\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.923437784376753 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{if}\;re \le 6.959884185585397 \cdot 10^{-191}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{if}\;re \le 4.324889483536473 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log re \cdot \left(--2\right)\right)\right)\\ \mathbf{if}\;re \le 2.312498063686259 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log re \cdot \left(--2\right)\right)\right)\\ \end{array}\]

Derivation

Split input into 3 regimes
if re < -4.923437784376753e+98
1. Initial program 50.1
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied add-sqr-sqrt50.1
  \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]
4. Applied pow1/250.1
  \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
5. Applied log-pow50.1
  \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
6. Applied times-frac50.1
  \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
7. Taylor expanded around -inf 8.9
  \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]
if -4.923437784376753e+98 < re < 6.959884185585397e-191 or 4.324889483536473e-171 < re < 2.312498063686259e+104
1. Initial program 20.8
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied clear-num20.8
  \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
if 6.959884185585397e-191 < re < 4.324889483536473e-171 or 2.312498063686259e+104 < re
1. Initial program 48.7
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied add-sqr-sqrt48.7
  \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]
4. Applied pow1/248.7
  \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
5. Applied log-pow48.7
  \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
6. Applied times-frac48.7
  \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
7. Taylor expanded around inf 11.8
  \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(-2 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]
8. Applied simplify11.8
  \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log re \cdot \left(--2\right)\right)\right)}\]
Recombined 3 regimes into one program.

Error

Derivation

`if re < -4.923437784376753e+98`

`if -4.923437784376753e+98 < re < 6.959884185585397e-191 or 4.324889483536473e-171 < re < 2.312498063686259e+104`

`if 6.959884185585397e-191 < re < 4.324889483536473e-171 or 2.312498063686259e+104 < re`

Runtime