Average Error: 32.0 → 17.3
Time: 4.4s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -2.1174862849080903 \cdot 10^{+60}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \leq 1.0865722709223886 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{2 \cdot \log re}{\sqrt{\log 10}}\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -2.1174862849080903 \cdot 10^{+60}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\

\mathbf{elif}\;re \leq 1.0865722709223886 \cdot 10^{+74}:\\
\;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\

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

\end{array}
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -2.1174862849080903e+60)
   (*
    (/ 0.5 (sqrt (log 10.0)))
    (* -2.0 (* (log (/ -1.0 re)) (sqrt (/ 1.0 (log 10.0))))))
   (if (<= re 1.0865722709223886e+74)
     (* (sqrt 0.5) (/ (sqrt 0.5) (/ (log 10.0) (log (+ (* re re) (* im im))))))
     (*
      (sqrt (/ 0.5 (sqrt (log 10.0))))
      (*
       (sqrt (/ 0.5 (sqrt (log 10.0))))
       (/ (* 2.0 (log re)) (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 (re <= -2.1174862849080903e+60) {
		tmp = (0.5 / sqrt(log(10.0))) * (-2.0 * (log(-1.0 / re) * sqrt(1.0 / log(10.0))));
	} else if (re <= 1.0865722709223886e+74) {
		tmp = sqrt(0.5) * (sqrt(0.5) / (log(10.0) / log((re * re) + (im * im))));
	} else {
		tmp = sqrt(0.5 / sqrt(log(10.0))) * (sqrt(0.5 / sqrt(log(10.0))) * ((2.0 * log(re)) / 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 3 regimes

  2. if re < -2.1174862849080903e60

    1. Initial program 46.1

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

    3. Applied add-sqr-sqrt_binary6446.1

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

    4. Applied pow1/2_binary6446.1

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

    5. Applied log-pow_binary6446.1

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

    6. Applied times-frac_binary6446.1

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

    7. Taylor expanded around -inf 10.3

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

    if -2.1174862849080903e60 < re < 1.086572270922389e74

    1. Initial program 22.0

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

    3. Applied add-sqr-sqrt_binary6422.0

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

    4. Applied pow1/2_binary6422.0

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

    5. Applied log-pow_binary6422.0

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

    6. Applied times-frac_binary6422.0

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

    8. Applied *-un-lft-identity_binary6422.0

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

    9. Applied add-sqr-sqrt_binary6422.0

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

    10. Applied times-frac_binary6422.0

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

    11. Applied associate-*l*_binary6421.9

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

    12. Simplified22.0

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

    if 1.086572270922389e74 < re

    1. Initial program 47.3

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

    3. Applied add-sqr-sqrt_binary6447.3

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

    4. Applied pow1/2_binary6447.3

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

    5. Applied log-pow_binary6447.3

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

    6. Applied times-frac_binary6447.3

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

    8. Applied add-sqr-sqrt_binary6447.3

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

    9. Applied associate-*l*_binary6447.3

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

    10. Simplified47.3

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

    11. Taylor expanded around inf 10.7

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

    12. Simplified10.7

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1174862849080903 \cdot 10^{+60}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \leq 1.0865722709223886 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{2 \cdot \log re}{\sqrt{\log 10}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))