Average Error: 31.6 → 6.9
Time: 7.8s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.4840638130856809 \cdot 10^{+75}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \leq -2.5396447653359938 \cdot 10^{-165}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.4840638130856809 \cdot 10^{+75}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{elif}\;re \leq -2.5396447653359938 \cdot 10^{-165}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\\

\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 -1.4840638130856809e+75)
   (* (/ 1.0 (sqrt (log 10.0))) (log (pow (- re) (/ 1.0 (sqrt (log 10.0))))))
   (if (<= re -2.5396447653359938e-165)
     (*
      (/ 0.5 (sqrt (log 10.0)))
      (/ (log (+ (* re re) (* im im))) (sqrt (log 10.0))))
     (/ (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 <= -1.4840638130856809e+75) {
		tmp = (1.0 / sqrt(log(10.0))) * log(pow(-re, (1.0 / sqrt(log(10.0)))));
	} else if (re <= -2.5396447653359938e-165) {
		tmp = (0.5 / sqrt(log(10.0))) * (log((re * re) + (im * im)) / sqrt(log(10.0)));
	} 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 < -1.48406381308568086e75

    1. Initial program 46.5

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

    2. Taylor expanded around -inf 5.7

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

    3. Simplified5.7

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

    5. Applied add-sqr-sqrt_binary64_7825.7

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

    6. Applied pow1_binary64_8215.7

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

    7. Applied log-pow_binary64_8495.7

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

    8. Applied times-frac_binary64_7665.6

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

    10. Applied add-log-exp_binary64_7995.6

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

    11. Simplified5.4

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

    if -1.48406381308568086e75 < re < -2.53964476533599376e-165

    1. Initial program 11.7

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

    3. Applied add-sqr-sqrt_binary64_78211.7

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

    4. Applied pow1/2_binary64_84011.7

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow_binary64_84911.7

      \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac_binary64_76611.7

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]

    if -2.53964476533599376e-165 < re

    1. Initial program 32.7

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

    2. Taylor expanded around 0 4.8

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.4840638130856809 \cdot 10^{+75}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \leq -2.5396447653359938 \cdot 10^{-165}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))