math.log10 on complex, real part

Average Error: 32.1 → 18.8

Time: 8.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.26753219211879359 \cdot 10^{116}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log 1 - 2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -1.47898343813311881 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{elif}\;re \le 1.99786214231465739 \cdot 10^{-154}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \le 3.84792093290837512 \cdot 10^{74}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)}}\\ \end{array}\]

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 <= -5.2675321921187936e+116)) {
		VAR = ((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(((double) (-1.0 / re))))))))))))));
	} else {
		double VAR_1;
		if ((re <= -1.4789834381331188e-232)) {
			VAR_1 = ((double) (((double) sqrt(0.5)) / ((double) (((double) (((double) log(10.0)) / ((double) log(((double) (((double) (re * re)) + ((double) (im * im)))))))) / ((double) sqrt(0.5))))));
		} else {
			double VAR_2;
			if ((re <= 1.9978621423146574e-154)) {
				VAR_2 = ((double) (((double) log(im)) / ((double) log(10.0))));
			} else {
				double VAR_3;
				if ((re <= 3.847920932908375e+74)) {
					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) (((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(((double) (1.0 / re))))))))))))));
				}
				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 5 regimes

2. if re < -5.26753219211879359e116

   1. Initial program 54.4

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

   3. Applied pow1/2 54.4

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

   4. Applied log-pow 54.4

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

   5. Applied associate-/l* 54.4

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

   7. Applied pow1 54.4

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

   8. Applied log-pow 54.4

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

   9. Applied add-sqr-sqrt 54.4

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

   10. Applied times-frac 54.5

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

   11. Applied add-sqr-sqrt 54.4

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

   12. Applied times-frac 54.4

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

   13. Simplified 54.4

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

   14. Taylor expanded around -inf 9.0

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

   if -5.26753219211879359e116 < re < -1.47898343813311881e-232

   1. Initial program 19.4

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

   3. Applied pow1/2 19.4

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

   4. Applied log-pow 19.4

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

   5. Applied associate-/l* 19.4

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

   7. Applied add-sqr-sqrt 19.5

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

   8. Applied associate-/l* 19.3

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

   if -1.47898343813311881e-232 < re < 1.99786214231465739e-154

   1. Initial program 32.5

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

   2. Taylor expanded around 0 34.9

      \[\leadsto \frac{\log \color{blue}{im}}{\log 10}\]

   if 1.99786214231465739e-154 < re < 3.84792093290837512e74

   1. Initial program 17.2

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

   3. Applied pow1/2 17.2

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

   4. Applied log-pow 17.2

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

   5. Applied associate-/l* 17.2

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

   7. Applied pow1 17.2

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

   8. Applied log-pow 17.2

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

   9. Applied add-sqr-sqrt 17.2

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

   10. Applied times-frac 17.4

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

   11. Applied add-sqr-sqrt 17.2

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

   12. Applied times-frac 17.1

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

   13. Simplified 17.1

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

   if 3.84792093290837512e74 < re

   1. Initial program 48.1

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

   3. Applied pow1/2 48.1

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

   4. Applied log-pow 48.1

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

   5. Applied associate-/l* 48.1

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

   7. Applied pow1 48.1

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

   8. Applied log-pow 48.1

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

   9. Applied add-sqr-sqrt 48.1

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

   10. Applied times-frac 48.2

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

   11. Applied add-sqr-sqrt 48.1

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

   12. Applied times-frac 48.0

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

   13. Simplified 48.0

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

   14. Taylor expanded around inf 11.3

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\color{blue}{\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.26753219211879359 \cdot 10^{116}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log 1 - 2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -1.47898343813311881 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{elif}\;re \le 1.99786214231465739 \cdot 10^{-154}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \le 3.84792093290837512 \cdot 10^{74}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))