if im < -9.2381495f+10

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   24.0
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   24.0
3. Using strategy rm
   24.0
4. Applied add-cube-cbrt to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\log base \cdot \log base}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{{\left(\sqrt[3]{\log base \cdot \log base}\right)}^3}}\]
   24.1
5. Applied add-cube-cbrt to get
   \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{{\left(\sqrt[3]{\log base \cdot \log base}\right)}^3} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}\right)}^3}}{{\left(\sqrt[3]{\log base \cdot \log base}\right)}^3}\]
   24.1
6. Applied cube-undiv to get
   \[\color{red}{\frac{{\left(\sqrt[3]{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}\right)}^3}{{\left(\sqrt[3]{\log base \cdot \log base}\right)}^3}} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\sqrt[3]{\log base \cdot \log base}}\right)}^3}\]
   24.1
7. Applied simplify to get
   \[{\color{red}{\left(\frac{\sqrt[3]{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\sqrt[3]{\log base \cdot \log base}}\right)}}^3 \leadsto {\color{blue}{\left(\frac{\sqrt[3]{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}}{\sqrt[3]{\log base \cdot \log base}}\right)}}^3\]
   24.1
8. Applied taylor to get
   \[{\left(\frac{\sqrt[3]{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}}{\sqrt[3]{\log base \cdot \log base}}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{\log \left(-1 \cdot im\right) \cdot \log base}}{\sqrt[3]{\log base \cdot \log base}}\right)}^3\]
   1.0
9. Taylor expanded around -inf to get
   \[{\left(\frac{\sqrt[3]{\log \color{red}{\left(-1 \cdot im\right)} \cdot \log base}}{\sqrt[3]{\log base \cdot \log base}}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{\log \color{blue}{\left(-1 \cdot im\right)} \cdot \log base}}{\sqrt[3]{\log base \cdot \log base}}\right)}^3\]
   1.0
10. Applied simplify to get
    \[{\left(\frac{\sqrt[3]{\log \left(-1 \cdot im\right) \cdot \log base}}{\sqrt[3]{\log base \cdot \log base}}\right)}^3 \leadsto \frac{\log \left(-im\right) \cdot \log base}{\log base \cdot \log base}\]
    0.4

---

11. Applied final simplification
12. Applied simplify to get
    \[\color{red}{\frac{\log \left(-im\right) \cdot \log base}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{\log \left(-im\right)}{\log base}}\]
    0.4

if -9.2381495f+10 < im < 3.2481306f+11

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   10.1
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   10.1
3. Using strategy rm
   10.1
4. Applied pow1 to get
   \[\frac{\log base \cdot \color{red}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \color{blue}{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^{1}} + 0}{\log base \cdot \log base}\]
   10.1
5. Applied pow1 to get
   \[\frac{\color{red}{\log base} \cdot {\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^{1} + 0}{\log base \cdot \log base} \leadsto \frac{\color{blue}{{\left(\log base\right)}^{1}} \cdot {\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^{1} + 0}{\log base \cdot \log base}\]
   10.1
6. Applied pow-prod-down to get
   \[\frac{\color{red}{{\left(\log base\right)}^{1} \cdot {\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^{1}} + 0}{\log base \cdot \log base} \leadsto \frac{\color{blue}{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^{1}} + 0}{\log base \cdot \log base}\]
   10.2

if 3.2481306f+11 < im

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   24.0
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   24.0
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log im + 0}{\log base \cdot \log base}\]
   0.4
4. Taylor expanded around 0 to get
   \[\frac{\log base \cdot \log \color{red}{im} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{im} + 0}{\log base \cdot \log base}\]
   0.4
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