math.log10 on complex, real part

Average Error: 31.9 → 17.2

Time: 50.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.5550581562043043 \cdot 10^{140}:\\ \;\;\;\;-2 \cdot \left(\frac{\log \left(\frac{-1}{re}\right) \cdot \left|\sqrt[3]{\frac{1}{2}}\right|}{\sqrt{2} \cdot \log 10} \cdot \sqrt{\sqrt[3]{\frac{1}{2}}}\right)\\ \mathbf{elif}\;re \le 8.21511175098157098 \cdot 10^{102}:\\ \;\;\;\;\frac{\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{1}}}{\sqrt{\frac{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}{\sqrt[3]{1}}}} \cdot \frac{\left|\sqrt[3]{\frac{1}{2}}\right| \cdot \sqrt{1}}{\sqrt{\frac{\sqrt{\sqrt[3]{\log 10}}}{\sqrt[3]{\frac{1}{2}}}} \cdot \left(\sqrt{\frac{\sqrt{\log 10}}{\frac{1}{2}}} \cdot \sqrt{\log 10}\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{\log \left(\frac{1}{re}\right) \cdot \left|\sqrt[3]{\frac{1}{2}}\right|}{\sqrt{2} \cdot \log 10} \cdot \sqrt{\sqrt[3]{\frac{1}{2}}}\right)\\ \end{array}\]

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 <= -4.555058156204304e+140)) {
		temp = (-2.0 * (((log((-1.0 / re)) * fabs(cbrt(0.5))) / (sqrt(2.0) * log(10.0))) * sqrt(cbrt(0.5))));
	} else {
		double temp_1;
		if ((re <= 8.215111750981571e+102)) {
			temp_1 = (((log(((re * re) + (im * im))) / sqrt(1.0)) / sqrt((sqrt((cbrt(log(10.0)) * cbrt(log(10.0)))) / cbrt(1.0)))) * ((fabs(cbrt(0.5)) * sqrt(1.0)) / (sqrt((sqrt(cbrt(log(10.0))) / cbrt(0.5))) * (sqrt((sqrt(log(10.0)) / 0.5)) * sqrt(log(10.0))))));
		} else {
			temp_1 = (-2.0 * (((log((1.0 / re)) * fabs(cbrt(0.5))) / (sqrt(2.0) * log(10.0))) * sqrt(cbrt(0.5))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -4.555058156204304e+140

   1. Initial program 60.4

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

   3. Applied add-sqr-sqrt 60.4

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

   4. Applied pow1/2 60.4

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

   5. Applied log-pow 60.4

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

   6. Applied times-frac 60.4

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

   8. Applied add-sqr-sqrt 60.4

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

   9. Applied associate-*l* 60.4

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

   11. Applied pow1 60.4

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

   12. Applied log-pow 60.4

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

   13. Applied sqrt-prod 60.4

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

   14. Applied associate-/r* 60.4

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

   15. Applied clear-num 60.4

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

   16. Applied sqrt-div 60.4

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

   17. Applied frac-times 60.4

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

   18. Applied add-cube-cbrt 60.4

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

   19. Applied associate-/l* 60.4

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

   20. Applied sqrt-div 60.4

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

   21. Applied frac-times 60.4

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

   22. Simplified 60.4

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

   23. Taylor expanded around -inf 7.2

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

   if -4.555058156204304e+140 < re < 8.215111750981571e+102

   1. Initial program 21.4

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

   3. Applied add-sqr-sqrt 21.4

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

   4. Applied pow1/2 21.4

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

   5. Applied log-pow 21.4

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

   6. Applied times-frac 21.4

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

   8. Applied add-sqr-sqrt 21.4

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

   9. Applied associate-*l* 21.3

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

   11. Applied pow1 21.3

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

   12. Applied log-pow 21.3

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

   13. Applied sqrt-prod 21.3

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

   14. Applied associate-/r* 21.3

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

   15. Applied clear-num 21.3

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

   16. Applied sqrt-div 21.3

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

   17. Applied frac-times 21.3

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

   18. Applied add-cube-cbrt 21.3

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

   19. Applied associate-/l* 21.3

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

   20. Applied sqrt-div 21.3

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

   21. Applied frac-times 21.3

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

   22. Simplified 21.3

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

   24. Applied *-un-lft-identity 21.3

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

   25. Applied cbrt-prod 21.3

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

   26. Applied add-cube-cbrt 21.3

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

   27. Applied sqrt-prod 21.3

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

   28. Applied times-frac 21.3

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

   29. Applied sqrt-prod 21.4

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

   30. Applied associate-*l* 21.3

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

   31. Applied *-commutative 21.3

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

   32. Applied times-frac 21.3

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

   if 8.215111750981571e+102 < re

   1. Initial program 52.1

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

   3. Applied add-sqr-sqrt 52.1

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

   4. Applied pow1/2 52.1

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

   5. Applied log-pow 52.1

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

   6. Applied times-frac 52.1

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

   8. Applied add-sqr-sqrt 52.1

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

   9. Applied associate-*l* 52.0

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

   11. Applied pow1 52.0

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

   12. Applied log-pow 52.0

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

   13. Applied sqrt-prod 52.0

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

   14. Applied associate-/r* 52.0

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

   15. Applied clear-num 52.0

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

   16. Applied sqrt-div 52.0

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

   17. Applied frac-times 52.0

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

   18. Applied add-cube-cbrt 52.0

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

   19. Applied associate-/l* 52.0

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

   20. Applied sqrt-div 52.0

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

   21. Applied frac-times 52.1

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

   22. Simplified 52.1

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

   23. Taylor expanded around inf 8.1

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

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.5550581562043043 \cdot 10^{140}:\\ \;\;\;\;-2 \cdot \left(\frac{\log \left(\frac{-1}{re}\right) \cdot \left|\sqrt[3]{\frac{1}{2}}\right|}{\sqrt{2} \cdot \log 10} \cdot \sqrt{\sqrt[3]{\frac{1}{2}}}\right)\\ \mathbf{elif}\;re \le 8.21511175098157098 \cdot 10^{102}:\\ \;\;\;\;\frac{\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{1}}}{\sqrt{\frac{\sqrt{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}}{\sqrt[3]{1}}}} \cdot \frac{\left|\sqrt[3]{\frac{1}{2}}\right| \cdot \sqrt{1}}{\sqrt{\frac{\sqrt{\sqrt[3]{\log 10}}}{\sqrt[3]{\frac{1}{2}}}} \cdot \left(\sqrt{\frac{\sqrt{\log 10}}{\frac{1}{2}}} \cdot \sqrt{\log 10}\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{\log \left(\frac{1}{re}\right) \cdot \left|\sqrt[3]{\frac{1}{2}}\right|}{\sqrt{2} \cdot \log 10} \cdot \sqrt{\sqrt[3]{\frac{1}{2}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))