math.log10 on complex, real part

Average Error: 32.5 → 17.4

Time: 4.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.06332324946270965 \cdot 10^{96}:\\ \;\;\;\;\frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \left(\frac{2}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right) + 0}{1}\right)\\ \mathbf{elif}\;re \le 2.0014054693044192 \cdot 10^{92}:\\ \;\;\;\;\log \left({\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{1}\right)}\right)}^{\left(\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \left(\frac{2}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{1}{re}\right) + 0}{1}\right)\\ \end{array}\]

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 <= -2.0633232494627097e+96)) {
		VAR = ((double) (((double) (0.25 / ((double) sqrt(((double) log(10.0)))))) * ((double) (((double) (2.0 / ((double) sqrt(((double) log(10.0)))))) * ((double) (((double) (((double) (-2.0 * ((double) log(((double) (-1.0 / re)))))) + 0.0)) / 1.0))))));
	} else {
		double VAR_1;
		if ((re <= 2.0014054693044192e+92)) {
			VAR_1 = ((double) log(((double) pow(((double) pow(((double) (((double) (re * re)) + ((double) (im * im)))), ((double) (((double) sqrt(((double) (0.5 / ((double) sqrt(((double) log(10.0)))))))) / 1.0)))), ((double) (((double) sqrt(((double) (0.5 / ((double) sqrt(((double) log(10.0)))))))) / ((double) sqrt(((double) log(10.0))))))))));
		} else {
			VAR_1 = ((double) (((double) (0.25 / ((double) sqrt(((double) log(10.0)))))) * ((double) (((double) (2.0 / ((double) sqrt(((double) log(10.0)))))) * ((double) (((double) (((double) (-2.0 * ((double) log(((double) (1.0 / re)))))) + 0.0)) / 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 < -2.06332324946270965e96

   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

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

   4. Applied sqrt-prod 51.4

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

   6. Applied add-sqr-sqrt 51.4

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

   7. Applied pow1/2 51.4

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

   8. Applied sqrt-pow1 51.4

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

   9. Applied pow1/2 51.4

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

   10. Applied sqrt-pow1 51.4

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

   11. Applied pow-prod-down 64.0

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

   12. Applied log-pow 64.0

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

   13. Applied times-frac 64.0

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

   14. Simplified 64.0

       \[\leadsto \color{blue}{\frac{\frac{1}{4}}{\sqrt{\log 10}}} \cdot \frac{\log \left(\left(re \cdot re + im \cdot im\right) \cdot \left(re \cdot re + im \cdot im\right)\right)}{\sqrt{\log 10}}\]

   15. Simplified 51.4

       \[\leadsto \frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\frac{2}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{1}\right)}\]

   16. Taylor expanded around -inf 9.7

       \[\leadsto \frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \left(\frac{2}{\sqrt{\log 10}} \cdot \frac{\color{blue}{\log 1 - 2 \cdot \log \left(\frac{-1}{re}\right)}}{1}\right)\]

   17. Simplified 9.7

       \[\leadsto \frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \left(\frac{2}{\sqrt{\log 10}} \cdot \frac{\color{blue}{-2 \cdot \log \left(\frac{-1}{re}\right) + 0}}{1}\right)\]

   if -2.06332324946270965e96 < re < 2.0014054693044192e92

   1. Initial program 22.1

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

   3. Applied add-sqr-sqrt 22.1

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

   4. Applied sqrt-prod 22.1

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

   6. Applied add-sqr-sqrt 22.1

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

   7. Applied pow1/2 22.1

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

   8. Applied sqrt-pow1 22.1

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

   9. Applied pow1/2 22.1

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

   10. Applied sqrt-pow1 22.1

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

   11. Applied pow-prod-down 39.8

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

   12. Applied log-pow 39.8

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

   13. Applied times-frac 39.8

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

   14. Simplified 39.8

       \[\leadsto \color{blue}{\frac{\frac{1}{4}}{\sqrt{\log 10}}} \cdot \frac{\log \left(\left(re \cdot re + im \cdot im\right) \cdot \left(re \cdot re + im \cdot im\right)\right)}{\sqrt{\log 10}}\]

   15. Simplified 22.0

       \[\leadsto \frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\frac{2}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{1}\right)}\]
   16. Using strategy rm

   17. Applied add-log-exp 22.0

       \[\leadsto \color{blue}{\log \left(e^{\frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \left(\frac{2}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{1}\right)}\right)}\]

   18. Simplified 21.9

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

   20. Applied pow1 21.9

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

   21. Applied log-pow 21.9

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

   22. Applied sqrt-prod 21.9

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

   23. Applied add-sqr-sqrt 22.2

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

   24. Applied times-frac 21.9

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

   25. Applied pow-unpow 21.8

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

   26. Simplified 21.8

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

   if 2.0014054693044192e92 < re

   1. Initial program 51.2

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

   3. Applied add-sqr-sqrt 51.2

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

   4. Applied sqrt-prod 51.2

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

   6. Applied add-sqr-sqrt 51.2

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

   7. Applied pow1/2 51.2

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

   8. Applied sqrt-pow1 51.2

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

   9. Applied pow1/2 51.2

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

   10. Applied sqrt-pow1 51.2

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

   11. Applied pow-prod-down 64.0

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

   12. Applied log-pow 64.0

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

   13. Applied times-frac 64.0

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

   14. Simplified 64.0

       \[\leadsto \color{blue}{\frac{\frac{1}{4}}{\sqrt{\log 10}}} \cdot \frac{\log \left(\left(re \cdot re + im \cdot im\right) \cdot \left(re \cdot re + im \cdot im\right)\right)}{\sqrt{\log 10}}\]

   15. Simplified 51.2

       \[\leadsto \frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\frac{2}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{1}\right)}\]

   16. Taylor expanded around inf 9.3

       \[\leadsto \frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \left(\frac{2}{\sqrt{\log 10}} \cdot \frac{\color{blue}{\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)}}{1}\right)\]

   17. Simplified 9.3

       \[\leadsto \frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \left(\frac{2}{\sqrt{\log 10}} \cdot \frac{\color{blue}{-2 \cdot \log \left(\frac{1}{re}\right) + 0}}{1}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.06332324946270965 \cdot 10^{96}:\\ \;\;\;\;\frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \left(\frac{2}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right) + 0}{1}\right)\\ \mathbf{elif}\;re \le 2.0014054693044192 \cdot 10^{92}:\\ \;\;\;\;\log \left({\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{1}\right)}\right)}^{\left(\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4}}{\sqrt{\log 10}} \cdot \left(\frac{2}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{1}{re}\right) + 0}{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))