math.log10 on complex, real part

Average Error: 31.8 → 24.9

Time: 18.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -2.957483221806522487658889360988388662853 \cdot 10^{136}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le -6.570997735276894967118260067026371622396 \cdot 10^{56}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;im \le -3054555687171828544312670244831232:\\ \;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;im \le -4.502640815266489300055229512512104070594 \cdot 10^{-262}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;im \le 9.677784630305516420633165389576243090092 \cdot 10^{-234}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le 3.867993196915089402122192166122080152846 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le 7.528125355310409892408520759625382180397 \cdot 10^{-199}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{elif}\;im \le 9.430941523167469074988219755431966183767 \cdot 10^{-169}:\\ \;\;\;\;\log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le 1.427553098473186275408804109016984211917 \cdot 10^{86}:\\ \;\;\;\;\log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{\frac{1}{2}}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

C

double f(double re, double im) {
        double r41926 = re;
        double r41927 = r41926 * r41926;
        double r41928 = im;
        double r41929 = r41928 * r41928;
        double r41930 = r41927 + r41929;
        double r41931 = sqrt(r41930);
        double r41932 = log(r41931);
        double r41933 = 10.0;
        double r41934 = log(r41933);
        double r41935 = r41932 / r41934;
        return r41935;
}

↓

double f(double re, double im) {
        double r41936 = im;
        double r41937 = -2.9574832218065225e+136;
        bool r41938 = r41936 <= r41937;
        double r41939 = 1.0;
        double r41940 = 10.0;
        double r41941 = log(r41940);
        double r41942 = r41939 / r41941;
        double r41943 = sqrt(r41942);
        double r41944 = re;
        double r41945 = log(r41944);
        double r41946 = r41943 * r41945;
        double r41947 = sqrt(r41941);
        double r41948 = r41939 / r41947;
        double r41949 = r41946 * r41948;
        double r41950 = -6.570997735276895e+56;
        bool r41951 = r41936 <= r41950;
        double r41952 = 0.5;
        double r41953 = sqrt(r41952);
        double r41954 = r41944 * r41944;
        double r41955 = r41936 * r41936;
        double r41956 = r41954 + r41955;
        double r41957 = log(r41956);
        double r41958 = r41941 / r41957;
        double r41959 = r41953 / r41958;
        double r41960 = r41953 * r41959;
        double r41961 = -3.0545556871718285e+33;
        bool r41962 = r41936 <= r41961;
        double r41963 = -1.0;
        double r41964 = r41963 / r41947;
        double r41965 = r41963 / r41944;
        double r41966 = log(r41965);
        double r41967 = r41966 * r41943;
        double r41968 = r41964 * r41967;
        double r41969 = -4.5026408152664893e-262;
        bool r41970 = r41936 <= r41969;
        double r41971 = cbrt(r41952);
        double r41972 = r41971 * r41971;
        double r41973 = sqrt(r41947);
        double r41974 = r41972 / r41973;
        double r41975 = r41971 / r41973;
        double r41976 = r41957 / r41947;
        double r41977 = r41975 * r41976;
        double r41978 = r41974 * r41977;
        double r41979 = 9.677784630305516e-234;
        bool r41980 = r41936 <= r41979;
        double r41981 = 3.867993196915089e-217;
        bool r41982 = r41936 <= r41981;
        double r41983 = r41952 / r41947;
        double r41984 = -2.0;
        double r41985 = r41984 * r41966;
        double r41986 = r41985 / r41947;
        double r41987 = r41983 * r41986;
        double r41988 = 7.52812535531041e-199;
        bool r41989 = r41936 <= r41988;
        double r41990 = r41945 / r41941;
        double r41991 = 9.430941523167469e-169;
        bool r41992 = r41936 <= r41991;
        double r41993 = -r41943;
        double r41994 = pow(r41965, r41993);
        double r41995 = log(r41994);
        double r41996 = r41995 * r41948;
        double r41997 = 1.4275530984731863e+86;
        bool r41998 = r41936 <= r41997;
        double r41999 = pow(r41956, r41983);
        double r42000 = log(r41999);
        double r42001 = r42000 * r41948;
        double r42002 = pow(r41936, r41943);
        double r42003 = log(r42002);
        double r42004 = r42003 * r41948;
        double r42005 = r41998 ? r42001 : r42004;
        double r42006 = r41992 ? r41996 : r42005;
        double r42007 = r41989 ? r41990 : r42006;
        double r42008 = r41982 ? r41987 : r42007;
        double r42009 = r41980 ? r41949 : r42008;
        double r42010 = r41970 ? r41978 : r42009;
        double r42011 = r41962 ? r41968 : r42010;
        double r42012 = r41951 ? r41960 : r42011;
        double r42013 = r41938 ? r41949 : r42012;
        return r42013;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 9 regimes

2. if im < -2.9574832218065225e+136 or -4.5026408152664893e-262 < im < 9.677784630305516e-234

