Average Error: 31.6 → 19.0
Time: 6.8s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.30410728802987954 \cdot 10^{54}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right)}{\log 10}\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}\\ \mathbf{elif}\;re \le 2.15180429374657097 \cdot 10^{-295}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{\frac{{\left(\log im\right)}^{9}}{{\left(\log 10\right)}^{9}}}}\\ \mathbf{elif}\;re \le 2.55703958874010118 \cdot 10^{92}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log re}{\log 10}\right)}^{3}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.30410728802987954 \cdot 10^{54}:\\
\;\;\;\;\frac{\log \left(-1 \cdot re\right)}{\log 10}\\

\mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}\\

\mathbf{elif}\;re \le 2.15180429374657097 \cdot 10^{-295}:\\
\;\;\;\;\sqrt[3]{\sqrt[3]{\frac{{\left(\log im\right)}^{9}}{{\left(\log 10\right)}^{9}}}}\\

\mathbf{elif}\;re \le 2.55703958874010118 \cdot 10^{92}:\\
\;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}\right)}^{3}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\log re}{\log 10}\right)}^{3}}\\

\end{array}
double code(double re, double im) {
	return (log(sqrt(((re * re) + (im * im)))) / log(10.0));
}
↓
double code(double re, double im) {
	double VAR;
	if ((re <= -5.3041072880298795e+54)) {
		VAR = (log((-1.0 * re)) / log(10.0));
	} else {
		double VAR_1;
		if ((re <= -3.8099669373079583e-103)) {
			VAR_1 = cbrt(pow((log(sqrt(((re * re) + (im * im)))) / log(10.0)), 3.0));
		} else {
			double VAR_2;
			if ((re <= 2.151804293746571e-295)) {
				VAR_2 = cbrt(cbrt((pow(log(im), 9.0) / pow(log(10.0), 9.0))));
			} else {
				double VAR_3;
				if ((re <= 2.5570395887401012e+92)) {
					VAR_3 = cbrt(cbrt(pow(pow((log(sqrt(((re * re) + (im * im)))) / log(10.0)), 3.0), 3.0)));
				} else {
					VAR_3 = cbrt(pow((log(re) / log(10.0)), 3.0));
				}
				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 5 regimes

  2. if re < -5.3041072880298795e+54

    1. Initial program 44.6

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

    2. Taylor expanded around -inf 11.1

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

    if -5.3041072880298795e+54 < re < -3.8099669373079583e-103

    1. Initial program 17.0

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

    3. Applied add-cbrt-cube17.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    4. Applied add-cbrt-cube17.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}\]

    5. Applied cbrt-undiv17.0

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    6. Simplified17.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}\]

    if -3.8099669373079583e-103 < re < 2.151804293746571e-295

    1. Initial program 28.8

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

    3. Applied add-cbrt-cube29.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    4. Applied add-cbrt-cube29.2

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}\]

    5. Applied cbrt-undiv28.8

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    6. Simplified28.8

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube28.8

      \[\leadsto \sqrt[3]{\color{blue}{\sqrt[3]{\left({\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3} \cdot {\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}\right) \cdot {\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}}\]

    9. Simplified28.8

      \[\leadsto \sqrt[3]{\sqrt[3]{\color{blue}{{\left({\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}\right)}^{3}}}}\]

    10. Taylor expanded around 0 36.0

      \[\leadsto \sqrt[3]{\sqrt[3]{\color{blue}{\frac{{\left(\log im\right)}^{9}}{{\left(\log 10\right)}^{9}}}}}\]

    if 2.151804293746571e-295 < re < 2.5570395887401012e+92

    1. Initial program 21.0

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

    3. Applied add-cbrt-cube21.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    4. Applied add-cbrt-cube21.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}\]

    5. Applied cbrt-undiv21.0

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    6. Simplified21.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube21.0

      \[\leadsto \sqrt[3]{\color{blue}{\sqrt[3]{\left({\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3} \cdot {\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}\right) \cdot {\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}}\]

    9. Simplified21.0

      \[\leadsto \sqrt[3]{\sqrt[3]{\color{blue}{{\left({\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}\right)}^{3}}}}\]

    if 2.5570395887401012e+92 < re

    1. Initial program 49.9

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

    3. Applied add-cbrt-cube50.1

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    4. Applied add-cbrt-cube50.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}\]

    5. Applied cbrt-undiv50.0

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    6. Simplified49.9

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}\]

    7. Taylor expanded around inf 9.4

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

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.30410728802987954 \cdot 10^{54}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right)}{\log 10}\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}\\ \mathbf{elif}\;re \le 2.15180429374657097 \cdot 10^{-295}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{\frac{{\left(\log im\right)}^{9}}{{\left(\log 10\right)}^{9}}}}\\ \mathbf{elif}\;re \le 2.55703958874010118 \cdot 10^{92}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log re}{\log 10}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))