Average Error: 32.1 → 17.8
Time: 6.1s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.24454759091797941 \cdot 10^{73}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{-1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \mathbf{elif}\;re \le 4.5299183249182523 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 5.64802436089789289 \cdot 10^{-175}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \mathbf{elif}\;re \le 1.7016760672680053 \cdot 10^{122}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.24454759091797941 \cdot 10^{73}:\\
\;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{-1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\

\mathbf{elif}\;re \le 4.5299183249182523 \cdot 10^{-190}:\\
\;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{elif}\;re \le 5.64802436089789289 \cdot 10^{-175}:\\
\;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\

\mathbf{elif}\;re \le 1.7016760672680053 \cdot 10^{122}:\\
\;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\

\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 <= -2.2445475909179794e+73)) {
		VAR = ((double) pow(((double) (((double) (((double) (-2.0 * ((double) log(((double) (-1.0 / re)))))) + 0.0)) * ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) / ((double) sqrt(((double) log(10.0)))))))), 1.0));
	} else {
		double VAR_1;
		if ((re <= 4.529918324918252e-190)) {
			VAR_1 = ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) * ((double) log(((double) pow(((double) (((double) (re * re)) + ((double) (im * im)))), ((double) (1.0 / ((double) sqrt(((double) log(10.0))))))))))));
		} else {
			double VAR_2;
			if ((re <= 5.648024360897893e-175)) {
				VAR_2 = ((double) pow(((double) (((double) (((double) (-2.0 * ((double) log(((double) (1.0 / re)))))) + 0.0)) * ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) / ((double) sqrt(((double) log(10.0)))))))), 1.0));
			} else {
				double VAR_3;
				if ((re <= 1.7016760672680053e+122)) {
					VAR_3 = ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) * ((double) log(((double) pow(((double) (((double) (re * re)) + ((double) (im * im)))), ((double) (1.0 / ((double) sqrt(((double) log(10.0))))))))))));
				} else {
					VAR_3 = ((double) pow(((double) (((double) (((double) (-2.0 * ((double) log(((double) (1.0 / re)))))) + 0.0)) * ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) / ((double) sqrt(((double) log(10.0)))))))), 1.0));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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 < -2.24454759091797941e73

    1. Initial program 47.4

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

    3. Applied add-sqr-sqrt47.4

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

    4. Applied pow1/247.4

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

    5. Applied log-pow47.4

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

    6. Applied times-frac47.4

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

    8. Applied add-log-exp47.4

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

    9. Simplified47.3

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

    11. Applied pow147.3

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

    12. Applied pow147.3

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

    13. Applied pow-prod-down47.3

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

    14. Simplified47.3

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

    15. Taylor expanded around -inf 10.8

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

    16. Simplified10.8

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

    if -2.24454759091797941e73 < re < 4.5299183249182523e-190 or 5.64802436089789289e-175 < re < 1.7016760672680053e122

    1. Initial program 21.9

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

    3. Applied add-sqr-sqrt21.9

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

    4. Applied pow1/221.9

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

    5. Applied log-pow21.9

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

    6. Applied times-frac21.9

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

    8. Applied add-log-exp21.9

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

    9. Simplified21.7

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

    if 4.5299183249182523e-190 < re < 5.64802436089789289e-175 or 1.7016760672680053e122 < re

    1. Initial program 55.0

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

    3. Applied add-sqr-sqrt55.0

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

    4. Applied pow1/255.0

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

    5. Applied log-pow55.0

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

    6. Applied times-frac55.0

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

    8. Applied add-log-exp55.0

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

    9. Simplified55.0

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

    11. Applied pow155.0

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

    12. Applied pow155.0

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

    13. Applied pow-prod-down55.0

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

    14. Simplified55.0

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

    15. Taylor expanded around inf 10.3

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

    16. Simplified10.3

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.24454759091797941 \cdot 10^{73}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{-1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \mathbf{elif}\;re \le 4.5299183249182523 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 5.64802436089789289 \cdot 10^{-175}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \mathbf{elif}\;re \le 1.7016760672680053 \cdot 10^{122}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{1}{re}\right) + 0\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020155 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))