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Derivation

1. Split input into 5 regimes

2. if re < -1.4267439596555745e+71

   1. Initial program 46.4

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

   2. Initial simplification46.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   3. Using strategy rm

   4. Applied pow1/246.4

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   5. Applied log-pow46.4

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   6. Applied associate-/l*46.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   7. Taylor expanded around -inf 10.4

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{-2 \cdot \log \left(\frac{-1}{re}\right)}}}\]

   8. Simplified10.4

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{\log \left(\frac{-1}{re}\right) \cdot -2}}}\]

   if -1.4267439596555745e+71 < re < -3.788944222453427e-299

   1. Initial program 20.4

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

   2. Initial simplification20.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   3. Using strategy rm

   4. Applied pow1/220.4

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   5. Applied log-pow20.4

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   6. Applied associate-/l*20.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity20.4

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt20.4

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac20.5

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied associate-/r*20.3

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   12. Using strategy rm

   13. Applied pow120.3

       \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\frac{\sqrt{\log 10}}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   14. Applied log-pow20.3

       \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\frac{\sqrt{\log 10}}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   15. Applied add-cube-cbrt20.8

       \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\frac{\sqrt{\color{blue}{\left(\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}\right) \cdot \sqrt[3]{\log 10}}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   16. Applied sqrt-prod20.8

       \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\frac{\color{blue}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \sqrt{\sqrt[3]{\log 10}}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   17. Applied times-frac20.7

       \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\color{blue}{\frac{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}{1} \cdot \frac{\sqrt{\sqrt[3]{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   18. Applied add-sqr-sqrt20.2

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}} \cdot \sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}}{\frac{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}{1} \cdot \frac{\sqrt{\sqrt[3]{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   19. Applied times-frac20.2

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\frac{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}{1}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   20. Simplified20.2

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   21. Simplified20.2

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \color{blue}{\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{\log \left(im \cdot im + re \cdot re\right)}}}\]

   if -3.788944222453427e-299 < re < 2.420049402955283e-227

   1. Initial program 27.4

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

   2. Initial simplification27.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   3. Using strategy rm

   4. Applied pow1/227.4

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   5. Applied log-pow27.4

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   6. Applied associate-/l*27.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity27.4

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt27.4

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac27.5

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied associate-/r*27.4

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   12. Using strategy rm

   13. Applied div-inv27.4

       \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\color{blue}{\sqrt{\log 10} \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   14. Applied add-sqr-sqrt27.4

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}} \cdot \sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}}{\sqrt{\log 10} \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\]

   15. Applied times-frac27.3

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   16. Simplified27.3

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\]

   17. Simplified27.3

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(im \cdot im + re \cdot re\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}\]

   18. Taylor expanded around -inf 32.0

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \left(\color{blue}{\left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\]

   19. Simplified32.0

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \left(\color{blue}{\left(\log \left(\frac{-1}{im}\right) \cdot -2\right)} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\]

   if 2.420049402955283e-227 < re < 2.0482039862590654e+83

   1. Initial program 21.1

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

   2. Initial simplification21.1

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   3. Using strategy rm

   4. Applied pow1/221.1

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   5. Applied log-pow21.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   6. Applied associate-/l*21.1

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity21.1

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt21.1

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac21.2

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied associate-/r*21.0

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   12. Using strategy rm

   13. Applied div-inv21.0

       \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\color{blue}{\sqrt{\log 10} \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   14. Applied add-sqr-sqrt21.1

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}} \cdot \sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}}{\sqrt{\log 10} \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\]

   15. Applied times-frac20.9

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   16. Simplified20.9

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\]

   17. Simplified20.9

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(im \cdot im + re \cdot re\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}\]
   18. Using strategy rm

   19. Applied div-inv20.9

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \left(\log \left(im \cdot im + re \cdot re\right) \cdot \sqrt{\color{blue}{\frac{1}{2} \cdot \frac{1}{\sqrt{\log 10}}}}\right)\]

   20. Applied sqrt-prod20.9

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \left(\log \left(im \cdot im + re \cdot re\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)}\right)\]

   21. Applied associate-*r*21.0

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\left(\log \left(im \cdot im + re \cdot re\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)}\]

   if 2.0482039862590654e+83 < re

   1. Initial program 47.6

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

   2. Initial simplification47.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   3. Using strategy rm

   4. Applied pow1/247.6

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   5. Applied log-pow47.6

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   6. Applied associate-/l*47.6

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity47.6

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt47.6

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac47.7

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied associate-/r*47.6

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   12. Using strategy rm

   13. Applied div-inv47.6

       \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}{\color{blue}{\sqrt{\log 10} \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   14. Applied add-sqr-sqrt47.6

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}} \cdot \sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}}{\sqrt{\log 10} \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\]

   15. Applied times-frac47.6

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   16. Simplified47.6

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\frac{\sqrt{\log 10}}{1}}}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\]

   17. Simplified47.6

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(im \cdot im + re \cdot re\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}\]

   18. Taylor expanded around 0 9.3

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \left(\color{blue}{\left(2 \cdot \log re\right)} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.4267439596555745 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -3.788944222453427 \cdot 10^{-299}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|}\\ \mathbf{elif}\;re \le 2.420049402955283 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \left(\left(\log \left(\frac{-1}{im}\right) \cdot -2\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;re \le 2.0482039862590654 \cdot 10^{+83}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \left(\left(\log re \cdot 2\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018277 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))