math.log10 on complex, real part

Average Error: 32.6 → 18.2

Time: 10.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.219332295965777137041720193068407814529 \cdot 10^{82}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{\sqrt{2}}}}{\log 10} \cdot \left|\sqrt[3]{\frac{1}{2}}\right|\right) \cdot \frac{\sqrt{\frac{1}{\sqrt{2}}}}{\frac{\frac{1}{-2 \cdot \log \left(\frac{-1}{re}\right)}}{\sqrt{\sqrt[3]{\frac{1}{2}}}}}\\ \mathbf{elif}\;re \le -3.743447547042940916879606925039648794356 \cdot 10^{-217}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{\sqrt{2}}}}{\log 10} \cdot \left|\sqrt[3]{\frac{1}{2}}\right|\right) \cdot \frac{\sqrt{\frac{1}{\sqrt{2}}}}{\frac{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\sqrt[3]{\frac{1}{2}}}}}\\ \mathbf{elif}\;re \le 2.623840975917302765619376361437223360048 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{1}{2} \cdot \frac{\log 10}{\log im}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{elif}\;re \le 124645931887550053482496:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{-1}{2} \cdot \frac{\log 10}{\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{2}}}}\\ \end{array}\]

C

double f(double re, double im) {
        double r119060 = re;
        double r119061 = r119060 * r119060;
        double r119062 = im;
        double r119063 = r119062 * r119062;
        double r119064 = r119061 + r119063;
        double r119065 = sqrt(r119064);
        double r119066 = log(r119065);
        double r119067 = 10.0;
        double r119068 = log(r119067);
        double r119069 = r119066 / r119068;
        return r119069;
}

↓

double f(double re, double im) {
        double r119070 = re;
        double r119071 = -4.219332295965777e+82;
        bool r119072 = r119070 <= r119071;
        double r119073 = 1.0;
        double r119074 = 2.0;
        double r119075 = sqrt(r119074);
        double r119076 = r119073 / r119075;
        double r119077 = sqrt(r119076);
        double r119078 = 10.0;
        double r119079 = log(r119078);
        double r119080 = r119077 / r119079;
        double r119081 = 0.5;
        double r119082 = cbrt(r119081);
        double r119083 = fabs(r119082);
        double r119084 = r119080 * r119083;
        double r119085 = -1.0;
        double r119086 = r119085 / r119070;
        double r119087 = log(r119086);
        double r119088 = r119074 * r119087;
        double r119089 = -r119088;
        double r119090 = r119073 / r119089;
        double r119091 = sqrt(r119082);
        double r119092 = r119090 / r119091;
        double r119093 = r119077 / r119092;
        double r119094 = r119084 * r119093;
        double r119095 = -3.743447547042941e-217;
        bool r119096 = r119070 <= r119095;
        double r119097 = r119070 * r119070;
        double r119098 = im;
        double r119099 = r119098 * r119098;
        double r119100 = r119097 + r119099;
        double r119101 = log(r119100);
        double r119102 = r119073 / r119101;
        double r119103 = r119102 / r119091;
        double r119104 = r119077 / r119103;
        double r119105 = r119084 * r119104;
        double r119106 = 2.623840975917303e-268;
        bool r119107 = r119070 <= r119106;
        double r119108 = sqrt(r119081);
        double r119109 = log(r119098);
        double r119110 = r119079 / r119109;
        double r119111 = r119081 * r119110;
        double r119112 = r119111 / r119108;
        double r119113 = r119108 / r119112;
        double r119114 = 1.2464593188755005e+23;
        bool r119115 = r119070 <= r119114;
        double r119116 = r119079 / r119101;
        double r119117 = r119116 / r119108;
        double r119118 = r119108 / r119117;
        double r119119 = -0.5;
        double r119120 = r119073 / r119070;
        double r119121 = log(r119120);
        double r119122 = r119121 * r119108;
        double r119123 = r119079 / r119122;
        double r119124 = r119119 * r119123;
        double r119125 = r119108 / r119124;
        double r119126 = r119115 ? r119118 : r119125;
        double r119127 = r119107 ? r119113 : r119126;
        double r119128 = r119096 ? r119105 : r119127;
        double r119129 = r119072 ? r119094 : r119128;
        return r119129;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -4.219332295965777e+82

