if re < -4.990674843592913e+108

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   51.7
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   51.7
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \left(-1 \cdot re\right) + 0}{\log base \cdot \log base}\]
   0.5
4. Taylor expanded around -inf to get
   \[\frac{\log base \cdot \log \color{red}{\left(-1 \cdot re\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{\left(-1 \cdot re\right)} + 0}{\log base \cdot \log base}\]
   0.5
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 -4.990674843592913e+108 < re < -3.965426564809338e-197

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   18.0
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   18.0
3. Using strategy rm
   18.0
4. Applied add-cbrt-cube to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{red}{\log base}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{blue}{\sqrt[3]{{\left(\log base\right)}^3}}}\]
   18.1
5. Applied add-cbrt-cube to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\log base} \cdot \sqrt[3]{{\left(\log base\right)}^3}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{\sqrt[3]{{\left(\log base\right)}^3}} \cdot \sqrt[3]{{\left(\log base\right)}^3}}\]
   18.3
6. Applied cbrt-unprod to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\sqrt[3]{{\left(\log base\right)}^3} \cdot \sqrt[3]{{\left(\log base\right)}^3}}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}}\]
   18.1
7. Applied add-cbrt-cube to get
   \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}}}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}\]
   18.2
8. Applied cbrt-undiv to get
   \[\color{red}{\frac{\sqrt[3]{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}}\]
   18.2
9. Applied simplify to get
   \[\sqrt[3]{\color{red}{\frac{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{\frac{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^3}{{\left(\log base\right)}^3}}}\]
   18.1
10. Applied simplify to get
    \[\sqrt[3]{\frac{\color{red}{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^3}}{{\left(\log base\right)}^3}} \leadsto \sqrt[3]{\frac{\color{blue}{{\left(\log \left(\sqrt{{im}^2 + re \cdot re}\right)\right)}^3}}{{\left(\log base\right)}^3}}\]
    18.1

if -3.965426564809338e-197 < re < 2.2020111063384302e-197

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   29.8
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   29.8
3. Using strategy rm
   29.8
4. Applied add-cube-cbrt to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}^3}\]
   30.2
5. Applied simplify to get
   \[{\color{red}{\left(\sqrt[3]{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}}^3\]
   30.2
6. Applied taylor to get
   \[{\left(\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}^3 \leadsto \frac{1}{2} \cdot \frac{{re}^2}{{im}^2 \cdot \log base} + \frac{\log im}{\log base}\]
   15.6
7. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{2} \cdot \frac{{re}^2}{{im}^2 \cdot \log base} + \frac{\log im}{\log base}} \leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^2}{{im}^2 \cdot \log base} + \frac{\log im}{\log base}}\]
   15.6
8. Applied simplify to get
   \[\frac{1}{2} \cdot \frac{{re}^2}{{im}^2 \cdot \log base} + \frac{\log im}{\log base} \leadsto \frac{\log im}{\log base} + \frac{\frac{1}{2}}{\log base} \cdot {\left(\frac{re}{im}\right)}^2\]
   0.4

---

9. Applied final simplification

if 2.2020111063384302e-197 < re < 2.9308959747651406e-60 or 1.1318552232567758e+17 < re < 1.432799947628018e+112

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   18.6
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   18.6
3. Using strategy rm
   18.6
4. Applied add-log-exp to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\log \left(e^{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}\]
   19.0
5. Applied simplify to get
   \[\log \color{red}{\left(e^{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)} \leadsto \log \color{blue}{\left(e^{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}\]
   18.9
6. Applied simplify to get
   \[\log \left(e^{\color{red}{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}}\right) \leadsto \log \left(e^{\color{blue}{\frac{\log \left(\sqrt{{im}^2 + re \cdot re}\right)}{\log base}}}\right)\]
   18.9

if 2.9308959747651406e-60 < re < 1.1318552232567758e+17

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   40.3
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   40.3
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{{\left(\log base\right)}^2}\]
   40.3
4. Taylor expanded around 0 to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{{\left(\log base\right)}^2}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{{\left(\log base\right)}^2}}\]
   40.3
5. Applied simplify to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{{\left(\log base\right)}^2}} \leadsto \color{blue}{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\]
   40.2
6. Applied taylor to get
   \[\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base} \leadsto \frac{\log \left(-1 \cdot im\right)}{\log base}\]
   0.4
7. Taylor expanded around -inf to get
   \[\frac{\log \color{red}{\left(-1 \cdot im\right)}}{\log base} \leadsto \frac{\log \color{blue}{\left(-1 \cdot im\right)}}{\log base}\]
   0.4
8. Applied simplify to get
   \[\color{red}{\frac{\log \left(-1 \cdot im\right)}{\log base}} \leadsto \color{blue}{\frac{\log \left(-im\right)}{\log base}}\]
   0.4

if 1.432799947628018e+112 < re

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   52.1
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   52.1
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log re + 0}{\log base \cdot \log base}\]
   0.5
4. Taylor expanded around inf to get
   \[\frac{\log base \cdot \log \color{red}{re} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{re} + 0}{\log base \cdot \log base}\]
   0.5
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