math.log10 on complex, real part

Average Error: 32.1 → 17.8

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.04737876035649095 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -5.04942056547374 \cdot 10^{-269}:\\ \;\;\;\;\log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \frac{2}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\\ \mathbf{elif}\;re \le 9.67584874540543507 \cdot 10^{-216}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 3.7629938528630668 \cdot 10^{58}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double temp;
	if ((re <= -5.047378760356491e-05)) {
		temp = ((1.0 / sqrt(log(10.0))) * log(pow((-1.0 / re), -sqrt((1.0 / log(10.0))))));
	} else {
		double temp_1;
		if ((re <= -5.04942056547374e-269)) {
			temp_1 = ((log(cbrt(pow(sqrt(((re * re) + (im * im))), (1.0 / sqrt(log(10.0)))))) * (2.0 / 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 temp_2;
			if ((re <= 9.675848745405435e-216)) {
				temp_2 = ((1.0 / sqrt(log(10.0))) * log(pow(im, sqrt((1.0 / log(10.0))))));
			} else {
				double temp_3;
				if ((re <= 3.762993852863067e+58)) {
					temp_3 = ((1.0 / sqrt(log(10.0))) * (log(sqrt(((re * re) + (im * im)))) * (1.0 / sqrt(log(10.0)))));
				} else {
					temp_3 = ((1.0 / sqrt(log(10.0))) * log(pow(re, (1.0 / sqrt(log(10.0))))));
				}
				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 < -5.047378760356491e-05

   1. Initial program 40.9

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

   3. Applied add-sqr-sqrt 40.9

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

   4. Applied pow1 40.9

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

   5. Applied log-pow 40.9

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

   6. Applied times-frac 40.9

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

   8. Applied div-inv 40.8

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

   10. Applied add-log-exp 40.8

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

   11. Simplified 40.8

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

   12. Taylor expanded around -inf 13.1

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

   13. Simplified 13.0

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

   if -5.047378760356491e-05 < re < -5.04942056547374e-269

   1. Initial program 22.4

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

   3. Applied add-sqr-sqrt 22.4

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

   4. Applied pow1 22.4

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

   5. Applied log-pow 22.4

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

   6. Applied times-frac 22.3

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

   8. Applied div-inv 22.2

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

   10. Applied add-log-exp 22.2

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

   11. Simplified 22.1

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

   13. Applied add-cube-cbrt 22.1

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

   14. Applied log-prod 22.2

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

   15. Applied distribute-lft-in 22.2

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

   16. Simplified 22.2

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

   if -5.04942056547374e-269 < re < 9.675848745405435e-216

   1. Initial program 31.8

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

   3. Applied add-sqr-sqrt 31.8

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

   4. Applied pow1 31.8

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

   5. Applied log-pow 31.8

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

   6. Applied times-frac 31.8

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

   8. Applied div-inv 31.7

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

   10. Applied add-log-exp 31.7

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

   11. Simplified 31.6

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

   12. Taylor expanded around 0 32.1

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

   13. Simplified 32.1

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

   if 9.675848745405435e-216 < re < 3.762993852863067e+58

   1. Initial program 18.9

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

   3. Applied add-sqr-sqrt 18.9

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

   4. Applied pow1 18.9

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

   5. Applied log-pow 18.9

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

   6. Applied times-frac 18.9

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

   8. Applied div-inv 18.8

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

   if 3.762993852863067e+58 < re

   1. Initial program 45.9

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

   3. Applied add-sqr-sqrt 45.9

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

   4. Applied pow1 45.9

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

   5. Applied log-pow 45.9

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

   6. Applied times-frac 45.9

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

   8. Applied div-inv 45.9

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

   10. Applied add-log-exp 45.9

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

   11. Simplified 45.9

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

   12. Taylor expanded around inf 10.7

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.04737876035649095 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -5.04942056547374 \cdot 10^{-269}:\\ \;\;\;\;\log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \frac{2}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\\ \mathbf{elif}\;re \le 9.67584874540543507 \cdot 10^{-216}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 3.7629938528630668 \cdot 10^{58}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))