Average Error: 32.2 → 6.9
Time: 13.1s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq 2.2514151755340617 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\ \mathbf{elif}\;im \leq 2.3275519908472327 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{1}{im}\right)\right)}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;im \leq 2.2514151755340617 \cdot 10^{-166}:\\
\;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\

\mathbf{elif}\;im \leq 2.3275519908472327 \cdot 10^{+131}:\\
\;\;\;\;\frac{\sqrt{0.5}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{0.5}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{1}{im}\right)\right)}}\\

\end{array}
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (if (<= im 2.2514151755340617e-166)
   (/ (sqrt 0.5) (/ (log 10.0) (* (sqrt 0.5) (* -2.0 (log (/ -1.0 re))))))
   (if (<= im 2.3275519908472327e+131)
     (/ (sqrt 0.5) (/ (/ (log 10.0) (log (+ (* re re) (* im im)))) (sqrt 0.5)))
     (/ (sqrt 0.5) (/ (log 10.0) (* (sqrt 0.5) (* -2.0 (log (/ 1.0 im)))))))))
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 <= 2.2514151755340617e-166) {
		tmp = sqrt(0.5) / (log(10.0) / (sqrt(0.5) * (-2.0 * log(-1.0 / re))));
	} else if (im <= 2.3275519908472327e+131) {
		tmp = sqrt(0.5) / ((log(10.0) / log((re * re) + (im * im))) / sqrt(0.5));
	} else {
		tmp = sqrt(0.5) / (log(10.0) / (sqrt(0.5) * (-2.0 * log(1.0 / im))));
	}
	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 < 2.2514151755340617e-166

    1. Initial program 33.8

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

    3. Applied pow1/2_binary64_15833.8

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

    4. Applied log-pow_binary64_16733.8

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

    5. Applied associate-/l*_binary64_2333.8

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

    7. Applied add-sqr-sqrt_binary64_10033.9

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

    8. Applied associate-/l*_binary64_2333.7

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

    9. Simplified33.7

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

    10. Taylor expanded around -inf 4.6

      \[\leadsto \frac{\sqrt{0.5}}{\frac{\log 10}{\color{blue}{\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)} \cdot \sqrt{0.5}}}\]

    if 2.2514151755340617e-166 < im < 2.32755199084723273e131

    1. Initial program 11.3

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

    3. Applied pow1/2_binary64_15811.3

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

    4. Applied log-pow_binary64_16711.3

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

    5. Applied associate-/l*_binary64_2311.3

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

    7. Applied add-sqr-sqrt_binary64_10011.5

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

    8. Applied associate-/l*_binary64_2311.2

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

    if 2.32755199084723273e131 < im

    1. Initial program 58.1

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

    3. Applied pow1/2_binary64_15858.1

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

    4. Applied log-pow_binary64_16758.1

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

    5. Applied associate-/l*_binary64_2358.1

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

    7. Applied add-sqr-sqrt_binary64_10058.2

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

    8. Applied associate-/l*_binary64_2358.1

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

    9. Simplified58.1

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

    10. Taylor expanded around inf 4.5

      \[\leadsto \frac{\sqrt{0.5}}{\frac{\log 10}{\color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right)\right)} \cdot \sqrt{0.5}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2514151755340617 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\ \mathbf{elif}\;im \leq 2.3275519908472327 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{1}{im}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))