if im < -3.7099458758044735e+109

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   52.4
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   52.4
3. Using strategy rm
   52.4
4. Applied add-log-exp to get
   \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\log \left(e^{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}\]
   52.4
5. Applied simplify to get
   \[\log \color{red}{\left(e^{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)} \leadsto \log \color{blue}{\left(e^{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}\]
   52.4
6. Applied taylor to get
   \[\log \left(e^{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right) \leadsto \log \left(e^{\frac{\log \left(-1 \cdot im\right)}{\log base}}\right)\]
   0.6
7. Taylor expanded around -inf to get
   \[\log \left(e^{\frac{\log \color{red}{\left(-1 \cdot im\right)}}{\log base}}\right) \leadsto \log \left(e^{\frac{\log \color{blue}{\left(-1 \cdot im\right)}}{\log base}}\right)\]
   0.6
8. Applied simplify to get
   \[\log \left(e^{\frac{\log \left(-1 \cdot im\right)}{\log base}}\right) \leadsto \frac{\log \left(-im\right)}{\log base}\]
   0.4

---

9. Applied final simplification

if -3.7099458758044735e+109 < im < 7.038626743233325e-267 or 3.664277580957564e-136 < im < 6.29031090221932e+57

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   19.9
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   19.9
3. Applied simplify to get
   \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\log base \cdot \log base} \leadsto \frac{\color{blue}{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}}{\log base \cdot \log base}\]
   19.9

if 7.038626743233325e-267 < im < 3.664277580957564e-136 or 6.29031090221932e+57 < im < 1.230140674236502e+103

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   38.3
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   38.3
3. Using strategy rm
   38.3
4. Applied add-sqr-sqrt to get
   \[\frac{\log base \cdot \color{red}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \color{blue}{{\left(\sqrt{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\right)}^2} + 0}{\log base \cdot \log base}\]
   49.2
5. Applied add-sqr-sqrt to get
   \[\frac{\color{red}{\log base} \cdot {\left(\sqrt{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\right)}^2 + 0}{\log base \cdot \log base} \leadsto \frac{\color{blue}{{\left(\sqrt{\log base}\right)}^2} \cdot {\left(\sqrt{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\right)}^2 + 0}{\log base \cdot \log base}\]
   54.9
6. Applied square-unprod to get
   \[\frac{\color{red}{{\left(\sqrt{\log base}\right)}^2 \cdot {\left(\sqrt{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\right)}^2} + 0}{\log base \cdot \log base} \leadsto \frac{\color{blue}{{\left(\sqrt{\log base} \cdot \sqrt{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\right)}^2} + 0}{\log base \cdot \log base}\]
   54.9
7. Applied taylor to get
   \[\frac{{\left(\sqrt{\log base} \cdot \sqrt{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\right)}^2 + 0}{\log base \cdot \log base} \leadsto \frac{{\left(\sqrt{\log base} \cdot \sqrt{\log re}\right)}^2 + 0}{\log base \cdot \log base}\]
   44.8
8. Taylor expanded around inf to get
   \[\frac{{\left(\sqrt{\log base} \cdot \sqrt{\log \color{red}{re}}\right)}^2 + 0}{\log base \cdot \log base} \leadsto \frac{{\left(\sqrt{\log base} \cdot \sqrt{\log \color{blue}{re}}\right)}^2 + 0}{\log base \cdot \log base}\]
   44.8
9. Applied simplify to get
   \[\frac{{\left(\sqrt{\log base} \cdot \sqrt{\log re}\right)}^2 + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log re}{{\left(\log base\right)}^2}\]
   0.5

---

10. Applied final simplification
11. Applied simplify to get
    \[\color{red}{\frac{\log base \cdot \log re}{{\left(\log base\right)}^2}} \leadsto \color{blue}{\frac{\log re}{\log base}}\]
    0.3

if 1.230140674236502e+103 < im

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   51.6
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
   51.6
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