Average Error: 31.7 → 17.7
Time: 8.2s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.26925823457863703 \cdot 10^{154}:\\ \;\;\;\;\sqrt[3]{-1 \cdot \frac{{\left(\log \left(\frac{-1}{re}\right)\right)}^{3}}{{\left(\log 10\right)}^{3}}}\\ \mathbf{elif}\;re \le 4.3252230304891542 \cdot 10^{77}:\\ \;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot e^{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.26925823457863703 \cdot 10^{154}:\\
\;\;\;\;\sqrt[3]{-1 \cdot \frac{{\left(\log \left(\frac{-1}{re}\right)\right)}^{3}}{{\left(\log 10\right)}^{3}}}\\

\mathbf{elif}\;re \le 4.3252230304891542 \cdot 10^{77}:\\
\;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot e^{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}\right)}{\log 10}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log re}{\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 <= -1.269258234578637e+154)) {
		VAR = ((double) cbrt(((double) (-1.0 * ((double) (((double) pow(((double) log(((double) (-1.0 / re)))), 3.0)) / ((double) pow(((double) log(10.0)), 3.0))))))));
	} else {
		double VAR_1;
		if ((re <= 4.325223030489154e+77)) {
			VAR_1 = ((double) (((double) log(((double) (((double) (((double) cbrt(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) * ((double) cbrt(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))) * ((double) exp(((double) log(((double) cbrt(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))))))))) / ((double) log(10.0))));
		} else {
			VAR_1 = ((double) (((double) log(re)) / ((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 < -1.269258234578637e+154

    1. Initial program 64.0

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

    3. Applied add-cbrt-cube64.0

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

    4. Applied add-cbrt-cube64.0

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

    5. Applied cbrt-undiv64.0

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

    6. Simplified64.0

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

    7. Taylor expanded around -inf 7.0

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

    if -1.269258234578637e+154 < re < 4.325223030489154e+77

    1. Initial program 21.4

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

    3. Applied add-cube-cbrt21.4

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

    5. Applied add-exp-log21.4

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

    if 4.325223030489154e+77 < re

    1. Initial program 49.4

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

    2. Taylor expanded around inf 10.9

      \[\leadsto \frac{\log \color{blue}{re}}{\log 10}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.26925823457863703 \cdot 10^{154}:\\ \;\;\;\;\sqrt[3]{-1 \cdot \frac{{\left(\log \left(\frac{-1}{re}\right)\right)}^{3}}{{\left(\log 10\right)}^{3}}}\\ \mathbf{elif}\;re \le 4.3252230304891542 \cdot 10^{77}:\\ \;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot e^{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))