math.log10 on complex, real part

Average Error: 32.1 → 18.4

Time: 7.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.10321569695692608 \cdot 10^{72}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(-1 \cdot re\right)}}\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log im}}\\ \mathbf{elif}\;re \le 1.13427855715340043 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{elif}\;re \le 1.30573406095301773 \cdot 10^{-191}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{im}\right)}}\\ \mathbf{elif}\;re \le 5.15621950091572796 \cdot 10^{39}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{-\sqrt{\log 10}}{\log \left(\frac{1}{re}\right)}}\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double temp;
	if ((re <= -1.103215696956926e+72)) {
		temp = ((1.0 / sqrt(log(10.0))) / (sqrt(log(10.0)) / log((-1.0 * re))));
	} else {
		double temp_1;
		if ((re <= -1.3504253849915568e-194)) {
			temp_1 = (((cbrt(1.0) * cbrt(1.0)) / sqrt(log(10.0))) * ((cbrt(1.0) / sqrt(log(10.0))) * log(sqrt(((re * re) + (im * im))))));
		} else {
			double temp_2;
			if ((re <= -2.968956980813959e-266)) {
				temp_2 = ((1.0 / sqrt(log(10.0))) / (sqrt(log(10.0)) / log(im)));
			} else {
				double temp_3;
				if ((re <= 1.1342785571534004e-228)) {
					temp_3 = (((cbrt(1.0) * cbrt(1.0)) / sqrt(log(10.0))) * ((cbrt(1.0) / sqrt(log(10.0))) * log(sqrt(((re * re) + (im * im))))));
				} else {
					double temp_4;
					if ((re <= 1.3057340609530177e-191)) {
						temp_4 = (3.0 / (log(10.0) / log(cbrt(im))));
					} else {
						double temp_5;
						if ((re <= 5.156219500915728e+39)) {
							temp_5 = (((cbrt(1.0) * cbrt(1.0)) / sqrt(log(10.0))) * ((cbrt(1.0) / sqrt(log(10.0))) * log(sqrt(((re * re) + (im * im))))));
						} else {
							temp_5 = ((1.0 / sqrt(log(10.0))) / (-sqrt(log(10.0)) / log((1.0 / re))));
						}
						temp_4 = temp_5;
					}
					temp_3 = temp_4;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -1.103215696956926e+72

   1. Initial program 47.0

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

   3. Applied add-cube-cbrt 47.0

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

   5. Applied pow1 47.0

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

   6. Applied pow1 47.0

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

   7. Applied pow1 47.0

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

   8. Applied pow-prod-up 47.0

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

   9. Applied pow-prod-up 47.0

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

   10. Applied log-pow 47.0

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

   11. Applied associate-/l* 47.0

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

   13. Applied pow1/3 47.1

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

   14. Applied log-pow 47.0

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

   15. Applied add-sqr-sqrt 47.0

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

   16. Applied times-frac 47.2

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

   17. Applied associate-/r* 47.2

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

   18. Simplified 47.0

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

   19. Taylor expanded around -inf 9.8

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

   if -1.103215696956926e+72 < re < -1.3504253849915568e-194 or -2.968956980813959e-266 < re < 1.1342785571534004e-228 or 1.3057340609530177e-191 < re < 5.156219500915728e+39

   1. Initial program 21.5

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

   3. Applied add-cube-cbrt 21.5

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

   5. Applied pow1 21.5

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

   6. Applied pow1 21.5

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

   7. Applied pow1 21.5

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

   8. Applied pow-prod-up 21.5

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

   9. Applied pow-prod-up 21.5

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

   10. Applied log-pow 21.5

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

   11. Applied associate-/l* 21.5

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

   13. Applied pow1/3 21.7

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

   14. Applied log-pow 21.7

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

   15. Applied add-sqr-sqrt 21.7

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

   16. Applied times-frac 22.1

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

   17. Applied associate-/r* 22.0

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

   18. Simplified 21.5

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

   20. Applied div-inv 21.5

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

   21. Applied pow1 21.5

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

   22. Applied log-pow 21.5

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

   23. Applied sqrt-prod 21.5

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

   24. Applied add-cube-cbrt 21.5

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

   25. Applied times-frac 21.5

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

   26. Applied times-frac 21.4

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

   27. Simplified 21.4

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

   28. Simplified 21.4

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

   if -1.3504253849915568e-194 < re < -2.968956980813959e-266

   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-cube-cbrt 32.4

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

   5. Applied pow1 32.4

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

   6. Applied pow1 32.4

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

   7. Applied pow1 32.4

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

   8. Applied pow-prod-up 32.4

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

   9. Applied pow-prod-up 32.4

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

   10. Applied log-pow 32.4

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

   11. Applied associate-/l* 32.4

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

   13. Applied pow1/3 32.5

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

   14. Applied log-pow 32.5

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

   15. Applied add-sqr-sqrt 32.5

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

   16. Applied times-frac 32.8

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

   17. Applied associate-/r* 32.8

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

   18. Simplified 32.4

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

   19. Taylor expanded around 0 35.6

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

   if 1.1342785571534004e-228 < re < 1.3057340609530177e-191

   1. Initial program 33.0

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

   3. Applied add-cube-cbrt 33.0

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

   5. Applied pow1 33.0

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

   6. Applied pow1 33.0

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

   7. Applied pow1 33.0

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

   8. Applied pow-prod-up 33.0

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

   9. Applied pow-prod-up 33.0

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

   10. Applied log-pow 33.0

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

   11. Applied associate-/l* 33.0

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

   12. Taylor expanded around 0 33.9

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

   if 5.156219500915728e+39 < re

   1. Initial program 44.1

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

   3. Applied add-cube-cbrt 44.0

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

   5. Applied pow1 44.0

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

   6. Applied pow1 44.0

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

   7. Applied pow1 44.0

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

   8. Applied pow-prod-up 44.0

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

   9. Applied pow-prod-up 44.0

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

   10. Applied log-pow 44.1

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

   11. Applied associate-/l* 44.1

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

   13. Applied pow1/3 44.2

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

   14. Applied log-pow 44.1

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

   15. Applied add-sqr-sqrt 44.1

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

   16. Applied times-frac 44.3

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

   17. Applied associate-/r* 44.3

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

   18. Simplified 44.0

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

   19. Taylor expanded around inf 12.1

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

   20. Simplified 12.1

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

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.10321569695692608 \cdot 10^{72}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(-1 \cdot re\right)}}\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log im}}\\ \mathbf{elif}\;re \le 1.13427855715340043 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{elif}\;re \le 1.30573406095301773 \cdot 10^{-191}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{im}\right)}}\\ \mathbf{elif}\;re \le 5.15621950091572796 \cdot 10^{39}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{-\sqrt{\log 10}}{\log \left(\frac{1}{re}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))