if im < -200.41609f0

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   21.0
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   21.0
3. Using strategy rm
   21.0
4. Applied add-cube-cbrt to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}^3}\]
   21.1
5. Applied simplify to get
   \[{\color{red}{\left(\sqrt[3]{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}}^3\]
   21.1
6. Applied taylor to get
   \[{\left(\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{\log \left(-1 \cdot im\right)}{\log base}}\right)}^3\]
   0.8
7. Taylor expanded around -inf to get
   \[{\left(\sqrt[3]{\frac{\log \color{red}{\left(-1 \cdot im\right)}}{\log base}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{\log \color{blue}{\left(-1 \cdot im\right)}}{\log base}}\right)}^3\]
   0.8
8. Applied simplify to get
   \[{\left(\sqrt[3]{\frac{\log \left(-1 \cdot im\right)}{\log base}}\right)}^3 \leadsto \frac{\log \left(-im\right)}{\log base}\]
   0.4

---

9. Applied final simplification

if -200.41609f0 < im < 2.6527427f+10

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   10.4
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   10.4
3. Using strategy rm
   10.4
4. Applied add-cube-cbrt to get
   \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right)}\right)}^3} + 0}{\log base \cdot \log base}\]
   10.7

if 2.6527427f+10 < im

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   23.2
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   23.2
3. Using strategy rm
   23.2
4. Applied add-cbrt-cube to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{red}{\log base}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{blue}{\sqrt[3]{{\left(\log base\right)}^3}}}\]
   23.2
5. Applied add-cbrt-cube to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\log base} \cdot \sqrt[3]{{\left(\log base\right)}^3}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{\sqrt[3]{{\left(\log base\right)}^3}} \cdot \sqrt[3]{{\left(\log base\right)}^3}}\]
   23.2
6. Applied cbrt-unprod to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\sqrt[3]{{\left(\log base\right)}^3} \cdot \sqrt[3]{{\left(\log base\right)}^3}}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}}\]
   23.2
7. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}} \leadsto \frac{\log base \cdot \log im + 0}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}\]
   0.4
8. Taylor expanded around 0 to get
   \[\frac{\log base \cdot \log \color{red}{im} + 0}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}} \leadsto \frac{\log base \cdot \log \color{blue}{im} + 0}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}\]
   0.4
9. Applied simplify to get
   \[\frac{\log base \cdot \log im + 0}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}} \leadsto \frac{\log im \cdot \log base}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}\]
   0.4

---

10. Applied final simplification
11. Applied simplify to get
    \[\color{red}{\frac{\log im \cdot \log base}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}} \leadsto \color{blue}{\frac{\log base \cdot \log im}{\log base \cdot \log base}}\]
    0.4