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 clear-num to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{1}{\frac{\log base \cdot \log base}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}}\]
   18.0
5. Applied simplify to get
   \[\frac{1}{\color{red}{\frac{\log base \cdot \log base}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}} \leadsto \frac{1}{\color{blue}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
   17.9
6. Using strategy rm
   17.9
7. Applied add-log-exp to get
   \[\frac{1}{\color{red}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \leadsto \frac{1}{\color{blue}{\log \left(e^{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\right)}}\]
   18.3

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

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   29.6
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   29.6
3. Applied taylor to get
   \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log im + 0}{\log base \cdot \log base}\]
   0.5
4. Taylor expanded around 0 to get
   \[\frac{\log base \cdot \log \color{red}{im} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{im} + 0}{\log base \cdot \log base}\]
   0.5
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

if 2.2020111063384302e-197 < re < 2.9308959747651406e-60

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   19.6
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   19.6
3. Using strategy rm
   19.6
4. Applied add-log-exp to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\log \left(e^{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}\]
   19.9
5. Applied simplify to get
   \[\log \color{red}{\left(e^{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)} \leadsto \log \color{blue}{\left(e^{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}\]
   19.8
6. Applied simplify to get
   \[\log \left(e^{\color{red}{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}}\right) \leadsto \log \left(e^{\color{blue}{\frac{\log \left(\sqrt{{im}^2 + re \cdot re}\right)}{\log base}}}\right)\]
   19.8

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 1.1318552232567758e+17 < re < 1.432799947628018e+112

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   17.2
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   17.2
3. Using strategy rm
   17.2
4. Applied add-log-exp to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\log \left(e^{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}\]
   17.6
5. Applied simplify to get
   \[\log \color{red}{\left(e^{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)} \leadsto \log \color{blue}{\left(e^{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}\]
   17.5
6. Applied simplify to get
   \[\log \left(e^{\color{red}{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}}\right) \leadsto \log \left(e^{\color{blue}{\frac{\log \left(\sqrt{{im}^2 + re \cdot re}\right)}{\log base}}}\right)\]
   17.5

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