if re < -7.494067f+15

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   26.9
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
   26.9
3. Using strategy rm
   26.9
4. Applied add-cube-cbrt to get
   \[\frac{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10}\]
   26.9
5. Using strategy rm
   26.9
6. Applied *-un-lft-identity to get
   \[\frac{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}{\color{red}{\log 10}} \leadsto \frac{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}{\color{blue}{1 \cdot \log 10}}\]
   26.9
7. Applied pow3 to get
   \[\frac{\log \color{red}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{1 \cdot \log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{1 \cdot \log 10}\]
   26.9
8. Applied log-pow to get
   \[\frac{\color{red}{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{1 \cdot \log 10} \leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}{1 \cdot \log 10}\]
   26.9
9. Applied times-frac to get
   \[\color{red}{\frac{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{1 \cdot \log 10}} \leadsto \color{blue}{\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}}\]
   26.9
10. Using strategy rm
    26.9
11. Applied *-un-lft-identity to get
    \[\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\color{red}{\log 10}} \leadsto \frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\color{blue}{1 \cdot \log 10}}\]
    26.9
12. Applied pow1/3 to get
    \[\frac{3}{1} \cdot \frac{\log \color{red}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}{1 \cdot \log 10} \leadsto \frac{3}{1} \cdot \frac{\log \color{blue}{\left({\left(\sqrt{{re}^2 + im \cdot im}\right)}^{\frac{1}{3}}\right)}}{1 \cdot \log 10}\]
    26.9
13. Applied log-pow to get
    \[\frac{3}{1} \cdot \frac{\color{red}{\log \left({\left(\sqrt{{re}^2 + im \cdot im}\right)}^{\frac{1}{3}}\right)}}{1 \cdot \log 10} \leadsto \frac{3}{1} \cdot \frac{\color{blue}{\frac{1}{3} \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right)}}{1 \cdot \log 10}\]
    26.9
14. Applied times-frac to get
    \[\frac{3}{1} \cdot \color{red}{\frac{\frac{1}{3} \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right)}{1 \cdot \log 10}} \leadsto \frac{3}{1} \cdot \color{blue}{\left(\frac{\frac{1}{3}}{1} \cdot \frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}\right)}\]
    26.9
15. Applied taylor to get
    \[\frac{3}{1} \cdot \left(\frac{\frac{1}{3}}{1} \cdot \frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}\right) \leadsto \frac{3}{1} \cdot \left(\frac{\frac{1}{3}}{1} \cdot \frac{\log \left(-1 \cdot re\right)}{\log 10}\right)\]
    0.3
16. Taylor expanded around -inf to get
    \[\frac{3}{1} \cdot \left(\frac{\frac{1}{3}}{1} \cdot \frac{\log \color{red}{\left(-1 \cdot re\right)}}{\log 10}\right) \leadsto \frac{3}{1} \cdot \left(\frac{\frac{1}{3}}{1} \cdot \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10}\right)\]
    0.3
17. Applied simplify to get
    \[\frac{3}{1} \cdot \left(\frac{\frac{1}{3}}{1} \cdot \frac{\log \left(-1 \cdot re\right)}{\log 10}\right) \leadsto \frac{3}{1} \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{1}{3}}{1}}}\]
    0.3

---

18. Applied final simplification
19. Applied simplify to get
    \[\color{red}{\frac{3}{1} \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{1}{3}}{1}}}} \leadsto \color{blue}{\frac{\log \left(-re\right)}{\frac{\frac{\log 10}{\frac{1}{3}}}{3}}}\]
    0.3

if -7.494067f+15 < re < 3.2777988f+12

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
   \[\frac{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10}\]
   9.9
5. Using strategy rm
   9.9
6. Applied *-un-lft-identity to get
   \[\frac{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}{\color{red}{\log 10}} \leadsto \frac{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}{\color{blue}{1 \cdot \log 10}}\]
   9.9
7. Applied pow3 to get
   \[\frac{\log \color{red}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{1 \cdot \log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{1 \cdot \log 10}\]
   9.9
8. Applied log-pow to get
   \[\frac{\color{red}{\log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^{3}\right)}}{1 \cdot \log 10} \leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}{1 \cdot \log 10}\]
   9.9
9. Applied times-frac to get
   \[\color{red}{\frac{3 \cdot \log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{1 \cdot \log 10}} \leadsto \color{blue}{\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\log 10}}\]
   9.9
10. Using strategy rm
    9.9
11. Applied *-un-lft-identity to get
    \[\frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\color{red}{\log 10}} \leadsto \frac{3}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}{\color{blue}{1 \cdot \log 10}}\]
    9.9
12. Applied pow1/3 to get
    \[\frac{3}{1} \cdot \frac{\log \color{red}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}{1 \cdot \log 10} \leadsto \frac{3}{1} \cdot \frac{\log \color{blue}{\left({\left(\sqrt{{re}^2 + im \cdot im}\right)}^{\frac{1}{3}}\right)}}{1 \cdot \log 10}\]
    9.9
13. Applied log-pow to get
    \[\frac{3}{1} \cdot \frac{\color{red}{\log \left({\left(\sqrt{{re}^2 + im \cdot im}\right)}^{\frac{1}{3}}\right)}}{1 \cdot \log 10} \leadsto \frac{3}{1} \cdot \frac{\color{blue}{\frac{1}{3} \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right)}}{1 \cdot \log 10}\]
    9.8
14. Applied times-frac to get
    \[\frac{3}{1} \cdot \color{red}{\frac{\frac{1}{3} \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right)}{1 \cdot \log 10}} \leadsto \frac{3}{1} \cdot \color{blue}{\left(\frac{\frac{1}{3}}{1} \cdot \frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}\right)}\]
    9.8

if 3.2777988f+12 < re

1. Started with
   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   24.8
2. Applied simplify to get
   \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
   24.8
3. Applied taylor to get
   \[\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10} \leadsto \frac{\log re}{\log 10}\]
   0.3
4. Taylor expanded around inf to get
   \[\frac{\log \color{red}{re}}{\log 10} \leadsto \frac{\log \color{blue}{re}}{\log 10}\]
   0.3