math.log10 on complex, real part

Average Error: 32.3 → 17.8

Time: 5.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -7.4478400426409315 \cdot 10^{96}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{-1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \mathbf{elif}\;re \le 4.7623623031765337 \cdot 10^{48}:\\ \;\;\;\;{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \end{array}\]

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 <= -7.447840042640931e+96)) {
		VAR = ((double) pow(((double) (((double) (((double) (-2.0 * ((double) log(((double) (-1.0 / re)))))) + 0.0)) * ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) / ((double) sqrt(((double) log(10.0)))))))), 1.0));
	} else {
		double VAR_1;
		if ((re <= 4.762362303176534e+48)) {
			VAR_1 = ((double) pow(((double) (((double) log(((double) (((double) (re * re)) + ((double) (im * im)))))) * ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) / ((double) sqrt(((double) log(10.0)))))))), 1.0));
		} else {
			VAR_1 = ((double) pow(((double) (((double) (((double) (-2.0 * ((double) log(((double) (1.0 / re)))))) + 0.0)) * ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) / ((double) sqrt(((double) log(10.0)))))))), 1.0));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -7.4478400426409315e96

   1. Initial program 50.7

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

   3. Applied add-sqr-sqrt 50.7

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

   4. Applied pow1/2 50.7

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

   5. Applied log-pow 50.7

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

   6. Applied times-frac 50.7

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

   8. Applied add-log-exp 50.7

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

   9. Simplified 50.6

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

   11. Applied pow1 50.6

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

   12. Applied pow1 50.6

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

   13. Applied pow-prod-down 50.6

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

   14. Simplified 50.6

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

   15. Taylor expanded around -inf 9.0

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

   16. Simplified 9.0

       \[\leadsto {\left(\color{blue}{\left(-2 \cdot \log \left(\frac{-1}{re}\right) + 0\right)} \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\]

   if -7.4478400426409315e96 < re < 4.7623623031765337e48

   1. Initial program 22.6

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

   3. Applied add-sqr-sqrt 22.6

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

   4. Applied pow1/2 22.6

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

   5. Applied log-pow 22.6

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

   6. Applied times-frac 22.6

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

   8. Applied add-log-exp 22.6

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

   9. Simplified 22.4

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

   11. Applied pow1 22.4

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

   12. Applied pow1 22.4

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

   13. Applied pow-prod-down 22.4

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

   14. Simplified 22.4

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

   if 4.7623623031765337e48 < re

   1. Initial program 46.7

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

   3. Applied add-sqr-sqrt 46.7

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

   4. Applied pow1/2 46.7

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

   5. Applied log-pow 46.7

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

   6. Applied times-frac 46.7

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

   8. Applied add-log-exp 46.7

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

   9. Simplified 46.6

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

   11. Applied pow1 46.6

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

   12. Applied pow1 46.6

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

   13. Applied pow-prod-down 46.6

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

   14. Simplified 46.6

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

   15. Taylor expanded around inf 11.1

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

   16. Simplified 11.1

       \[\leadsto {\left(\color{blue}{\left(-2 \cdot \log \left(\frac{1}{re}\right) + 0\right)} \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.4478400426409315 \cdot 10^{96}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{-1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \mathbf{elif}\;re \le 4.7623623031765337 \cdot 10^{48}:\\ \;\;\;\;{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))