Average Error: 32.3 → 18.0
Time: 6.6s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.10203345612408853 \cdot 10^{84}:\\ \;\;\;\;\log \left({\left(e^{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}^{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)\right)}\right)\\ \mathbf{elif}\;re \le -1.15865348838344793 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 7.30470516084041271 \cdot 10^{-295}:\\ \;\;\;\;\log \left({\left(e^{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}^{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(2 \cdot \log im\right)\right)}\right)\\ \mathbf{elif}\;re \le 1.2700176216873086 \cdot 10^{67}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}^{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(-2 \cdot \log \left(\frac{1}{re}\right)\right)\right)}\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.10203345612408853 \cdot 10^{84}:\\
\;\;\;\;\log \left({\left(e^{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}^{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)\right)}\right)\\

\mathbf{elif}\;re \le -1.15865348838344793 \cdot 10^{-182}:\\
\;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\

\mathbf{elif}\;re \le 7.30470516084041271 \cdot 10^{-295}:\\
\;\;\;\;\log \left({\left(e^{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}^{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(2 \cdot \log im\right)\right)}\right)\\

\mathbf{elif}\;re \le 1.2700176216873086 \cdot 10^{67}:\\
\;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\

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

\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 <= -9.102033456124089e+84)) {
		VAR = ((double) log(((double) pow(((double) exp(((double) (0.5 / ((double) sqrt(((double) log(10.0)))))))), ((double) (((double) sqrt(((double) (1.0 / ((double) log(10.0)))))) * ((double) (-2.0 * ((double) log(((double) (-1.0 / re))))))))))));
	} else {
		double VAR_1;
		if ((re <= -1.158653488383448e-182)) {
			VAR_1 = ((double) (((double) sqrt(((double) (0.5 / ((double) sqrt(((double) log(10.0)))))))) * ((double) (((double) sqrt(((double) (0.5 / ((double) sqrt(((double) log(10.0)))))))) * ((double) (((double) log(((double) (((double) (re * re)) + ((double) (im * im)))))) / ((double) sqrt(((double) log(10.0))))))))));
		} else {
			double VAR_2;
			if ((re <= 7.304705160840413e-295)) {
				VAR_2 = ((double) log(((double) pow(((double) exp(((double) (0.5 / ((double) sqrt(((double) log(10.0)))))))), ((double) (((double) sqrt(((double) (1.0 / ((double) log(10.0)))))) * ((double) (2.0 * ((double) log(im))))))))));
			} else {
				double VAR_3;
				if ((re <= 1.2700176216873086e+67)) {
					VAR_3 = ((double) (((double) sqrt(((double) (0.5 / ((double) sqrt(((double) log(10.0)))))))) * ((double) (((double) sqrt(((double) (0.5 / ((double) sqrt(((double) log(10.0)))))))) * ((double) (((double) log(((double) (((double) (re * re)) + ((double) (im * im)))))) / ((double) sqrt(((double) log(10.0))))))))));
				} else {
					VAR_3 = ((double) log(((double) pow(((double) exp(((double) (0.5 / ((double) sqrt(((double) log(10.0)))))))), ((double) (((double) sqrt(((double) (1.0 / ((double) log(10.0)))))) * ((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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if re < -9.10203345612408853e84

    1. Initial program 50.0

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

    3. Applied add-sqr-sqrt50.0

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

    4. Applied pow1/250.0

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

    5. Applied log-pow50.0

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

    6. Applied times-frac50.0

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

    8. Applied add-sqr-sqrt50.0

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

    9. Applied associate-*l*50.0

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

    11. Applied add-log-exp50.0

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

    12. Simplified50.0

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

    13. Taylor expanded around -inf 9.4

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

    14. Simplified9.4

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

    if -9.10203345612408853e84 < re < -1.15865348838344793e-182 or 7.30470516084041271e-295 < re < 1.2700176216873086e67

    1. Initial program 19.9

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

    3. Applied add-sqr-sqrt19.9

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

    4. Applied pow1/219.9

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

    5. Applied log-pow19.9

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

    6. Applied times-frac19.9

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

    8. Applied add-sqr-sqrt19.9

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

    9. Applied associate-*l*19.8

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

    if -1.15865348838344793e-182 < re < 7.30470516084041271e-295

    1. Initial program 34.2

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

    3. Applied add-sqr-sqrt34.2

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

    4. Applied pow1/234.2

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

    5. Applied log-pow34.2

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

    6. Applied times-frac34.2

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

    8. Applied add-sqr-sqrt34.2

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

    9. Applied associate-*l*34.2

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

    11. Applied add-log-exp34.2

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

    12. Simplified34.1

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

    13. Taylor expanded around 0 36.3

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

    14. Simplified36.3

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

    if 1.2700176216873086e67 < re

    1. Initial program 47.2

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

    3. Applied add-sqr-sqrt47.2

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

    4. Applied pow1/247.2

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

    5. Applied log-pow47.2

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

    6. Applied times-frac47.2

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

    8. Applied add-sqr-sqrt47.2

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

    9. Applied associate-*l*47.2

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

    11. Applied add-log-exp47.2

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

    12. Simplified47.1

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

    13. Taylor expanded around inf 10.6

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

    14. Simplified10.6

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.10203345612408853 \cdot 10^{84}:\\ \;\;\;\;\log \left({\left(e^{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}^{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)\right)}\right)\\ \mathbf{elif}\;re \le -1.15865348838344793 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 7.30470516084041271 \cdot 10^{-295}:\\ \;\;\;\;\log \left({\left(e^{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}^{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(2 \cdot \log im\right)\right)}\right)\\ \mathbf{elif}\;re \le 1.2700176216873086 \cdot 10^{67}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)}^{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(-2 \cdot \log \left(\frac{1}{re}\right)\right)\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))