Average Error: 32.5 → 7.6
Time: 8.4s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -7.143276769119883 \cdot 10^{+148}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -7.144782444788284 \cdot 10^{-138}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -7.143276769119883 \cdot 10^{+148}:\\
\;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\

\mathbf{elif}\;re \leq -7.144782444788284 \cdot 10^{-138}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log im}{\log 10}\\

\end{array}
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -7.143276769119883e+148)
   (/ (log (- re)) (log 10.0))
   (if (<= re -7.144782444788284e-138)
     (cbrt (pow (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)) 3.0))
     (/ (log im) (log 10.0)))))
double code(double re, double im) {
	return log(sqrt((re * re) + (im * im))) / log(10.0);
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -7.143276769119883e+148) {
		tmp = log(-re) / log(10.0);
	} else if (re <= -7.144782444788284e-138) {
		tmp = cbrt(pow((log(sqrt((re * re) + (im * im))) / log(10.0)), 3.0));
	} else {
		tmp = log(im) / log(10.0);
	}
	return tmp;
}

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 < -7.1432767691198832e148

    1. Initial program 62.5

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

    2. Taylor expanded around -inf 3.9

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

    3. Simplified3.9

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)}}{\log 10}\]
    4. Using strategy rm

    5. Applied pow1_binary643.9

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

    6. Applied log-pow_binary643.9

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

    7. Applied associate-/r*_binary643.9

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

    if -7.1432767691198832e148 < re < -7.1447824447882838e-138

    1. Initial program 11.6

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

    3. Applied add-cbrt-cube_binary6411.7

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

    4. Simplified11.7

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

    if -7.1447824447882838e-138 < re

    1. Initial program 32.9

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

    2. Taylor expanded around 0 6.3

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.143276769119883 \cdot 10^{+148}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -7.144782444788284 \cdot 10^{-138}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))