Average Error: 31.9 → 17.7
Time: 15.3s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq -7.061285158928872 \cdot 10^{+113}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{im}\right)\right)\right)\\ \mathbf{elif}\;im \leq -5.535169434540731 \cdot 10^{-301}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(im \cdot im + re \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;im \leq 3.643016007571221 \cdot 10^{-180}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right)\right)\\ \mathbf{elif}\;im \leq 5.179540222444118 \cdot 10^{+90}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(im \cdot im + re \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{1}{im}\right)}{\sqrt{\log 10}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;im \leq -7.061285158928872 \cdot 10^{+113}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{im}\right)\right)\right)\\

\mathbf{elif}\;im \leq -5.535169434540731 \cdot 10^{-301}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(im \cdot im + re \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{elif}\;im \leq 3.643016007571221 \cdot 10^{-180}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right)\right)\\

\mathbf{elif}\;im \leq 5.179540222444118 \cdot 10^{+90}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(im \cdot im + re \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{1}{im}\right)}{\sqrt{\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 -7.061285158928872e+113)
   (*
    (/ 0.5 (sqrt (log 10.0)))
    (* -2.0 (* (sqrt (/ 1.0 (log 10.0))) (log (/ -1.0 im)))))
   (if (<= im -5.535169434540731e-301)
     (*
      (/ 0.5 (sqrt (log 10.0)))
      (log (pow (+ (* im im) (* re re)) (/ 1.0 (sqrt (log 10.0))))))
     (if (<= im 3.643016007571221e-180)
       (*
        (/ 0.5 (sqrt (log 10.0)))
        (* -2.0 (* (sqrt (/ 1.0 (log 10.0))) (log (/ -1.0 re)))))
       (if (<= im 5.179540222444118e+90)
         (*
          (/ 0.5 (sqrt (log 10.0)))
          (log (pow (+ (* im im) (* re re)) (/ 1.0 (sqrt (log 10.0))))))
         (*
          (/ 0.5 (sqrt (log 10.0)))
          (/ (* -2.0 (log (/ 1.0 im))) (sqrt (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 <= -7.061285158928872e+113) {
		tmp = (0.5 / sqrt(log(10.0))) * (-2.0 * (sqrt(1.0 / log(10.0)) * log(-1.0 / im)));
	} else if (im <= -5.535169434540731e-301) {
		tmp = (0.5 / sqrt(log(10.0))) * log(pow(((im * im) + (re * re)), (1.0 / sqrt(log(10.0)))));
	} else if (im <= 3.643016007571221e-180) {
		tmp = (0.5 / sqrt(log(10.0))) * (-2.0 * (sqrt(1.0 / log(10.0)) * log(-1.0 / re)));
	} else if (im <= 5.179540222444118e+90) {
		tmp = (0.5 / sqrt(log(10.0))) * log(pow(((im * im) + (re * re)), (1.0 / sqrt(log(10.0)))));
	} else {
		tmp = (0.5 / sqrt(log(10.0))) * ((-2.0 * log(1.0 / im)) / sqrt(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 im < -7.0612851589288723e113

    1. Initial program 53.3

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

    3. Applied add-sqr-sqrt_binary64_78253.3

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

    4. Applied pow1/2_binary64_84053.3

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

    5. Applied log-pow_binary64_84953.3

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

    6. Applied times-frac_binary64_76653.3

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

    7. Taylor expanded around -inf 8.8

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

    if -7.0612851589288723e113 < im < -5.53516943454073139e-301 or 3.643016007571221e-180 < im < 5.1795402224441176e90

    1. Initial program 19.8

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

    3. Applied add-sqr-sqrt_binary64_78219.8

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

    4. Applied pow1/2_binary64_84019.8

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

    5. Applied log-pow_binary64_84919.8

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

    6. Applied times-frac_binary64_76619.8

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

    8. Applied add-log-exp_binary64_79919.8

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

    9. Simplified19.6

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

    if -5.53516943454073139e-301 < im < 3.643016007571221e-180

    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-sqr-sqrt_binary64_78232.7

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

    4. Applied pow1/2_binary64_84032.7

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

    5. Applied log-pow_binary64_84932.7

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

    6. Applied times-frac_binary64_76632.7

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

    7. Taylor expanded around -inf 33.8

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

    if 5.1795402224441176e90 < im

    1. Initial program 50.4

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

    3. Applied add-sqr-sqrt_binary64_78250.4

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

    4. Applied pow1/2_binary64_84050.4

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

    5. Applied log-pow_binary64_84950.4

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

    6. Applied times-frac_binary64_76650.4

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

    7. Taylor expanded around inf 9.3

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \frac{\color{blue}{-2 \cdot \log \left(\frac{1}{im}\right)}}{\sqrt{\log 10}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.061285158928872 \cdot 10^{+113}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{im}\right)\right)\right)\\ \mathbf{elif}\;im \leq -5.535169434540731 \cdot 10^{-301}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(im \cdot im + re \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;im \leq 3.643016007571221 \cdot 10^{-180}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right)\right)\\ \mathbf{elif}\;im \leq 5.179540222444118 \cdot 10^{+90}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(im \cdot im + re \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{1}{im}\right)}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))