Error

Derivation

1. Split input into 6 regimes

2. if im < -6.860130959618919e+100

   1. Initial program 51.0

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

   3. Applied pow151.0

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

   4. Applied sqrt-pow151.0

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

   5. Applied log-pow51.0

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

   6. Applied associate-/l*51.0

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

   7. Simplified51.0

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

   9. Applied pow151.0

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

   10. Applied log-pow51.0

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

   11. Applied add-sqr-sqrt51.0

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

   12. Applied times-frac51.1

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

   13. Applied add-sqr-sqrt51.0

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

   14. Applied times-frac51.0

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

   15. Simplified51.0

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

   16. Simplified51.0

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

   17. Taylor expanded around -inf 9.1

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

   if -6.860130959618919e+100 < im < -2.20945520994778e-32 or -7.828804558990274e-268 < im < 1.2181563798430822e-210

   1. Initial program 24.0

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

   3. Applied pow124.0

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

   4. Applied sqrt-pow124.0

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

   5. Applied log-pow24.0

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

   6. Applied associate-/l*24.0

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

   7. Simplified24.0

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

   9. Applied pow124.0

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

   10. Applied log-pow24.0

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

   11. Applied add-sqr-sqrt24.0

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

   12. Applied times-frac24.1

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

   13. Applied add-sqr-sqrt24.0

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

   14. Applied times-frac23.9

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

   15. Simplified23.9

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

   16. Simplified23.9

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

   18. Applied associate-*r/23.9

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

   19. Applied associate-*r/23.9

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

   if -2.20945520994778e-32 < im < -7.828804558990274e-268 or 1.2181563798430822e-210 < im < 4.692347688006194e-180

   1. Initial program 23.1

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

   2. Taylor expanded around -inf 38.4

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

   3. Simplified38.4

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

   if 4.692347688006194e-180 < im < 3.7140034679671814e-140

   1. Initial program 23.7

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

   3. Applied pow123.7

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

   4. Applied sqrt-pow123.7

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

   5. Applied log-pow23.7

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

   6. Applied associate-/l*23.7

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

   7. Simplified23.7

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

   9. Applied pow123.7

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

   10. Applied log-pow23.7

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

   11. Applied add-sqr-sqrt23.7

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

   12. Applied times-frac23.8

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

   13. Applied add-sqr-sqrt23.7

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

   14. Applied times-frac23.6

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

   15. Simplified23.6

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

   16. Simplified23.6

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

   17. Taylor expanded around 0 39.3

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

   18. Simplified39.2

       \[\leadsto \color{blue}{\frac{\log re}{\log 10}}\]

   if 3.7140034679671814e-140 < im < 1.941145076499788e+113

   1. Initial program 14.6

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

   3. Applied pow114.6

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

   4. Applied sqrt-pow114.6

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

   5. Applied log-pow14.6

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

   6. Applied associate-/l*14.7

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

   7. Simplified14.7

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

   9. Applied add-sqr-sqrt14.8

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

   10. Applied associate-/l*14.5

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

   if 1.941145076499788e+113 < im

   1. Initial program 51.7

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

   3. Applied pow151.7

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

   4. Applied sqrt-pow151.7

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

   5. Applied log-pow51.7

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

   6. Applied associate-/l*51.7

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

   7. Simplified51.7

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

   9. Applied pow151.7

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

   10. Applied log-pow51.7

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

   11. Applied add-sqr-sqrt51.7

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

   12. Applied times-frac51.8

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

   13. Applied add-sqr-sqrt51.7

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

   14. Applied times-frac51.7

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

   15. Simplified51.7

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

   16. Simplified51.7

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

   17. Taylor expanded around inf 9.1

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

   18. Simplified9.1

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

4. Final simplification20.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -6.860130959618919 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{2}} \cdot \log \left(\frac{-1}{im}\right)\right) \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot -2\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le -2.20945520994778 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{\frac{1}{2}}\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le -7.828804558990274 \cdot 10^{-268}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \le 1.2181563798430822 \cdot 10^{-210}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{\frac{1}{2}}\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le 4.692347688006194 \cdot 10^{-180}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \le 3.7140034679671814 \cdot 10^{-140}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{elif}\;im \le 1.941145076499788 \cdot 10^{+113}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \log im\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))