if re < -5.5866455f+09

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   23.1
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   23.1
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \left(-1 \cdot re\right) + 0}{\log base \cdot \log base}\]
   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 -5.5866455f+09 < re < -1.5735045f-29 or 3.3031373f-09 < re < 13.069779f0

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   8.7
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   8.7
3. Using strategy rm
   8.7
4. Applied add-cbrt-cube to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{red}{\log base}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{blue}{\sqrt[3]{{\left(\log base\right)}^3}}}\]
   8.7
5. Applied add-cbrt-cube to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\log base} \cdot \sqrt[3]{{\left(\log base\right)}^3}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{\sqrt[3]{{\left(\log base\right)}^3}} \cdot \sqrt[3]{{\left(\log base\right)}^3}}\]
   8.7
6. Applied cbrt-unprod to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\sqrt[3]{{\left(\log base\right)}^3} \cdot \sqrt[3]{{\left(\log base\right)}^3}}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}}\]
   8.8

if -1.5735045f-29 < re < 3.3031373f-09

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   14.1
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   14.1
3. Using strategy rm
   14.1
4. Applied add-cube-cbrt to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{red}{\log base}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{blue}{{\left(\sqrt[3]{\log base}\right)}^3}}\]
   14.3
5. Applied add-cube-cbrt to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\log base} \cdot {\left(\sqrt[3]{\log base}\right)}^3} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{{\left(\sqrt[3]{\log base}\right)}^3} \cdot {\left(\sqrt[3]{\log base}\right)}^3}\]
   14.5
6. Applied cube-unprod to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{{\left(\sqrt[3]{\log base}\right)}^3 \cdot {\left(\sqrt[3]{\log base}\right)}^3}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{{\left(\sqrt[3]{\log base} \cdot \sqrt[3]{\log base}\right)}^3}}\]
   14.5
7. Applied add-cube-cbrt to get
   \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{{\left(\sqrt[3]{\log base} \cdot \sqrt[3]{\log base}\right)}^3} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}\right)}^3}}{{\left(\sqrt[3]{\log base} \cdot \sqrt[3]{\log base}\right)}^3}\]
   14.6
8. Applied cube-undiv to get
   \[\color{red}{\frac{{\left(\sqrt[3]{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}\right)}^3}{{\left(\sqrt[3]{\log base} \cdot \sqrt[3]{\log base}\right)}^3}} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\sqrt[3]{\log base} \cdot \sqrt[3]{\log base}}\right)}^3}\]
   14.6
9. Applied simplify to get
   \[{\color{red}{\left(\frac{\sqrt[3]{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\sqrt[3]{\log base} \cdot \sqrt[3]{\log base}}\right)}}^3 \leadsto {\color{blue}{\left(\frac{\sqrt[3]{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}}{{\left(\sqrt[3]{\log base}\right)}^2}\right)}}^3\]
   14.6
10. Applied taylor to get
    \[{\left(\frac{\sqrt[3]{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}}{{\left(\sqrt[3]{\log base}\right)}^2}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{\log \left(-1 \cdot im\right) \cdot \log base}}{{\left(\sqrt[3]{\log base}\right)}^2}\right)}^3\]
    1.4
11. Taylor expanded around -inf to get
    \[{\left(\frac{\sqrt[3]{\log \color{red}{\left(-1 \cdot im\right)} \cdot \log base}}{{\left(\sqrt[3]{\log base}\right)}^2}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{\log \color{blue}{\left(-1 \cdot im\right)} \cdot \log base}}{{\left(\sqrt[3]{\log base}\right)}^2}\right)}^3\]
    1.4
12. Applied simplify to get
    \[{\left(\frac{\sqrt[3]{\log \left(-1 \cdot im\right) \cdot \log base}}{{\left(\sqrt[3]{\log base}\right)}^2}\right)}^3 \leadsto \frac{\log base}{\log base} \cdot \frac{\log \left(-im\right)}{\log base}\]
    0.3

---

13. Applied final simplification
14. Applied simplify to get
    \[\color{red}{\frac{\log base}{\log base} \cdot \frac{\log \left(-im\right)}{\log base}} \leadsto \color{blue}{\frac{\log \left(-im\right)}{\log base}}\]
    0.3

if 13.069779f0 < re

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   20.5
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   20.5
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