Derivation

Split input into 3 regimes
if re < -3.432314412220065e+101
1. Initial program 50.8
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Taylor expanded around -inf 8.6
  \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10}\]
3. Applied simplify8.6
  \[\leadsto \color{blue}{\frac{\log \left(-re\right)}{\log 10}}\]
if -3.432314412220065e+101 < re < 1.3347446688487232e+95
1. Initial program 21.3
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied add-sqr-sqrt21.3
  \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]
4. Applied pow1/221.3
  \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
5. Applied log-pow21.3
  \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
6. Applied times-frac21.3
  \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
if 1.3347446688487232e+95 < re
1. Initial program 49.1
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied add-cube-cbrt49.1
  \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}{\log 10}\]
4. Using strategy rm
5. Applied *-un-lft-identity49.1
  \[\leadsto \frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}{\color{blue}{1 \cdot \log 10}}\]
6. Applied pow149.1
  \[\leadsto \frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{1}}\right)}{1 \cdot \log 10}\]
7. Applied pow149.1
  \[\leadsto \frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{1}\right)}{1 \cdot \log 10}\]
8. Applied pow149.1
  \[\leadsto \frac{\log \left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{1}} \cdot {\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{1}\right) \cdot {\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{1}\right)}{1 \cdot \log 10}\]
9. Applied pow-prod-up49.1
  \[\leadsto \frac{\log \left(\color{blue}{{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{\left(1 + 1\right)}} \cdot {\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{1}\right)}{1 \cdot \log 10}\]
10. Applied pow-prod-up49.1
  \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{\left(\left(1 + 1\right) + 1\right)}\right)}}{1 \cdot \log 10}\]
11. Applied log-pow49.1
  \[\leadsto \frac{\color{blue}{\left(\left(1 + 1\right) + 1\right) \cdot \log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}{1 \cdot \log 10}\]
12. Applied times-frac49.1
  \[\leadsto \color{blue}{\frac{\left(1 + 1\right) + 1}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}{\log 10}}\]
13. Applied simplify49.1
  \[\leadsto \color{blue}{\frac{2 + 1}{1}} \cdot \frac{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}{\log 10}\]
14. Taylor expanded around inf 9.1
  \[\leadsto \frac{2 + 1}{1} \cdot \frac{\log \left(\sqrt[3]{\color{blue}{re}}\right)}{\log 10}\]
15. Applied simplify9.0
  \[\leadsto \color{blue}{\frac{\log \left(\sqrt[3]{re}\right)}{\frac{\log 10}{1 + 2}}}\]
Recombined 3 regimes into one program.

Error

Derivation

`if re < -3.432314412220065e+101`

`if -3.432314412220065e+101 < re < 1.3347446688487232e+95`

`if 1.3347446688487232e+95 < re`

Runtime