math.log10 on complex, real part

Average Error: 32.0 → 17.5

Time: 5.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.1469124341159173 \cdot 10^{94}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{-1}{2} \cdot \frac{\log 10}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -3.19651783577238613 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \le 1.1868703169839283 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{2 \cdot \log im}}\\ \mathbf{elif}\;re \le 3.51842597900795008 \cdot 10^{92}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \left(-2 \cdot \frac{\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{2}}}{\log 10}\right)\\ \end{array}\]

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 <= -2.1469124341159173e+94)) {
		VAR = ((double) (((double) sqrt(0.5)) * ((double) (((double) sqrt(0.5)) / ((double) (-0.5 * ((double) (((double) log(10.0)) / ((double) log(((double) (-1.0 / re))))))))))));
	} else {
		double VAR_1;
		if ((re <= -3.196517835772386e-203)) {
			VAR_1 = ((double) (((double) sqrt(0.5)) * ((double) (((double) sqrt(0.5)) / ((double) (((double) log(10.0)) / ((double) log(((double) (((double) (re * re)) + ((double) (im * im))))))))))));
		} else {
			double VAR_2;
			if ((re <= 1.1868703169839283e-289)) {
				VAR_2 = ((double) (((double) sqrt(0.5)) * ((double) (((double) sqrt(0.5)) / ((double) (((double) log(10.0)) / ((double) (2.0 * ((double) log(im))))))))));
			} else {
				double VAR_3;
				if ((re <= 3.51842597900795e+92)) {
					VAR_3 = ((double) (((double) sqrt(0.5)) * ((double) (((double) sqrt(0.5)) / ((double) (((double) log(10.0)) / ((double) log(((double) (((double) (re * re)) + ((double) (im * im))))))))))));
				} else {
					VAR_3 = ((double) (((double) sqrt(0.5)) * ((double) (-2.0 * ((double) (((double) (((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 < -2.1469124341159173e+94

   1. Initial program 50.9

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

   3. Applied pow1/2 50.9

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

   4. Applied log-pow 50.9

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

   5. Applied associate-/l* 50.9

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

   7. Applied pow1 50.9

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

   8. Applied log-pow 50.9

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

   9. Applied pow1 50.9

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

   10. Applied log-pow 50.9

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

   11. Applied times-frac 50.9

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

   12. Applied add-sqr-sqrt 51.0

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

   13. Applied times-frac 50.9

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

   14. Simplified 50.9

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

   15. Taylor expanded around -inf 8.5

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

   if -2.1469124341159173e+94 < re < -3.196517835772386e-203 or 1.1868703169839283e-289 < re < 3.51842597900795e+92

   1. Initial program 19.9

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

   3. Applied pow1/2 19.9

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

   4. Applied log-pow 19.9

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

   5. Applied associate-/l* 19.9

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

   7. Applied pow1 19.9

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

   8. Applied log-pow 19.9

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

   9. Applied pow1 19.9

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

   10. Applied log-pow 19.9

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

   11. Applied times-frac 19.9

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

   12. Applied add-sqr-sqrt 20.0

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

   13. Applied times-frac 19.8

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

   14. Simplified 19.8

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

   if -3.196517835772386e-203 < re < 1.1868703169839283e-289

   1. Initial program 31.7

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

   3. Applied pow1/2 31.7

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

   4. Applied log-pow 31.7

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

   5. Applied associate-/l* 31.7

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

   7. Applied pow1 31.7

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

   8. Applied log-pow 31.7

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

   9. Applied pow1 31.7

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

   10. Applied log-pow 31.7

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

   11. Applied times-frac 31.7

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

   12. Applied add-sqr-sqrt 31.7

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

   13. Applied times-frac 31.6

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

   14. Simplified 31.6

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

   15. Taylor expanded around 0 35.1

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

   if 3.51842597900795e+92 < re

   1. Initial program 50.0

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

   3. Applied pow1/2 50.0

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

   4. Applied log-pow 50.0

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

   5. Applied associate-/l* 50.0

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

   7. Applied pow1 50.0

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

   8. Applied log-pow 50.0

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

   9. Applied pow1 50.0

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

   10. Applied log-pow 50.0

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

   11. Applied times-frac 50.0

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

   12. Applied add-sqr-sqrt 50.1

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

   13. Applied times-frac 50.0

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

   14. Simplified 50.0

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

   15. Taylor expanded around inf 9.6

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

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.1469124341159173 \cdot 10^{94}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{-1}{2} \cdot \frac{\log 10}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -3.19651783577238613 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \le 1.1868703169839283 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{2 \cdot \log im}}\\ \mathbf{elif}\;re \le 3.51842597900795008 \cdot 10^{92}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \left(-2 \cdot \frac{\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{2}}}{\log 10}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))