Error

Derivation

1. Split input into 6 regimes

2. if re < -1.3873125929143802e+125

   1. Initial program 54.4

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

   2. Taylor expanded around -inf 8.0

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

   3. Simplified8.0

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

   if -1.3873125929143802e+125 < re < -6.029881316946268e-188

   1. Initial program 17.3

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

   3. Applied pow1/217.3

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

   4. Applied log-pow17.3

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

   5. Applied associate-/l*17.3

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

   7. Applied *-un-lft-identity17.3

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

   8. Applied *-un-lft-identity17.3

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

   9. Applied times-frac17.3

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

   10. Simplified17.3

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

   12. Applied *-un-lft-identity17.3

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

   13. Applied add-sqr-sqrt17.3

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

   14. Applied times-frac17.4

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

   15. Applied add-sqr-sqrt17.3

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

   16. Applied times-frac17.2

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

   17. Simplified17.2

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

   18. Simplified17.1

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

   if -6.029881316946268e-188 < re < -1.1645012368360103e-224

   1. Initial program 31.9

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

   3. Applied pow1/231.9

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

   4. Applied log-pow31.9

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

   5. Applied associate-/l*31.9

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

   7. Applied *-un-lft-identity31.9

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

   8. Applied add-cube-cbrt32.3

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

   9. Applied times-frac32.3

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

   10. Applied add-cube-cbrt31.9

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

   11. Applied times-frac31.8

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

   12. Simplified31.8

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

   13. Simplified31.8

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

   14. Taylor expanded around -inf 31.7

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

   15. Simplified31.7

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

   if -1.1645012368360103e-224 < re < 2.01444127271811e-294 or 1.7290291236201859e-270 < re < 9.88552434217187e+29

   1. Initial program 23.1

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

   3. Applied pow1/223.1

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

   4. Applied log-pow23.1

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

   5. Applied associate-/l*23.1

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

   7. Applied *-un-lft-identity23.1

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

   8. Applied add-cube-cbrt23.6

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

   9. Applied times-frac23.6

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

   10. Applied add-cube-cbrt23.0

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

   11. Applied times-frac23.0

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

   12. Simplified23.0

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

   13. Simplified22.9

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

   if 2.01444127271811e-294 < re < 1.7290291236201859e-270

   1. Initial program 35.1

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

   3. Applied pow1/235.1

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

   4. Applied log-pow35.1

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

   5. Applied associate-/l*35.1

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

   7. Applied *-un-lft-identity35.1

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

   8. Applied add-cube-cbrt35.5

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

   9. Applied times-frac35.5

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

   10. Applied add-cube-cbrt35.1

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

   11. Applied times-frac35.0

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

   12. Simplified35.0

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

   13. Simplified35.0

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

   14. Taylor expanded around inf 32.6

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

   15. Simplified32.6

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

   if 9.88552434217187e+29 < re

   1. Initial program 41.3

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

   3. Applied pow1/241.3

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

   4. Applied log-pow41.3

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

   5. Applied associate-/l*41.3

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

   7. Applied *-un-lft-identity41.3

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

   8. Applied add-cube-cbrt41.6

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

   9. Applied times-frac41.6

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

   10. Applied add-cube-cbrt41.3

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

   11. Applied times-frac41.3

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

   12. Simplified41.3

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

   13. Simplified41.2

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

   14. Taylor expanded around 0 12.8

       \[\leadsto \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\color{blue}{\left(2 \cdot \log re\right)} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\]
3. Recombined 6 regimes into one program.

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3873125929143802 \cdot 10^{+125}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le -6.029881316946268 \cdot 10^{-188}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -1.1645012368360103 \cdot 10^{-224}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)\right)\\ \mathbf{elif}\;re \le 2.01444127271811 \cdot 10^{-294}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{elif}\;re \le 1.7290291236201859 \cdot 10^{-270}:\\ \;\;\;\;\left(\left(\log im \cdot 2\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{elif}\;re \le 9.88552434217187 \cdot 10^{+29}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\left(2 \cdot \log re\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))