Average Error: 32.2 → 18.2
Time: 5.5s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -7.300218245200566 \cdot 10^{+92}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\left(\log 1 + \log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \leq 3.112931712533811 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{0.5} \cdot \left(\left(\log 1 + 2 \cdot \log re\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{0.25}\right)\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -7.300218245200566 \cdot 10^{+92}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\left(\log 1 + \log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\

\mathbf{elif}\;re \leq 3.112931712533811 \cdot 10^{+80}:\\
\;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{0.5} \cdot \left(\left(\log 1 + 2 \cdot \log re\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{0.25}\right)\right)\\

\end{array}
double code(double re, double im) {
	return (((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 <= -7.300218245200566e+92)) {
		VAR = ((double) ((0.5 / ((double) sqrt(((double) log(10.0))))) * ((double) (((double) (((double) log(1.0)) + ((double) (((double) log((-1.0 / re))) * -2.0)))) * ((double) sqrt((1.0 / ((double) log(10.0)))))))));
	} else {
		double VAR_1;
		if ((re <= 3.112931712533811e+80)) {
			VAR_1 = ((double) (((double) sqrt((0.5 / ((double) sqrt(((double) log(10.0))))))) * ((double) (((double) sqrt((0.5 / ((double) sqrt(((double) log(10.0))))))) * (((double) log(((double) (((double) (re * re)) + ((double) (im * im)))))) / ((double) sqrt(((double) log(10.0)))))))));
		} else {
			VAR_1 = ((double) (((double) sqrt((0.5 / ((double) sqrt(((double) log(10.0))))))) * ((double) (((double) sqrt(0.5)) * ((double) (((double) (((double) log(1.0)) + ((double) (2.0 * ((double) log(re)))))) * ((double) pow((1.0 / ((double) pow(((double) log(10.0)), 3.0))), 0.25))))))));
		}
		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 < -7.30021824520056552e92

    1. Initial program 50.6

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

    3. Applied add-sqr-sqrt50.6

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

    4. Applied pow1/250.6

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow50.6

      \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac50.6

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]

    7. Taylor expanded around -inf 10.0

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \color{blue}{\left(\left(\log 1 - 2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \sqrt{\frac{1}{\log 10}}\right)}\]

    8. Simplified10.0

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \color{blue}{\left(\left(\log 1 + \log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)}\]

    if -7.30021824520056552e92 < re < 3.112931712533811e80

    1. Initial program 22.9

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

    3. Applied add-sqr-sqrt22.9

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

    4. Applied pow1/222.9

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow22.9

      \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac22.9

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt22.9

      \[\leadsto \color{blue}{\left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \sqrt{\frac{0.5}{\sqrt{\log 10}}}\right)} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]

    9. Applied associate-*l*22.8

      \[\leadsto \color{blue}{\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)}\]

    10. Simplified22.8

      \[\leadsto \sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{0.5}{\sqrt{\log 10}}}\right)}\]

    if 3.112931712533811e80 < re

    1. Initial program 47.3

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

    3. Applied add-sqr-sqrt47.3

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

    4. Applied pow1/247.3

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow47.3

      \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac47.3

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt47.3

      \[\leadsto \color{blue}{\left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \sqrt{\frac{0.5}{\sqrt{\log 10}}}\right)} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]

    9. Applied associate-*l*47.3

      \[\leadsto \color{blue}{\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)}\]

    10. Simplified47.3

      \[\leadsto \sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{0.5}{\sqrt{\log 10}}}\right)}\]

    11. Taylor expanded around inf 10.2

      \[\leadsto \sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\left(\left(\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)\right) \cdot \sqrt{0.5}\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{0.25}\right)}\]

    12. Simplified10.2

      \[\leadsto \sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\sqrt{0.5} \cdot \left(\left(\log 1 + 2 \cdot \log re\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{0.25}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.300218245200566 \cdot 10^{+92}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\left(\log 1 + \log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \leq 3.112931712533811 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{0.5} \cdot \left(\left(\log 1 + 2 \cdot \log re\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{0.25}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))