math.log10 on complex, real part

Average Error: 31.9 → 18.3

Time: 8.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.6998950373020774 \cdot 10^{87}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\left(\log 1 - 2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \sqrt{\frac{1}{2}}}{\log 10}\\ \mathbf{elif}\;re \le -1.1243171280702968 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \le 2.4669184868707975 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log 1 + 2 \cdot \log im}}\\ \mathbf{elif}\;re \le 9.355465747616013 \cdot 10^{131}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\left(\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)\right) \cdot \sqrt{\frac{1}{2}}}{\log 10}\\ \end{array}\]

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 <= -1.6998950373020774e+87)) {
		VAR = ((double) (((double) sqrt(0.5)) * ((double) (((double) (((double) (((double) log(1.0)) - ((double) (2.0 * ((double) log(((double) (-1.0 / re)))))))) * ((double) sqrt(0.5)))) / ((double) log(10.0))))));
	} else {
		double VAR_1;
		if ((re <= -1.1243171280702968e-257)) {
			VAR_1 = ((double) (((double) (((double) sqrt(0.5)) / ((double) sqrt(((double) log(10.0)))))) * ((double) (((double) sqrt(0.5)) / ((double) (((double) sqrt(((double) log(10.0)))) / ((double) log(((double) (((double) (re * re)) + ((double) (im * im))))))))))));
		} else {
			double VAR_2;
			if ((re <= 2.4669184868707975e-173)) {
				VAR_2 = ((double) (((double) (((double) sqrt(0.5)) / ((double) sqrt(((double) log(10.0)))))) * ((double) (((double) sqrt(0.5)) / ((double) (((double) sqrt(((double) log(10.0)))) / ((double) (((double) log(1.0)) + ((double) (2.0 * ((double) log(im))))))))))));
			} else {
				double VAR_3;
				if ((re <= 9.355465747616013e+131)) {
					VAR_3 = ((double) (((double) (((double) sqrt(0.5)) / ((double) sqrt(((double) log(10.0)))))) * ((double) (((double) sqrt(0.5)) / ((double) (((double) sqrt(((double) log(10.0)))) / ((double) log(((double) (((double) (re * re)) + ((double) (im * im))))))))))));
				} else {
					VAR_3 = ((double) (((double) sqrt(0.5)) * ((double) (((double) (((double) (((double) log(1.0)) - ((double) (2.0 * ((double) log(((double) (1.0 / re)))))))) * ((double) sqrt(0.5)))) / ((double) 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 < -1.6998950373020774e87

   1. Initial program 49.0

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

   3. Applied pow1/2 49.0

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

   4. Applied log-pow 49.0

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

   5. Applied associate-/l* 49.0

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

   7. Applied pow1 49.0

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

   8. Applied log-pow 49.0

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

   9. Applied pow1 49.0

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

   10. Applied log-pow 49.0

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

   11. Applied times-frac 49.0

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

   12. Applied add-sqr-sqrt 49.0

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

   13. Applied times-frac 48.9

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

   14. Simplified 48.9

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

   15. Taylor expanded around -inf 9.3

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

   if -1.6998950373020774e87 < re < -1.1243171280702968e-257 or 2.4669184868707975e-173 < re < 9.355465747616013e131

   1. Initial program 19.6

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

   3. Applied pow1/2 19.6

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

   4. Applied log-pow 19.6

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

   5. Applied associate-/l* 19.6

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

   7. Applied pow1 19.6

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

   8. Applied log-pow 19.6

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

   9. Applied add-sqr-sqrt 19.6

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

   10. Applied times-frac 19.7

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

   11. Applied add-sqr-sqrt 19.6

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

   12. Applied times-frac 19.5

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

   13. Simplified 19.5

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

   if -1.1243171280702968e-257 < re < 2.4669184868707975e-173

   1. Initial program 31.5

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

   3. Applied pow1/2 31.5

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

   4. Applied log-pow 31.5

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

   5. Applied associate-/l* 31.5

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

   7. Applied pow1 31.5

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

   8. Applied log-pow 31.5

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

   9. Applied add-sqr-sqrt 31.5

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

   10. Applied times-frac 31.6

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

   11. Applied add-sqr-sqrt 31.5

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

   12. Applied times-frac 31.4

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

   13. Simplified 31.4

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

   14. Taylor expanded around 0 34.4

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\color{blue}{\log 1 + 2 \cdot \log im}}}\]

   if 9.355465747616013e131 < re

   1. Initial program 58.0

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

   3. Applied pow1/2 58.0

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

   4. Applied log-pow 58.0

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

   5. Applied associate-/l* 57.9

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

   7. Applied pow1 57.9

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

   8. Applied log-pow 57.9

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

   9. Applied pow1 57.9

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

   10. Applied log-pow 57.9

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

   11. Applied times-frac 57.9

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

   12. Applied add-sqr-sqrt 58.0

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

   13. Applied times-frac 57.9

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

   14. Simplified 57.9

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

   15. Taylor expanded around inf 7.2

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

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.6998950373020774 \cdot 10^{87}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\left(\log 1 - 2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \sqrt{\frac{1}{2}}}{\log 10}\\ \mathbf{elif}\;re \le -1.1243171280702968 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \le 2.4669184868707975 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log 1 + 2 \cdot \log im}}\\ \mathbf{elif}\;re \le 9.355465747616013 \cdot 10^{131}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\left(\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)\right) \cdot \sqrt{\frac{1}{2}}}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))