math.log10 on complex, real part

Average Error: 32.1 → 18.1

Time: 5.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -8.7623037487145096 \cdot 10^{143}:\\ \;\;\;\;-1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\ \;\;\;\;3 \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\right)\\ \mathbf{elif}\;re \le -3.39914851247206701 \cdot 10^{-257}:\\ \;\;\;\;3 \cdot \frac{\log \left(\sqrt[3]{im}\right)}{\log 10}\\ \mathbf{elif}\;re \le 9.71398425882254737 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \frac{\log \left(\sqrt[3]{re}\right)}{\log 10}\\ \end{array}\]

C

double f(double re, double im) {
        double r85402 = re;
        double r85403 = r85402 * r85402;
        double r85404 = im;
        double r85405 = r85404 * r85404;
        double r85406 = r85403 + r85405;
        double r85407 = sqrt(r85406);
        double r85408 = log(r85407);
        double r85409 = 10.0;
        double r85410 = log(r85409);
        double r85411 = r85408 / r85410;
        return r85411;
}

↓

double f(double re, double im) {
        double r85412 = re;
        double r85413 = -8.76230374871451e+143;
        bool r85414 = r85412 <= r85413;
        double r85415 = -1.0;
        double r85416 = r85415 / r85412;
        double r85417 = log(r85416);
        double r85418 = 10.0;
        double r85419 = log(r85418);
        double r85420 = r85417 / r85419;
        double r85421 = r85415 * r85420;
        double r85422 = -1.180833083521909e-161;
        bool r85423 = r85412 <= r85422;
        double r85424 = 3.0;
        double r85425 = 1.0;
        double r85426 = sqrt(r85419);
        double r85427 = r85425 / r85426;
        double r85428 = r85412 * r85412;
        double r85429 = im;
        double r85430 = r85429 * r85429;
        double r85431 = r85428 + r85430;
        double r85432 = sqrt(r85431);
        double r85433 = cbrt(r85432);
        double r85434 = log(r85433);
        double r85435 = r85434 * r85427;
        double r85436 = r85427 * r85435;
        double r85437 = r85424 * r85436;
        double r85438 = -3.399148512472067e-257;
        bool r85439 = r85412 <= r85438;
        double r85440 = cbrt(r85429);
        double r85441 = log(r85440);
        double r85442 = r85441 / r85419;
        double r85443 = r85424 * r85442;
        double r85444 = 9.713984258822547e+53;
        bool r85445 = r85412 <= r85444;
        double r85446 = log(r85432);
        double r85447 = r85446 / r85426;
        double r85448 = r85427 * r85447;
        double r85449 = cbrt(r85412);
        double r85450 = log(r85449);
        double r85451 = r85450 / r85419;
        double r85452 = r85424 * r85451;
        double r85453 = r85445 ? r85448 : r85452;
        double r85454 = r85439 ? r85443 : r85453;
        double r85455 = r85423 ? r85437 : r85454;
        double r85456 = r85414 ? r85421 : r85455;
        return r85456;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -8.76230374871451e+143

   1. Initial program 60.9

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

   2. Taylor expanded around -inf 7.2

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

   if -8.76230374871451e+143 < re < -1.180833083521909e-161

   1. Initial program 17.2

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

   3. Applied add-cube-cbrt 17.2

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

   5. Applied pow1 17.2

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

   6. Applied log-pow 17.2

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

   7. Applied pow1 17.2

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

   8. Applied pow1 17.2

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

   9. Applied pow1 17.2

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

   10. Applied pow-prod-up 17.2

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

   11. Applied pow-prod-up 17.2

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

   12. Applied log-pow 17.2

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

   13. Applied times-frac 17.3

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

   14. Simplified 17.3

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

   16. Applied add-cube-cbrt 17.3

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

   18. Applied add-sqr-sqrt 17.3

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

   19. Applied pow1 17.3

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

   20. Applied log-pow 17.3

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

   21. Applied times-frac 17.2

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

   22. Simplified 17.1

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

   if -1.180833083521909e-161 < re < -3.399148512472067e-257

   1. Initial program 32.6

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

   3. Applied add-cube-cbrt 32.6

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

   5. Applied pow1 32.6

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

   6. Applied log-pow 32.6

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

   7. Applied pow1 32.6

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

   8. Applied pow1 32.6

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

   9. Applied pow1 32.6

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

   10. Applied pow-prod-up 32.6

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

   11. Applied pow-prod-up 32.6

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

   12. Applied log-pow 32.6

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

   13. Applied times-frac 32.6

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

   14. Simplified 32.6

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

   15. Taylor expanded around 0 35.9

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

   if -3.399148512472067e-257 < re < 9.713984258822547e+53

   1. Initial program 24.1

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

   3. Applied add-cube-cbrt 24.1

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

   5. Applied add-sqr-sqrt 24.1

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

   6. Applied pow1/3 24.1

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

   7. Applied pow1/3 24.2

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

   8. Applied pow1/3 24.2

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

   9. Applied pow-prod-up 24.2

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

   10. Applied pow-prod-up 24.1

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

   11. Applied log-pow 24.1

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

   12. Applied times-frac 24.0

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

   13. Simplified 24.0

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

   if 9.713984258822547e+53 < re

   1. Initial program 44.0

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

   3. Applied add-cube-cbrt 44.0

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

   5. Applied pow1 44.0

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

   6. Applied log-pow 44.0

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

   7. Applied pow1 44.0

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

   8. Applied pow1 44.0

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

   9. Applied pow1 44.0

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

   10. Applied pow-prod-up 44.0

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

   11. Applied pow-prod-up 44.0

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

   12. Applied log-pow 44.0

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

   13. Applied times-frac 44.0

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

   14. Simplified 44.0

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

   15. Taylor expanded around inf 11.2

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.7623037487145096 \cdot 10^{143}:\\ \;\;\;\;-1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\ \;\;\;\;3 \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\right)\\ \mathbf{elif}\;re \le -3.39914851247206701 \cdot 10^{-257}:\\ \;\;\;\;3 \cdot \frac{\log \left(\sqrt[3]{im}\right)}{\log 10}\\ \mathbf{elif}\;re \le 9.71398425882254737 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \frac{\log \left(\sqrt[3]{re}\right)}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))