if im < -8.943399f+18

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   29.5
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   29.5
3. Using strategy rm
   29.5
4. Applied add-sqr-sqrt to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{{\left(\sqrt{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}^2}\]
   30.5
5. Applied simplify to get
   \[{\color{red}{\left(\sqrt{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}}^2 \leadsto {\color{blue}{\left(\sqrt{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}}^2\]
   30.5
6. Applied taylor to get
   \[{\left(\sqrt{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}^2 \leadsto {\left(\sqrt{\frac{\log \left(-1 \cdot im\right)}{\log base}}\right)}^2\]
   15.4
7. Taylor expanded around -inf to get
   \[{\left(\sqrt{\frac{\log \color{red}{\left(-1 \cdot im\right)}}{\log base}}\right)}^2 \leadsto {\left(\sqrt{\frac{\log \color{blue}{\left(-1 \cdot im\right)}}{\log base}}\right)}^2\]
   15.4
8. Applied simplify to get
   \[{\left(\sqrt{\frac{\log \left(-1 \cdot im\right)}{\log base}}\right)}^2 \leadsto \frac{\log \left(-im\right)}{\log base}\]
   0.4

---

9. Applied final simplification

if -8.943399f+18 < im < 7.8802614f+18

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   9.2
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   9.2
3. Using strategy rm
   9.2
4. Applied add-cube-cbrt to get
   \[\frac{\log base \cdot \log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)} + 0}{\log base \cdot \log base}\]
   9.3
5. Using strategy rm
   9.3
6. Applied cube-mult to get
   \[\frac{\log base \cdot \log \color{red}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)\right)} + 0}{\log base \cdot \log base}\]
   9.3
7. Applied log-prod to get
   \[\frac{\log base \cdot \color{red}{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)\right)} + 0}{\log base \cdot \log base}\]
   9.2
8. Applied distribute-lft-in to get
   \[\frac{\color{red}{\log base \cdot \left(\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\color{blue}{\left(\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)\right)} + 0}{\log base \cdot \log base}\]
   9.2
9. Applied simplify to get
   \[\frac{\left(\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \color{red}{\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\left(\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \color{blue}{\log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right) \cdot \left(\log base + \log base\right)}\right) + 0}{\log base \cdot \log base}\]
   9.3
10. Applied simplify to get
    \[\frac{\left(\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \color{red}{\log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)} \cdot \left(\log base + \log base\right)\right) + 0}{\log base \cdot \log base} \leadsto \frac{\left(\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \color{blue}{\log \left(\sqrt[3]{\sqrt{{im}^2 + re \cdot re}}\right)} \cdot \left(\log base + \log base\right)\right) + 0}{\log base \cdot \log base}\]
    9.3

if 7.8802614f+18 < im

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   29.7
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   29.7
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log im + 0}{\log base \cdot \log base}\]
   0.3
4. Taylor expanded around 0 to get
   \[\frac{\log base \cdot \log \color{red}{im} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{im} + 0}{\log base \cdot \log base}\]
   0.3
5. Applied simplify to get
   \[\color{red}{\frac{\log base \cdot \log im + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{\log im}{\log base}}\]
   0.3