math.log10 on complex, real part

Average Error: 32.5 → 18.1

Time: 17.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.711194027210843773517449104208961424863 \cdot 10^{119}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{-1}{2} \cdot \frac{\log 10}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -4.206533095177794323245903402127634125049 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{elif}\;re \le 7.904117357981244010954183845189744893902 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}} \cdot 1}{\sqrt{\log 10}} \cdot \left(2 \cdot \left(\left(\log im \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \le 1.775590008512004516315298419200411685417 \cdot 10^{61}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\log re \cdot 2}}\\ \end{array}\]

C

double f(double re, double im) {
        double r41945 = re;
        double r41946 = r41945 * r41945;
        double r41947 = im;
        double r41948 = r41947 * r41947;
        double r41949 = r41946 + r41948;
        double r41950 = sqrt(r41949);
        double r41951 = log(r41950);
        double r41952 = 10.0;
        double r41953 = log(r41952);
        double r41954 = r41951 / r41953;
        return r41954;
}

↓

double f(double re, double im) {
        double r41955 = re;
        double r41956 = -1.7111940272108438e+119;
        bool r41957 = r41955 <= r41956;
        double r41958 = 0.5;
        double r41959 = sqrt(r41958);
        double r41960 = -0.5;
        double r41961 = 10.0;
        double r41962 = log(r41961);
        double r41963 = -1.0;
        double r41964 = r41963 / r41955;
        double r41965 = log(r41964);
        double r41966 = r41962 / r41965;
        double r41967 = r41960 * r41966;
        double r41968 = r41959 / r41967;
        double r41969 = r41959 * r41968;
        double r41970 = -4.206533095177794e-204;
        bool r41971 = r41955 <= r41970;
        double r41972 = sqrt(r41962);
        double r41973 = r41958 / r41972;
        double r41974 = r41973 / r41972;
        double r41975 = r41955 * r41955;
        double r41976 = im;
        double r41977 = r41976 * r41976;
        double r41978 = r41975 + r41977;
        double r41979 = log(r41978);
        double r41980 = r41974 * r41979;
        double r41981 = 7.904117357981244e-205;
        bool r41982 = r41955 <= r41981;
        double r41983 = 1.0;
        double r41984 = r41959 * r41983;
        double r41985 = r41984 / r41972;
        double r41986 = 2.0;
        double r41987 = log(r41976);
        double r41988 = r41987 * r41959;
        double r41989 = r41983 / r41962;
        double r41990 = sqrt(r41989);
        double r41991 = r41988 * r41990;
        double r41992 = r41986 * r41991;
        double r41993 = r41985 * r41992;
        double r41994 = 1.7755900085120045e+61;
        bool r41995 = r41955 <= r41994;
        double r41996 = log(r41955);
        double r41997 = r41996 * r41986;
        double r41998 = r41962 / r41997;
        double r41999 = r41959 / r41998;
        double r42000 = r41959 * r41999;
        double r42001 = r41995 ? r41980 : r42000;
        double r42002 = r41982 ? r41993 : r42001;
        double r42003 = r41971 ? r41980 : r42002;
        double r42004 = r41957 ? r41969 : r42003;
        return r42004;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.7111940272108438e+119

   1. Initial program 54.8

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

   3. Applied pow1/2 54.8

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

   4. Applied log-pow 54.8

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

   5. Applied associate-/l* 54.8

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

   7. Applied pow1 54.8

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

   8. Applied log-pow 54.8

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

   9. Applied pow1 54.8

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

   10. Applied log-pow 54.8

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

   11. Applied times-frac 54.8

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

   12. Applied add-sqr-sqrt 54.8

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

   13. Applied times-frac 54.7

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

   14. Simplified 54.7

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

   15. Taylor expanded around -inf 7.9

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

   if -1.7111940272108438e+119 < re < -4.206533095177794e-204 or 7.904117357981244e-205 < re < 1.7755900085120045e+61

   1. Initial program 19.3

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

   3. Applied pow1/2 19.3

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

   4. Applied log-pow 19.3

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

   5. Applied associate-/l* 19.3

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

   7. Applied pow1 19.3

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

   8. Applied log-pow 19.3

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

   9. Applied pow1 19.3

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

   10. Applied log-pow 19.3

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

   11. Applied times-frac 19.3

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

   12. Applied add-sqr-sqrt 19.4

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

   13. Applied times-frac 19.2

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

   14. Simplified 19.2

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

   16. Applied pow1 19.2

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

   17. Applied log-pow 19.2

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

   18. Applied add-sqr-sqrt 19.2

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

   19. Applied times-frac 19.2

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

   20. Applied *-un-lft-identity 19.2

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

   21. Applied sqrt-prod 19.2

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

   22. Applied times-frac 19.2

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

   23. Applied associate-*r* 19.3

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

   24. Simplified 19.2

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

   26. Applied associate-/r/ 19.1

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

   27. Applied associate-*r* 19.1

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

   28. Simplified 19.1

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

   if -4.206533095177794e-204 < re < 7.904117357981244e-205

   1. Initial program 32.1

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

   3. Applied pow1/2 32.1

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

   4. Applied log-pow 32.1

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

   5. Applied associate-/l* 32.1

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

   7. Applied pow1 32.1

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

   8. Applied log-pow 32.1

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

   9. Applied pow1 32.1

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

   10. Applied log-pow 32.1

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

   11. Applied times-frac 32.1

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

   12. Applied add-sqr-sqrt 32.2

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

   13. Applied times-frac 32.0

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

   14. Simplified 32.0

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

   16. Applied pow1 32.0

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

   17. Applied log-pow 32.0

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

   18. Applied add-sqr-sqrt 32.0

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

   19. Applied times-frac 32.0

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

   20. Applied *-un-lft-identity 32.0

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

   21. Applied sqrt-prod 32.0

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

   22. Applied times-frac 32.1

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

   23. Applied associate-*r* 32.1

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

   24. Simplified 32.0

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

   25. Taylor expanded around 0 33.6

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

   if 1.7755900085120045e+61 < re

   1. Initial program 47.0

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

   3. Applied pow1/2 47.0

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

   4. Applied log-pow 47.0

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

   5. Applied associate-/l* 47.0

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

   7. Applied pow1 47.0

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

   8. Applied log-pow 47.0

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

   9. Applied pow1 47.0

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

   10. Applied log-pow 47.0

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

   11. Applied times-frac 47.0

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

   12. Applied add-sqr-sqrt 47.1

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

   13. Applied times-frac 47.0

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

   14. Simplified 47.0

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

   15. Taylor expanded around inf 11.0

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

   16. Simplified 11.0

       \[\leadsto \sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\color{blue}{\log re \cdot 2}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.711194027210843773517449104208961424863 \cdot 10^{119}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{-1}{2} \cdot \frac{\log 10}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -4.206533095177794323245903402127634125049 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{elif}\;re \le 7.904117357981244010954183845189744893902 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}} \cdot 1}{\sqrt{\log 10}} \cdot \left(2 \cdot \left(\left(\log im \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \le 1.775590008512004516315298419200411685417 \cdot 10^{61}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\log re \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))