Error

Try it out

Derivation

1. Split input into 6 regimes

2. if re < -2.139770464496479e+66

   1. Initial program 43.9

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

   2. Initial simplification43.9

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

   4. Applied add-cbrt-cube44.2

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

   5. Applied add-cbrt-cube44.1

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

   6. Applied cbrt-undiv44.0

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

   7. Simplified44.0

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

   9. Applied add-sqr-sqrt44.0

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

   10. Applied pow1/244.0

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

   11. Applied log-pow44.0

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

   12. Applied times-frac44.0

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

   13. Taylor expanded around -inf 10.7

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

   if -2.139770464496479e+66 < re < -1.7386973165922955e-170

   1. Initial program 15.7

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

   2. Initial simplification15.7

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

   4. Applied add-cbrt-cube16.3

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

   5. Applied add-cbrt-cube16.2

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

   6. Applied cbrt-undiv15.8

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

   7. Simplified15.7

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

   9. Applied add-cube-cbrt16.2

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

   10. Applied unpow-prod-down16.2

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

   11. Simplified15.8

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

   if -1.7386973165922955e-170 < re < 4.948363111980352e-288

   1. Initial program 30.8

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

   2. Initial simplification30.8

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

   4. Applied add-cbrt-cube31.2

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

   5. Applied add-cbrt-cube31.2

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

   6. Applied cbrt-undiv30.9

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

   7. Simplified30.9

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

   8. Taylor expanded around 0 33.6

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

   if 4.948363111980352e-288 < re < 5.406204743658871e-224 or 5.280060465538948e-177 < re < 920024787383512.5

   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 add-cbrt-cube21.6

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

   5. Applied add-cbrt-cube21.5

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

   6. Applied cbrt-undiv21.1

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

   7. Simplified21.1

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

   9. Applied add-sqr-sqrt21.1

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

   10. Applied pow1/221.1

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

   11. Applied log-pow21.1

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

   12. Applied times-frac21.1

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

   if 5.406204743658871e-224 < re < 5.280060465538948e-177

   1. Initial program 28.1

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

   2. Initial simplification28.1

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

   4. Applied pow1/228.1

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

   5. Applied log-pow28.1

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

   6. Applied associate-/l*28.1

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

   7. Taylor expanded around 0 37.3

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{1}{2} \cdot \frac{\log 10}{\log im}}}\]

   if 920024787383512.5 < re

   1. Initial program 40.4

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

   2. Initial simplification40.4

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

   4. Applied add-cbrt-cube40.7

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

   5. Applied add-cbrt-cube40.6

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

   6. Applied cbrt-undiv40.4

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

   7. Simplified40.4

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

   9. Applied add-sqr-sqrt40.4

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

   10. Applied pow1/240.4

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

   11. Applied log-pow40.4

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

   12. Applied times-frac40.4

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

   13. Taylor expanded around inf 12.6

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

   14. Simplified12.6

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

4. Final simplification18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.139770464496479 \cdot 10^{+66}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\right)}^{3}}\\ \mathbf{elif}\;re \le -1.7386973165922955 \cdot 10^{-170}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}}\right)}^{3} \cdot \left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}\\ \mathbf{elif}\;re \le 4.948363111980352 \cdot 10^{-288}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log im}{\log 10}\right)}^{3}}\\ \mathbf{elif}\;re \le 5.406204743658871 \cdot 10^{-224}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right)}^{3}}\\ \mathbf{elif}\;re \le 5.280060465538948 \cdot 10^{-177}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log 10}{\log im} \cdot \frac{1}{2}}\\ \mathbf{elif}\;re \le 920024787383512.5:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\left(\left(\log re \cdot 2\right) \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right)}^{3}}\\ \end{array}\]

Runtime

Time bar (total: 38.2s)Debug logProfile

herbie shell --seed 2018340 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))