Error

Try it out

Derivation

1. Split input into 4 regimes

2. if re < -5.536892084933209e+91

   1. Initial program 48.6

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

   3. Applied add-sqr-sqrt48.6

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

   4. Applied sqrt-prod48.6

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

   6. Applied add-sqr-sqrt48.6

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

   7. Applied pow1/248.6

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

   8. Applied pow1/248.6

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

   9. Applied pow-prod-down48.6

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

   10. Applied log-pow48.6

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

   11. Applied times-frac48.6

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

   12. Simplified48.6

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

   14. Applied div-inv48.6

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

   15. Applied associate-*r*48.6

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

   16. Taylor expanded around -inf 9.6

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

   17. Simplified9.6

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

   if -5.536892084933209e+91 < re < -2.786453322648853e-243 or 2.217402866571808e-285 < re < 4.711515390940537e+66

   1. Initial program 20.1

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

   3. Applied add-sqr-sqrt20.1

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

   4. Applied sqrt-prod20.1

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

   6. Applied add-sqr-sqrt20.1

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

   7. Applied pow1/220.1

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

   8. Applied pow1/220.1

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

   9. Applied pow-prod-down20.1

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

   10. Applied log-pow20.1

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

   11. Applied times-frac20.1

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

   12. Simplified20.1

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

   14. Applied add-sqr-sqrt20.1

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

   15. Applied associate-*l*20.0

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

   if -2.786453322648853e-243 < re < 2.217402866571808e-285

   1. Initial program 30.9

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

   3. Applied add-sqr-sqrt30.9

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

   4. Applied sqrt-prod30.9

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

   6. Applied add-sqr-sqrt30.9

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

   7. Applied pow1/230.9

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

   8. Applied pow1/230.9

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

   9. Applied pow-prod-down30.9

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

   10. Applied log-pow30.9

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

   11. Applied times-frac30.9

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

   12. Simplified30.9

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

   14. Applied div-inv30.8

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

   15. Applied associate-*r*30.8

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

   16. Taylor expanded around 0 33.0

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

   if 4.711515390940537e+66 < re

   1. Initial program 45.2

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

   3. Applied add-sqr-sqrt45.2

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

   4. Applied sqrt-prod45.2

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

   6. Applied add-sqr-sqrt45.2

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

   7. Applied pow1/245.2

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

   8. Applied pow1/245.2

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

   9. Applied pow-prod-down45.2

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

   10. Applied log-pow45.2

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

   11. Applied times-frac45.2

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

   12. Simplified45.2

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

   14. Applied div-inv45.2

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

   15. Applied associate-*r*45.2

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

   16. Taylor expanded around inf 10.2

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

   17. Simplified10.2

       \[\leadsto \color{blue}{\left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)} \cdot \frac{1}{\sqrt{\log 10}}\]
3. Recombined 4 regimes into one program.

4. Final simplification17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.536892084933209 \cdot 10^{+91}:\\ \;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -2.786453322648853 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 2.217402866571808 \cdot 10^{-285}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log im\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 4.711515390940537 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \end{array}\]

Runtime

Time bar (total: 33.3s)Debug logProfile

herbie shell --seed 2018250 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))