if im < -7.479207f+13

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   25.7
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   25.7
3. Using strategy rm
   25.7
4. Applied add-exp-log to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{red}{\log base}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{blue}{e^{\log \left(\log base\right)}}}\]
   28.6
5. Applied add-exp-log to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\log base} \cdot e^{\log \left(\log base\right)}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{e^{\log \left(\log base\right)}} \cdot e^{\log \left(\log base\right)}}\]
   28.7
6. Applied prod-exp to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{e^{\log \left(\log base\right)} \cdot e^{\log \left(\log base\right)}}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{e^{\log \left(\log base\right) + \log \left(\log base\right)}}}\]
   28.7
7. Applied add-exp-log to get
   \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{e^{\log \left(\log base\right) + \log \left(\log base\right)}} \leadsto \frac{\color{blue}{e^{\log \left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}}}{e^{\log \left(\log base\right) + \log \left(\log base\right)}}\]
   28.7
8. Applied div-exp to get
   \[\color{red}{\frac{e^{\log \left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}}{e^{\log \left(\log base\right) + \log \left(\log base\right)}}} \leadsto \color{blue}{e^{\log \left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right) - \left(\log \left(\log base\right) + \log \left(\log base\right)\right)}}\]
   28.7
9. Applied simplify to get
   \[e^{\color{red}{\log \left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right) - \left(\log \left(\log base\right) + \log \left(\log base\right)\right)}} \leadsto e^{\color{blue}{\log \left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right) - \log \left(\log base\right)}}\]
   28.6
10. Applied taylor to get
    \[e^{\log \left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right) - \log \left(\log base\right)} \leadsto e^{\log \left(\log \left(-1 \cdot im\right)\right) - \log \left(\log base\right)}\]
    15.6
11. Taylor expanded around -inf to get
    \[e^{\log \left(\log \color{red}{\left(-1 \cdot im\right)}\right) - \log \left(\log base\right)} \leadsto e^{\log \left(\log \color{blue}{\left(-1 \cdot im\right)}\right) - \log \left(\log base\right)}\]
    15.6
12. Applied simplify to get
    \[e^{\log \left(\log \left(-1 \cdot im\right)\right) - \log \left(\log base\right)} \leadsto \frac{\log \left(-im\right)}{\log base}\]
    0.4

---

13. Applied final simplification

if -7.479207f+13 < im < 4804.8477f0

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   9.5
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   9.5
3. Using strategy rm
   9.5
4. Applied add-cube-cbrt to get
   \[\frac{\log base \cdot \log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)} + 0}{\log base \cdot \log base}\]
   9.5
5. Using strategy rm
   9.5
6. Applied cube-mult to get
   \[\frac{\log base \cdot \log \color{red}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)\right)} + 0}{\log base \cdot \log base}\]
   9.5
7. Applied log-prod to get
   \[\frac{\log base \cdot \color{red}{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)\right)} + 0}{\log base \cdot \log base}\]
   9.5
8. Applied distribute-lft-in to get
   \[\frac{\color{red}{\log base \cdot \left(\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\color{blue}{\left(\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)\right)} + 0}{\log base \cdot \log base}\]
   9.5
9. Applied simplify to get
   \[\frac{\left(\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \color{red}{\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}} \cdot \sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\left(\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \color{blue}{\log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right) \cdot \left(\log base + \log base\right)}\right) + 0}{\log base \cdot \log base}\]
   9.5
10. Applied simplify to get
    \[\frac{\left(\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \color{red}{\log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)} \cdot \left(\log base + \log base\right)\right) + 0}{\log base \cdot \log base} \leadsto \frac{\left(\log base \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right) + \color{blue}{\log \left(\sqrt[3]{\sqrt{{im}^2 + re \cdot re}}\right)} \cdot \left(\log base + \log base\right)\right) + 0}{\log base \cdot \log base}\]
    9.5

if 4804.8477f0 < im

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   21.5
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   21.5
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log im + 0}{\log base \cdot \log base}\]
   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