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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.5444508077594253 \cdot 10^{+135}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{if}\;re \le -1.6471939554857127 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{if}\;re \le 1.3032773090778938 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(2 \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{if}\;re \le 3.513824930285112 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Derivation

Split input into 4 regimes
if re < -2.5444508077594253e+135
1. Initial program 56.4
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Taylor expanded around -inf 7.6
  \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10}\]
3. Applied simplify7.6
  \[\leadsto \color{blue}{\frac{\log \left(-re\right)}{\log 10}}\]
if -2.5444508077594253e+135 < re < -1.6471939554857127e-238 or 1.3032773090778938e-265 < re < 3.513824930285112e+77
1. Initial program 19.5
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied add-sqr-sqrt19.5
  \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]
4. Applied pow1/219.5
  \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
5. Applied log-pow19.5
  \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
6. Applied times-frac19.5
  \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
7. Using strategy rm
8. Applied add-sqr-sqrt19.5
  \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]
9. Applied associate-*l*19.4
  \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)}\]
if -1.6471939554857127e-238 < re < 1.3032773090778938e-265
1. Initial program 30.0
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied add-sqr-sqrt30.0
  \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]
4. Applied pow1/230.0
  \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
5. Applied log-pow30.0
  \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
6. Applied times-frac30.0
  \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
7. Taylor expanded around 0 33.1
  \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(2 \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]
if 3.513824930285112e+77 < re
1. Initial program 45.9
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Taylor expanded around inf 9.9
  \[\leadsto \frac{\log \color{blue}{re}}{\log 10}\]
Recombined 4 regimes into one program.
Removed slow pow expressions.

Error

Derivation

`if re < -2.5444508077594253e+135`

`if -2.5444508077594253e+135 < re < -1.6471939554857127e-238 or 1.3032773090778938e-265 < re < 3.513824930285112e+77`

`if -1.6471939554857127e-238 < re < 1.3032773090778938e-265`

`if 3.513824930285112e+77 < re`

Runtime