math.log10 on complex, real part

Average Error: 31.6 → 17.3

Time: 6.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.9954377646446586 \cdot 10^{97}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 9.860139717477679 \cdot 10^{-245}:\\ \;\;\;\;\frac{\left(2 \cdot \log \left(\sqrt[3]{\sqrt{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}}\right)\right) \cdot 1}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\\ \mathbf{elif}\;re \le 4.31887266363036965 \cdot 10^{-169}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 1.3030602997104563 \cdot 10^{144}:\\ \;\;\;\;\frac{\left(2 \cdot \log \left(\sqrt[3]{\sqrt{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}}\right)\right) \cdot 1}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -2.9954377646446586e+97)) {
		VAR = ((1.0 / sqrt(log(10.0))) * log(pow((-1.0 / re), -sqrt((1.0 / log(10.0))))));
	} else {
		double VAR_1;
		if ((re <= 9.860139717477679e-245)) {
			VAR_1 = ((((2.0 * log(cbrt((sqrt(pow(sqrt(((re * re) + (im * im))), (1.0 / sqrt(log(10.0))))) * sqrt(pow(sqrt(((re * re) + (im * im))), (1.0 / sqrt(log(10.0))))))))) * 1.0) / sqrt(log(10.0))) + ((1.0 / sqrt(log(10.0))) * log(cbrt(pow(sqrt(((re * re) + (im * im))), (1.0 / sqrt(log(10.0))))))));
		} else {
			double VAR_2;
			if ((re <= 4.3188726636303697e-169)) {
				VAR_2 = ((1.0 / sqrt(log(10.0))) * log(pow(im, sqrt((1.0 / log(10.0))))));
			} else {
				double VAR_3;
				if ((re <= 1.3030602997104563e+144)) {
					VAR_3 = ((((2.0 * log(cbrt((sqrt(pow(sqrt(((re * re) + (im * im))), (1.0 / sqrt(log(10.0))))) * sqrt(pow(sqrt(((re * re) + (im * im))), (1.0 / sqrt(log(10.0))))))))) * 1.0) / sqrt(log(10.0))) + ((1.0 / sqrt(log(10.0))) * log(cbrt(pow(sqrt(((re * re) + (im * im))), (1.0 / sqrt(log(10.0))))))));
				} else {
					VAR_3 = ((1.0 / sqrt(log(10.0))) * log(pow(re, (1.0 / sqrt(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 < -2.9954377646446586e+97

   1. Initial program 51.4

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

   3. Applied add-sqr-sqrt 51.4

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

   4. Applied pow1 51.4

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

   5. Applied log-pow 51.4

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

   6. Applied times-frac 51.3

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

   8. Applied add-log-exp 51.3

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

   9. Simplified 51.3

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

   10. Taylor expanded around -inf 9.2

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

   11. Simplified 9.1

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

   if -2.9954377646446586e+97 < re < 9.860139717477679e-245 or 4.3188726636303697e-169 < re < 1.3030602997104563e+144

   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

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

   12. Applied log-prod 20.2

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

   13. Applied distribute-lft-in 20.2

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

   14. Simplified 20.2

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

   16. Applied add-sqr-sqrt 20.2

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

   if 9.860139717477679e-245 < re < 4.3188726636303697e-169

   1. Initial program 32.2

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

   3. Applied add-sqr-sqrt 32.2

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

   4. Applied pow1 32.2

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

   5. Applied log-pow 32.2

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

   6. Applied times-frac 32.2

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

   8. Applied add-log-exp 32.2

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

   9. Simplified 32.1

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

   10. Taylor expanded around 0 35.0

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

   11. Simplified 34.9

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

   if 1.3030602997104563e+144 < re

   1. Initial program 61.3

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

   3. Applied add-sqr-sqrt 61.3

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

   4. Applied pow1 61.3

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

   5. Applied log-pow 61.3

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

   6. Applied times-frac 61.3

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

   8. Applied add-log-exp 61.3

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

   9. Simplified 61.3

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

   10. Taylor expanded around inf 5.4

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

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.9954377646446586 \cdot 10^{97}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 9.860139717477679 \cdot 10^{-245}:\\ \;\;\;\;\frac{\left(2 \cdot \log \left(\sqrt[3]{\sqrt{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}}\right)\right) \cdot 1}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\\ \mathbf{elif}\;re \le 4.31887266363036965 \cdot 10^{-169}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 1.3030602997104563 \cdot 10^{144}:\\ \;\;\;\;\frac{\left(2 \cdot \log \left(\sqrt[3]{\sqrt{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}}\right)\right) \cdot 1}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))