.2

if -6.407630827637074e-218 < im < 2.3918651035698753e-213

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   32.8
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   32.8
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 1.3041855436404e+134 < im

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   56.6
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   56.6
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