math.log10 on complex, real part

Average Error: 31.0 → 17.4

Time: 17.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.7109531485520302 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -5.7290891404837934 \cdot 10^{-198}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}\\ \mathbf{elif}\;re \le -6.376038522303672 \cdot 10^{-268}:\\ \;\;\;\;\left(\frac{\log im + \log im}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}\\ \mathbf{elif}\;re \le 1.7061266383920343 \cdot 10^{+78}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\log re + \log re}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}\\ \end{array}\]

C

double f(double re, double im) {
        double r1292030 = re;
        double r1292031 = r1292030 * r1292030;
        double r1292032 = im;
        double r1292033 = r1292032 * r1292032;
        double r1292034 = r1292031 + r1292033;
        double r1292035 = sqrt(r1292034);
        double r1292036 = log(r1292035);
        double r1292037 = 10.0;
        double r1292038 = log(r1292037);
        double r1292039 = r1292036 / r1292038;
        return r1292039;
}

↓

double f(double re, double im) {
        double r1292040 = re;
        double r1292041 = -1.7109531485520302e+90;
        bool r1292042 = r1292040 <= r1292041;
        double r1292043 = -1.0;
        double r1292044 = r1292043 / r1292040;
        double r1292045 = log(r1292044);
        double r1292046 = -2.0;
        double r1292047 = r1292045 * r1292046;
        double r1292048 = 1.0;
        double r1292049 = 10.0;
        double r1292050 = log(r1292049);
        double r1292051 = sqrt(r1292050);
        double r1292052 = r1292048 / r1292051;
        double r1292053 = r1292047 * r1292052;
        double r1292054 = 0.5;
        double r1292055 = r1292054 / r1292051;
        double r1292056 = r1292053 * r1292055;
        double r1292057 = -5.7290891404837934e-198;
        bool r1292058 = r1292040 <= r1292057;
        double r1292059 = cbrt(r1292054);
        double r1292060 = cbrt(r1292050);
        double r1292061 = sqrt(r1292060);
        double r1292062 = r1292059 / r1292061;
        double r1292063 = r1292040 * r1292040;
        double r1292064 = im;
        double r1292065 = r1292064 * r1292064;
        double r1292066 = r1292063 + r1292065;
        double r1292067 = log(r1292066);
        double r1292068 = r1292067 / r1292051;
        double r1292069 = r1292062 * r1292068;
        double r1292070 = r1292059 * r1292059;
        double r1292071 = r1292060 * r1292060;
        double r1292072 = sqrt(r1292071);
        double r1292073 = r1292070 / r1292072;
        double r1292074 = r1292069 * r1292073;
        double r1292075 = -6.376038522303672e-268;
        bool r1292076 = r1292040 <= r1292075;
        double r1292077 = log(r1292064);
        double r1292078 = r1292077 + r1292077;
        double r1292079 = r1292078 / r1292051;
        double r1292080 = r1292079 * r1292062;
        double r1292081 = r1292080 * r1292073;
        double r1292082 = 1.7061266383920343e+78;
        bool r1292083 = r1292040 <= r1292082;
        double r1292084 = r1292052 * r1292067;
        double r1292085 = r1292084 * r1292055;
        double r1292086 = log(r1292040);
        double r1292087 = r1292086 + r1292086;
        double r1292088 = r1292087 / r1292051;
        double r1292089 = r1292088 * r1292062;
        double r1292090 = r1292089 * r1292073;
        double r1292091 = r1292083 ? r1292085 : r1292090;
        double r1292092 = r1292076 ? r1292081 : r1292091;
        double r1292093 = r1292058 ? r1292074 : r1292092;
        double r1292094 = r1292042 ? r1292056 : r1292093;
        return r1292094;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -1.7109531485520302e+90

   1. Initial program 48.4

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

   3. Applied add-sqr-sqrt 48.4

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

   4. Applied pow1/2 48.4

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

   5. Applied log-pow 48.4

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

   6. Applied times-frac 48.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
   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. Taylor expanded around -inf 8.9

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

   10. Simplified 8.9

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

   if -1.7109531485520302e+90 < re < -5.7290891404837934e-198

   1. Initial program 18.0

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

   3. Applied add-sqr-sqrt 18.0

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

   4. Applied pow1/2 18.0

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

   5. Applied log-pow 18.0

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

   6. Applied times-frac 17.9

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

   8. Applied div-inv 17.8

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

   10. Applied add-cube-cbrt 18.3

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

   11. Applied sqrt-prod 18.3

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

   12. Applied add-cube-cbrt 17.8

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

   13. Applied times-frac 18.2

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

   14. Applied associate-*l* 18.0

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

   15. Simplified 17.9

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

   if -5.7290891404837934e-198 < re < -6.376038522303672e-268

   1. Initial program 29.7

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

   3. Applied add-sqr-sqrt 29.7

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

   4. Applied pow1/2 29.7

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

   5. Applied log-pow 29.7

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

   6. Applied times-frac 29.7

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

   8. Applied div-inv 29.6

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

   10. Applied add-cube-cbrt 30.0

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

   11. Applied sqrt-prod 30.0

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

   12. Applied add-cube-cbrt 29.6

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

   13. Applied times-frac 29.8

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

   14. Applied associate-*l* 29.7

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

   15. Simplified 29.6

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

   16. Taylor expanded around 0 34.9

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

   17. Simplified 34.9

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

   if -6.376038522303672e-268 < re < 1.7061266383920343e+78

   1. Initial program 22.5

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

   3. Applied add-sqr-sqrt 22.5

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

   4. Applied pow1/2 22.5

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

   5. Applied log-pow 22.5

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

   6. Applied times-frac 22.5

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

   8. Applied div-inv 22.4

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

   if 1.7061266383920343e+78 < re

   1. Initial program 46.5

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

   3. Applied add-sqr-sqrt 46.5

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

   4. Applied pow1/2 46.5

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

   5. Applied log-pow 46.5

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

   6. Applied times-frac 46.4

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

   8. Applied div-inv 46.4

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

   10. Applied add-cube-cbrt 46.6

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

   11. Applied sqrt-prod 46.6

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

   12. Applied add-cube-cbrt 46.4

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

   13. Applied times-frac 46.5

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

   14. Applied associate-*l* 46.5

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

   15. Simplified 46.4

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

   16. Taylor expanded around inf 10.7

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

   17. Simplified 10.7

       \[\leadsto \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}} \cdot \left(\frac{\color{blue}{\log re + \log re}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}}\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.7109531485520302 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -5.7290891404837934 \cdot 10^{-198}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}\\ \mathbf{elif}\;re \le -6.376038522303672 \cdot 10^{-268}:\\ \;\;\;\;\left(\frac{\log im + \log im}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}\\ \mathbf{elif}\;re \le 1.7061266383920343 \cdot 10^{+78}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\log re + \log re}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))