Average Error: 32.6 → 7.2
Time: 8.6s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -9.645547492180995 \cdot 10^{+94}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(-re\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \leq -2.7017925900492935 \cdot 10^{-149}:\\ \;\;\;\;\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log \left(\frac{1}{im}\right)}{\log 10}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -9.645547492180995 \cdot 10^{+94}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(-re\right)}{\sqrt{\log 10}}\\

\mathbf{elif}\;re \leq -2.7017925900492935 \cdot 10^{-149}:\\
\;\;\;\;\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log 10}\\

\mathbf{else}:\\
\;\;\;\;-\frac{\log \left(\frac{1}{im}\right)}{\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 -9.645547492180995e+94)
   (* (/ 1.0 (sqrt (log 10.0))) (/ (log (- re)) (sqrt (log 10.0))))
   (if (<= re -2.7017925900492935e-149)
     (/
      (log
       (*
        (fabs (cbrt (+ (* re re) (* im im))))
        (sqrt (cbrt (+ (* re re) (* im im))))))
      (log 10.0))
     (- (/ (log (/ 1.0 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 <= -9.645547492180995e+94) {
		tmp = (1.0 / sqrt(log(10.0))) * (log(-re) / sqrt(log(10.0)));
	} else if (re <= -2.7017925900492935e-149) {
		tmp = log(fabs(cbrt((re * re) + (im * im))) * sqrt(cbrt((re * re) + (im * im)))) / log(10.0);
	} else {
		tmp = -(log(1.0 / 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 < -9.6455474921809946e94

    1. Initial program 50.7

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

    2. Taylor expanded around -inf 5.2

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

    3. Simplified5.2

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

    5. Applied add-sqr-sqrt_binary645.2

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

    6. Applied pow1_binary645.2

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

    7. Applied log-pow_binary645.2

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

    8. Applied times-frac_binary645.2

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

    if -9.6455474921809946e94 < re < -2.70179259004929345e-149

    1. Initial program 11.3

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

    3. Applied add-cube-cbrt_binary6411.3

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

    4. Applied sqrt-prod_binary6411.3

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

    5. Simplified11.3

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

    if -2.70179259004929345e-149 < re

    1. Initial program 33.0

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

    2. Taylor expanded around inf 5.8

      \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.645547492180995 \cdot 10^{+94}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(-re\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \leq -2.7017925900492935 \cdot 10^{-149}:\\ \;\;\;\;\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log \left(\frac{1}{im}\right)}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))