   1. Initial program 48.3

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

   3. Applied add-sqr-sqrt 48.3

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

   4. Applied pow1/2 48.3

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

   5. Applied log-pow 48.3

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

   6. Applied times-frac 48.3

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

   8. Applied div-inv 48.3

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

   9. Applied associate-*r* 48.3

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

   11. Applied add-log-exp 48.3

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

   12. Simplified 48.2

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

   13. Taylor expanded around inf 46.8

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

   14. Simplified 46.8

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

   if -2.9574832218065225e+136 < im < -6.570997735276895e+56

   1. Initial program 13.3

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

   3. Applied add-sqr-sqrt 13.3

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

   4. Applied pow1/2 13.3

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

   5. Applied log-pow 13.3

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

   6. Applied times-frac 13.2

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

   8. Applied *-un-lft-identity 13.2

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

   9. Applied add-sqr-sqrt 13.3

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

   10. Applied times-frac 13.2

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

   11. Applied associate-*l* 13.1

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

   12. Simplified 13.1

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

   if -6.570997735276895e+56 < im < -3.0545556871718285e+33

   1. Initial program 24.8

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

   3. Applied add-sqr-sqrt 24.8

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

   4. Applied pow1/2 24.8

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

   5. Applied log-pow 24.8

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

   6. Applied times-frac 24.8

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

   8. Applied div-inv 24.7

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

   9. Applied associate-*r* 24.7

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

   11. Applied add-log-exp 24.7

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

   12. Simplified 24.6

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

   13. Taylor expanded around -inf 44.1

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

   14. Simplified 44.1

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

   if -3.0545556871718285e+33 < im < -4.5026408152664893e-262

   1. Initial program 21.3

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

   3. Applied add-sqr-sqrt 21.3

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

   4. Applied pow1/2 21.3

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

   5. Applied log-pow 21.3

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

   6. Applied times-frac 21.3

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

   8. Applied add-sqr-sqrt 21.3

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

   9. Applied sqrt-prod 21.8

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

   10. Applied add-cube-cbrt 21.3

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

   11. Applied times-frac 21.3

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

   12. Applied associate-*l* 21.2

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

   if 9.677784630305516e-234 < im < 3.867993196915089e-217

   1. Initial program 30.2

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

   3. Applied add-sqr-sqrt 30.2

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

   4. Applied pow1/2 30.2

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

   5. Applied log-pow 30.2

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

   6. Applied times-frac 30.1

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

   7. Taylor expanded around -inf 33.4

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

   8. Simplified 33.4

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

   if 3.867993196915089e-217 < im < 7.52812535531041e-199

   1. Initial program 34.6

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

   2. Taylor expanded around inf 33.0

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

   if 7.52812535531041e-199 < im < 9.430941523167469e-169

   1. Initial program 34.2

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

   3. Applied add-sqr-sqrt 34.2

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

   4. Applied pow1/2 34.2

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

   5. Applied log-pow 34.2

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

   6. Applied times-frac 34.2

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

   8. Applied div-inv 34.1

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

   9. Applied associate-*r* 34.1

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

   11. Applied add-log-exp 34.1

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

   12. Simplified 34.1

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

   13. Taylor expanded around -inf 38.3

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

   14. Simplified 38.3

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

   if 9.430941523167469e-169 < im < 1.4275530984731863e+86

   1. Initial program 16.6

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

   3. Applied add-sqr-sqrt 16.6

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

   4. Applied pow1/2 16.6

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

   5. Applied log-pow 16.6

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

   6. Applied times-frac 16.5

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

   8. Applied div-inv 16.4

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

   9. Applied associate-*r* 16.4

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

   11. Applied add-log-exp 16.4

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

   12. Simplified 16.4

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

   if 1.4275530984731863e+86 < im

   1. Initial program 48.3

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

   3. Applied add-sqr-sqrt 48.3

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

   4. Applied pow1/2 48.3

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

   5. Applied log-pow 48.3

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

   6. Applied times-frac 48.3

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

   8. Applied div-inv 48.3

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

   9. Applied associate-*r* 48.3

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

   11. Applied add-log-exp 48.3

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

   12. Simplified 48.2

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

   13. Taylor expanded around 0 9.8

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

   14. Simplified 9.7

       \[\leadsto \log \color{blue}{\left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)} \cdot \frac{1}{\sqrt{\log 10}}\]
3. Recombined 9 regimes into one program.

4. Final simplification 24.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -2.957483221806522487658889360988388662853 \cdot 10^{136}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le -6.570997735276894967118260067026371622396 \cdot 10^{56}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;im \le -3054555687171828544312670244831232:\\ \;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;im \le -4.502640815266489300055229512512104070594 \cdot 10^{-262}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;im \le 9.677784630305516420633165389576243090092 \cdot 10^{-234}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le 3.867993196915089402122192166122080152846 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le 7.528125355310409892408520759625382180397 \cdot 10^{-199}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{elif}\;im \le 9.430941523167469074988219755431966183767 \cdot 10^{-169}:\\ \;\;\;\;\log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le 1.427553098473186275408804109016984211917 \cdot 10^{86}:\\ \;\;\;\;\log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{\frac{1}{2}}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))