Average Error: 31.7 → 18.1
Time: 5.7s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -6.504888998661915 \cdot 10^{+102}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -8.510846260708714 \cdot 10^{-176}:\\ \;\;\;\;\frac{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\log 10}\\ \mathbf{elif}\;re \leq -3.735603583061054 \cdot 10^{-219}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(-re\right)}{\log 10}\right)}^{3}}\\ \mathbf{elif}\;re \leq 5.153255919931365 \cdot 10^{+39}:\\ \;\;\;\;\frac{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -6.504888998661915 \cdot 10^{+102}:\\
\;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\

\mathbf{elif}\;re \leq -8.510846260708714 \cdot 10^{-176}:\\
\;\;\;\;\frac{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\log 10}\\

\mathbf{elif}\;re \leq -3.735603583061054 \cdot 10^{-219}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\log \left(-re\right)}{\log 10}\right)}^{3}}\\

\mathbf{elif}\;re \leq 5.153255919931365 \cdot 10^{+39}:\\
\;\;\;\;\frac{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\log 10}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log re}{\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 -6.504888998661915e+102)
   (/ (log (- re)) (log 10.0))
   (if (<= re -8.510846260708714e-176)
     (/
      (log
       (*
        (cbrt (sqrt (+ (* re re) (* im im))))
        (*
         (cbrt (sqrt (+ (* re re) (* im im))))
         (cbrt (sqrt (+ (* re re) (* im im)))))))
      (log 10.0))
     (if (<= re -3.735603583061054e-219)
       (cbrt (pow (/ (log (- re)) (log 10.0)) 3.0))
       (if (<= re 5.153255919931365e+39)
         (/
          (log
           (*
            (cbrt (sqrt (+ (* re re) (* im im))))
            (*
             (cbrt (sqrt (+ (* re re) (* im im))))
             (cbrt (sqrt (+ (* re re) (* im im)))))))
          (log 10.0))
         (/ (log re) (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 <= -6.504888998661915e+102) {
		tmp = log(-re) / log(10.0);
	} else if (re <= -8.510846260708714e-176) {
		tmp = log(cbrt(sqrt((re * re) + (im * im))) * (cbrt(sqrt((re * re) + (im * im))) * cbrt(sqrt((re * re) + (im * im))))) / log(10.0);
	} else if (re <= -3.735603583061054e-219) {
		tmp = cbrt(pow((log(-re) / log(10.0)), 3.0));
	} else if (re <= 5.153255919931365e+39) {
		tmp = log(cbrt(sqrt((re * re) + (im * im))) * (cbrt(sqrt((re * re) + (im * im))) * cbrt(sqrt((re * re) + (im * im))))) / log(10.0);
	} else {
		tmp = log(re) / 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 4 regimes

  2. if re < -6.504888998661915e102

    1. Initial program 52.1

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

    2. Taylor expanded around -inf 7.6

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

    3. Simplified7.6

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

    if -6.504888998661915e102 < re < -8.5108462607087144e-176 or -3.7356035830610541e-219 < re < 5.15325591993136483e39

    1. Initial program 21.9

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

    3. Applied add-cube-cbrt_binary64_11421.9

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

    if -8.5108462607087144e-176 < re < -3.7356035830610541e-219

    1. Initial program 32.7

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

    3. Applied add-cbrt-cube_binary64_11532.8

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

    4. Simplified32.7

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

    5. Taylor expanded around -inf 44.5

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

    6. Simplified44.5

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

    if 5.15325591993136483e39 < re

    1. Initial program 42.7

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

    2. Taylor expanded around inf 11.4

      \[\leadsto \frac{\log \color{blue}{re}}{\log 10}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.504888998661915 \cdot 10^{+102}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -8.510846260708714 \cdot 10^{-176}:\\ \;\;\;\;\frac{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\log 10}\\ \mathbf{elif}\;re \leq -3.735603583061054 \cdot 10^{-219}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(-re\right)}{\log 10}\right)}^{3}}\\ \mathbf{elif}\;re \leq 5.153255919931365 \cdot 10^{+39}:\\ \;\;\;\;\frac{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2020289 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))