Average Error: 32.0 → 17.9
Time: 6.4s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.6998950373020774 \cdot 10^{87}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right) + 0}}\\ \mathbf{elif}\;re \le 1.6013000626986049 \cdot 10^{132}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\left(\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)\right) \cdot \sqrt{\frac{1}{2}}}{\log 10}\\ \end{array}\]
\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{\sqrt{\frac{1}{2}}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right) + 0}}\\

\mathbf{elif}\;re \le 1.6013000626986049 \cdot 10^{132}:\\
\;\;\;\;\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 \frac{\left(\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)\right) \cdot \sqrt{\frac{1}{2}}}{\log 10}\\

\end{array}
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) sqrt(0.5)) / ((double) (((double) log(10.0)) / ((double) (((double) (-2.0 * ((double) log(((double) (-1.0 / re)))))) + 0.0))))))));
	} else {
		double VAR_1;
		if ((re <= 1.601300062698605e+132)) {
			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 {
			VAR_1 = ((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 = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -1.6998950373020774e87

    1. Initial program 48.9

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

    3. Applied pow1/248.9

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

    4. Applied log-pow48.9

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

    5. Applied associate-/l*48.9

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

    7. Applied pow148.9

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

    8. Applied log-pow48.9

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

    9. Applied pow148.9

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

    10. Applied log-pow48.9

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

    11. Applied times-frac48.9

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

    12. Applied add-sqr-sqrt48.9

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

    13. Applied times-frac48.8

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

    14. Simplified48.8

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

    15. Taylor expanded around -inf 9.3

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

    16. Simplified9.3

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

    if -1.6998950373020774e87 < re < 1.6013000626986049e132

    1. Initial program 22.3

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

    3. Applied pow1/222.3

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

    4. Applied log-pow22.3

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

    5. Applied associate-/l*22.4

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

    7. Applied pow122.4

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

    8. Applied log-pow22.4

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

    9. Applied pow122.4

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

    10. Applied log-pow22.4

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

    11. Applied times-frac22.4

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

    12. Applied add-sqr-sqrt22.5

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

    13. Applied times-frac22.3

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

    14. Simplified22.3

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

    if 1.6013000626986049e132 < re

    1. Initial program 58.4

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

    3. Applied pow1/258.4

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

    4. Applied log-pow58.4

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

    5. Applied associate-/l*58.4

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

    7. Applied pow158.4

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

    8. Applied log-pow58.4

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

    9. Applied pow158.4

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

    10. Applied log-pow58.4

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

    11. Applied times-frac58.4

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

    12. Applied add-sqr-sqrt58.4

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

    13. Applied times-frac58.4

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

    14. Simplified58.4

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

    15. Taylor expanded around inf 7.3

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.6998950373020774 \cdot 10^{87}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right) + 0}}\\ \mathbf{elif}\;re \le 1.6013000626986049 \cdot 10^{132}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\left(\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)\right) \cdot \sqrt{\frac{1}{2}}}{\log 10}\\ \end{array}\]

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)))