if re < -5.2596218f+17

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   28.3
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
   28.3
3. Applied taylor to get
   \[\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10} \leadsto \frac{\log \left(-1 \cdot re\right)}{\log 10}\]
   0.3
4. Taylor expanded around -inf to get
   \[\frac{\log \color{red}{\left(-1 \cdot re\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10}\]
   0.3
5. Applied simplify to get
   \[\color{red}{\frac{\log \left(-1 \cdot re\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(-re\right)}{\log 10}}\]
   0.3

if -5.2596218f+17 < re < 9.018499f+07

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   9.8
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
   9.8
3. Using strategy rm
   9.8
4. Applied add-cube-cbrt to get
   \[\frac{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10}\]
   9.8
5. Using strategy rm
   9.8
6. Applied pow1/3 to get
   \[\frac{\log \left({\color{red}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}^3\right)}{\log 10} \leadsto \frac{\log \left({\color{blue}{\left({\left(\sqrt{{re}^2 + im \cdot im}\right)}^{\frac{1}{3}}\right)}}^3\right)}{\log 10}\]
   9.8

if 9.018499f+07 < re

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   21.7
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
   21.7
3. Using strategy rm
   21.7
4. Applied add-cube-cbrt to get
   \[\frac{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10}\]
   21.7
5. Using strategy rm
   21.7
6. Applied *-un-lft-identity to get
   \[\frac{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}{\color{red}{\log 10}} \leadsto \frac{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}{\color{blue}{1 \cdot \log 10}}\]
   21.7
7. Applied pow3 to get
   \[\frac{\log \color{red}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{1 \cdot \log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{1 \cdot \log 10}\]
   21.7
8. Applied log-pow to get
   \[\frac{\color{red}{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{1 \cdot \log 10} \leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}{1 \cdot \log 10}\]
   21.7
9. Applied times-frac to get
   \[\color{red}{\frac{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{1 \cdot \log 10}} \leadsto \color{blue}{\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}}\]
   21.7
10. Applied taylor to get
    \[\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10} \leadsto \frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{re}\right)}{\log 10}\]
    0.3
11. Taylor expanded around inf to get
    \[\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\color{red}{re}}\right)}{\log 10} \leadsto \frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\color{blue}{re}}\right)}{\log 10}\]
    0.3
12. Applied simplify to get
    \[\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{re}\right)}{\log 10} \leadsto \frac{\log \left(\sqrt[3]{re}\right)}{\frac{\log 10}{\frac{3}{1}}}\]
    0.3

---

13. Applied final simplification
14. Applied simplify to get
    \[\color{red}{\frac{\log \left(\sqrt[3]{re}\right)}{\frac{\log 10}{\frac{3}{1}}}} \leadsto \color{blue}{\frac{\log \left(\sqrt[3]{re}\right)}{\frac{\log 10}{3}}}\]
    0.3