Average Error: 31.8 → 7.8
Time: 9.7s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq 3.1954262360162713 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(-re\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \leq 6.578330190231049 \cdot 10^{+55}:\\ \;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;im \leq 3.1954262360162713 \cdot 10^{-124}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(-re\right)}{\sqrt{\log 10}}\\

\mathbf{elif}\;im \leq 6.578330190231049 \cdot 10^{+55}:\\
\;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\

\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 (<= im 3.1954262360162713e-124)
   (* (/ 1.0 (sqrt (log 10.0))) (/ (log (- re)) (sqrt (log 10.0))))
   (if (<= im 6.578330190231049e+55)
     (/ 0.5 (/ (log 10.0) (log (+ (* re re) (* im im)))))
     (/ (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 (im <= 3.1954262360162713e-124) {
		tmp = (1.0 / sqrt(log(10.0))) * (log(-re) / sqrt(log(10.0)));
	} else if (im <= 6.578330190231049e+55) {
		tmp = 0.5 / (log(10.0) / log((re * re) + (im * im)));
	} 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 im < 3.19542623601627129e-124

    1. Initial program 31.0

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

    2. Taylor expanded around -inf 7.1

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

    3. Simplified7.1

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

    5. Applied add-sqr-sqrt_binary647.1

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

    6. Applied pow1_binary647.1

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

    7. Applied log-pow_binary647.1

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

    8. Applied times-frac_binary647.1

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

    if 3.19542623601627129e-124 < im < 6.57833019023104904e55

    1. Initial program 11.1

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

    3. Applied pow1/2_binary6411.1

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

    4. Applied log-pow_binary6411.1

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

    5. Applied associate-/l*_binary6411.1

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

    if 6.57833019023104904e55 < im

    1. Initial program 44.7

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

    2. Taylor expanded around 0 6.7

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.1954262360162713 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(-re\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \leq 6.578330190231049 \cdot 10^{+55}:\\ \;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))