if re < -1.674345f+14

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   25.6
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   25.6
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \left(-1 \cdot re\right) + 0}{\log base \cdot \log base}\]
   0.4
4. Taylor expanded around -inf to get
   \[\frac{\log base \cdot \log \color{red}{\left(-1 \cdot re\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{\left(-1 \cdot re\right)} + 0}{\log base \cdot \log base}\]
   0.4
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 -1.674345f+14 < re < 1.695064f+08

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   9.8
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   9.8
3. Using strategy rm
   9.8
4. Applied pow1 to get
   \[\frac{\log base \cdot \color{red}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \color{blue}{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^{1}} + 0}{\log base \cdot \log base}\]
   9.8
5. Applied pow1 to get
   \[\frac{\color{red}{\log base} \cdot {\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^{1} + 0}{\log base \cdot \log base} \leadsto \frac{\color{blue}{{\left(\log base\right)}^{1}} \cdot {\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^{1} + 0}{\log base \cdot \log base}\]
   9.8
6. Applied pow-prod-down to get
   \[\frac{\color{red}{{\left(\log base\right)}^{1} \cdot {\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^{1}} + 0}{\log base \cdot \log base} \leadsto \frac{\color{blue}{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^{1}} + 0}{\log base \cdot \log base}\]
   9.8

if 1.695064f+08 < re

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   22.1
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   22.1
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log re + 0}{\log base \cdot \log base}\]
   0.4
4. Taylor expanded around inf to get
   \[\frac{\log base \cdot \log \color{red}{re} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{re} + 0}{\log base \cdot \log base}\]
   0.4
5. Applied simplify to get
   \[\color{red}{\frac{\log base \cdot \log re + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{\log re}{\log base}}\]
   0.4