   1. Initial program 49.2

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

   3. Applied pow1 49.2

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

   4. Applied sqrt-pow1 49.2

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

   5. Applied log-pow 49.2

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

   6. Applied associate-/l* 49.2

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

   8. Applied add-sqr-sqrt 49.3

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

   9. Applied associate-/l* 49.2

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

   10. Simplified 49.2

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

   12. Applied add-cube-cbrt 49.2

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

   13. Applied sqrt-prod 49.4

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

   14. Applied div-inv 49.4

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

   15. Applied times-frac 49.4

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

   16. Applied add-sqr-sqrt 49.2

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

   17. Applied add-sqr-sqrt 49.2

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

   18. Applied times-frac 49.2

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

   19. Applied sqrt-prod 49.2

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

   20. Applied times-frac 49.2

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

   21. Simplified 49.2

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

   22. Simplified 49.2

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

   23. Taylor expanded around -inf 10.4

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

   if -4.219332295965777e+82 < re < -3.743447547042941e-217

   1. Initial program 18.8

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

   3. Applied pow1 18.8

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

   4. Applied sqrt-pow1 18.8

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

   5. Applied log-pow 18.8

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

   6. Applied associate-/l* 18.8

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

   8. Applied add-sqr-sqrt 18.9

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

   9. Applied associate-/l* 18.7

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

   10. Simplified 18.7

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

   12. Applied add-cube-cbrt 18.7

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

   13. Applied sqrt-prod 19.1

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

   14. Applied div-inv 19.1

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

   15. Applied times-frac 19.1

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

   16. Applied add-sqr-sqrt 18.7

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

   17. Applied add-sqr-sqrt 18.7

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

   18. Applied times-frac 18.7

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

   19. Applied sqrt-prod 18.7

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

   20. Applied times-frac 18.7

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

   21. Simplified 18.7

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

   22. Simplified 18.7

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

   if -3.743447547042941e-217 < re < 2.623840975917303e-268

   1. Initial program 34.4

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

   3. Applied pow1 34.4

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

   4. Applied sqrt-pow1 34.4

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

   5. Applied log-pow 34.4

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

   6. Applied associate-/l* 34.4

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

   8. Applied add-sqr-sqrt 34.5

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

   9. Applied associate-/l* 34.3

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

   10. Simplified 34.3

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

   11. Taylor expanded around 0 34.6

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

   if 2.623840975917303e-268 < re < 1.2464593188755005e+23

   1. Initial program 23.0

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

   3. Applied pow1 23.0

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

   4. Applied sqrt-pow1 23.0

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

   5. Applied log-pow 23.0

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

   6. Applied associate-/l* 23.0

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

   8. Applied add-sqr-sqrt 23.1

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

   9. Applied associate-/l* 22.9

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

   10. Simplified 22.9

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

   if 1.2464593188755005e+23 < re

   1. Initial program 41.9

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

   3. Applied pow1 41.9

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

   4. Applied sqrt-pow1 41.9

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

   5. Applied log-pow 41.9

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

   6. Applied associate-/l* 42.0

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

   8. Applied add-sqr-sqrt 42.0

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

   9. Applied associate-/l* 41.9

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

   10. Simplified 41.9

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

   11. Taylor expanded around inf 12.2

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

4. Final simplification 18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.219332295965777137041720193068407814529 \cdot 10^{82}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{\sqrt{2}}}}{\log 10} \cdot \left|\sqrt[3]{\frac{1}{2}}\right|\right) \cdot \frac{\sqrt{\frac{1}{\sqrt{2}}}}{\frac{\frac{1}{-2 \cdot \log \left(\frac{-1}{re}\right)}}{\sqrt{\sqrt[3]{\frac{1}{2}}}}}\\ \mathbf{elif}\;re \le -3.743447547042940916879606925039648794356 \cdot 10^{-217}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{\sqrt{2}}}}{\log 10} \cdot \left|\sqrt[3]{\frac{1}{2}}\right|\right) \cdot \frac{\sqrt{\frac{1}{\sqrt{2}}}}{\frac{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\sqrt[3]{\frac{1}{2}}}}}\\ \mathbf{elif}\;re \le 2.623840975917302765619376361437223360048 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{1}{2} \cdot \frac{\log 10}{\log im}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{elif}\;re \le 124645931887550053482496:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{-1}{2} \cdot \frac{\log 10}{\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{2}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))