math.log10 on complex, real part

Average Error: 31.7 → 18.2

Time: 9.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.30805639812844361 \cdot 10^{37}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -2.3437866053113895 \cdot 10^{-235}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right) \cdot \log \left({\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10}}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10}}}}\right)}\right)\\ \mathbf{elif}\;re \le 4.543347056569168 \cdot 10^{-225}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right) \cdot \log \left({im}^{\left(\sqrt[3]{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right)}\right)\\ \mathbf{elif}\;re \le 2.2414384362599133 \cdot 10^{83}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right) \cdot \log \left({\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10}}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10}}}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

C

double code(double re, double im) {
	return ((double) (((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) / ((double) log(10.0))));
}

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -2.3080563981284436e+37)) {
		VAR = ((double) (((double) (1.0 / ((double) sqrt(((double) log(10.0)))))) * ((double) log(((double) pow(((double) (-1.0 / re)), ((double) -(((double) sqrt(((double) (1.0 / ((double) log(10.0))))))))))))));
	} else {
		double VAR_1;
		if ((re <= -2.3437866053113895e-235)) {
			VAR_1 = ((double) (((double) (((double) (1.0 / ((double) sqrt(((double) log(10.0)))))) * ((double) cbrt(((double) (1.0 / ((double) sqrt(((double) log(10.0)))))))))) * ((double) log(((double) pow(((double) pow(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))), ((double) (((double) cbrt(((double) (1.0 / ((double) sqrt(((double) (((double) cbrt(((double) log(10.0)))) * ((double) cbrt(((double) log(10.0)))))))))))) * ((double) cbrt(((double) (1.0 / ((double) sqrt(((double) (((double) cbrt(((double) log(10.0)))) * ((double) cbrt(((double) log(10.0)))))))))))))))), ((double) (((double) cbrt(((double) (1.0 / ((double) sqrt(((double) cbrt(((double) log(10.0)))))))))) * ((double) cbrt(((double) (1.0 / ((double) sqrt(((double) cbrt(((double) log(10.0))))))))))))))))));
		} else {
			double VAR_2;
			if ((re <= 4.543347056569168e-225)) {
				VAR_2 = ((double) (((double) (((double) (1.0 / ((double) sqrt(((double) log(10.0)))))) * ((double) cbrt(((double) (1.0 / ((double) sqrt(((double) log(10.0)))))))))) * ((double) log(((double) pow(im, ((double) (((double) cbrt(((double) (1.0 / ((double) sqrt(((double) log(10.0)))))))) * ((double) cbrt(((double) (1.0 / ((double) sqrt(((double) log(10.0))))))))))))))));
			} else {
				double VAR_3;
				if ((re <= 2.2414384362599133e+83)) {
					VAR_3 = ((double) (((double) (((double) (1.0 / ((double) sqrt(((double) log(10.0)))))) * ((double) cbrt(((double) (1.0 / ((double) sqrt(((double) log(10.0)))))))))) * ((double) log(((double) pow(((double) pow(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))), ((double) (((double) cbrt(((double) (1.0 / ((double) sqrt(((double) (((double) cbrt(((double) log(10.0)))) * ((double) cbrt(((double) log(10.0)))))))))))) * ((double) cbrt(((double) (1.0 / ((double) sqrt(((double) (((double) cbrt(((double) log(10.0)))) * ((double) cbrt(((double) log(10.0)))))))))))))))), ((double) (((double) cbrt(((double) (1.0 / ((double) sqrt(((double) cbrt(((double) log(10.0)))))))))) * ((double) cbrt(((double) (1.0 / ((double) sqrt(((double) cbrt(((double) log(10.0))))))))))))))))));
				} else {
					VAR_3 = ((double) (((double) (1.0 / ((double) sqrt(((double) log(10.0)))))) * ((double) log(((double) pow(re, ((double) (1.0 / ((double) sqrt(((double) log(10.0))))))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -2.3080563981284436e+37

   1. Initial program 42.4

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

   3. Applied add-sqr-sqrt 42.4

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

   4. Applied pow1 42.4

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

   5. Applied log-pow 42.4

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

   6. Applied times-frac 42.4

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

   8. Applied add-log-exp 42.4

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

   9. Simplified 42.3

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

   10. Taylor expanded around -inf 11.8

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

   11. Simplified 11.7

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

   if -2.3080563981284436e+37 < re < -2.3437866053113895e-235 or 4.543347056569168e-225 < re < 2.2414384362599133e+83

   1. Initial program 20.3

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

   3. Applied add-sqr-sqrt 20.3

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

   4. Applied pow1 20.3

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

   5. Applied log-pow 20.3

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

   6. Applied times-frac 20.3

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

   8. Applied add-log-exp 20.3

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

   9. Simplified 20.1

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

   11. Applied add-cube-cbrt 20.1

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

   12. Applied pow-unpow 20.3

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

   13. Applied log-pow 20.3

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

   14. Applied associate-*r* 20.1

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

   16. Applied add-cube-cbrt 20.5

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

   17. Applied sqrt-prod 20.5

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

   18. Applied *-un-lft-identity 20.5

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

   19. Applied times-frac 20.1

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

   20. Applied cbrt-prod 20.1

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

   21. Applied add-cube-cbrt 20.5

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

   22. Applied sqrt-prod 20.5

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

   23. Applied *-un-lft-identity 20.5

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

   24. Applied times-frac 20.1

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

   25. Applied cbrt-prod 20.1

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

   26. Applied swap-sqr 20.1

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

   27. Applied pow-unpow 20.1

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

   if -2.3437866053113895e-235 < re < 4.543347056569168e-225

   1. Initial program 32.4

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

   3. Applied add-sqr-sqrt 32.4

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

   4. Applied pow1 32.4

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

   5. Applied log-pow 32.4

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

   6. Applied times-frac 32.3

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

   8. Applied add-log-exp 32.3

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

   9. Simplified 32.2

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

   11. Applied add-cube-cbrt 32.2

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

   12. Applied pow-unpow 32.3

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

   13. Applied log-pow 32.3

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

   14. Applied associate-*r* 32.2

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

   15. Taylor expanded around 0 33.9

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

   if 2.2414384362599133e+83 < re

   1. Initial program 48.6

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

   3. Applied add-sqr-sqrt 48.6

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

   4. Applied pow1 48.6

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

   5. Applied log-pow 48.6

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

   6. Applied times-frac 48.6

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

   8. Applied add-log-exp 48.6

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

   9. Simplified 48.6

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

   10. Taylor expanded around inf 10.1

       \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left({\color{blue}{re}}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.30805639812844361 \cdot 10^{37}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -2.3437866053113895 \cdot 10^{-235}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right) \cdot \log \left({\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10}}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10}}}}\right)}\right)\\ \mathbf{elif}\;re \le 4.543347056569168 \cdot 10^{-225}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right) \cdot \log \left({im}^{\left(\sqrt[3]{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right)}\right)\\ \mathbf{elif}\;re \le 2.2414384362599133 \cdot 10^{83}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right) \cdot \log \left({\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10}}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\sqrt[3]{\log 10}}}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))