Average Error: 32.0 → 17.5
Time: 4.3s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -4.685897936873047 \cdot 10^{+105}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(\frac{-1}{re}\right) \cdot -2}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;re \leq 1.6162243242536524 \cdot 10^{+94}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{2 \cdot \log re}{\sqrt{\log 10}}\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -4.685897936873047 \cdot 10^{+105}:\\
\;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(\frac{-1}{re}\right) \cdot -2}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\\

\mathbf{elif}\;re \leq 1.6162243242536524 \cdot 10^{+94}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\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 -4.685897936873047e+105)
   (*
    (/ (* (cbrt 0.5) (cbrt 0.5)) (sqrt (sqrt (log 10.0))))
    (*
     (/ (* (log (/ -1.0 re)) -2.0) (sqrt (log 10.0)))
     (/ (cbrt 0.5) (sqrt (sqrt (log 10.0))))))
   (if (<= re 1.6162243242536524e+94)
     (*
      (/ 0.5 (sqrt (log 10.0)))
      (log (pow (+ (* re re) (* im im)) (/ 1.0 (sqrt (log 10.0))))))
     (*
      (/ (* (cbrt 0.5) (cbrt 0.5)) (sqrt (sqrt (log 10.0))))
      (*
       (/ (cbrt 0.5) (sqrt (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 <= -4.685897936873047e+105) {
		tmp = ((cbrt(0.5) * cbrt(0.5)) / sqrt(sqrt(log(10.0)))) * (((log(-1.0 / re) * -2.0) / sqrt(log(10.0))) * (cbrt(0.5) / sqrt(sqrt(log(10.0)))));
	} else if (re <= 1.6162243242536524e+94) {
		tmp = (0.5 / sqrt(log(10.0))) * log(pow(((re * re) + (im * im)), (1.0 / sqrt(log(10.0)))));
	} else {
		tmp = ((cbrt(0.5) * cbrt(0.5)) / sqrt(sqrt(log(10.0)))) * ((cbrt(0.5) / sqrt(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 < -4.68589793687304724e105

    1. Initial program 54.0

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

    3. Applied add-sqr-sqrt_binary6454.0

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

    4. Applied pow1/2_binary6454.0

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

    5. Applied log-pow_binary6454.0

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

    6. Applied times-frac_binary6454.0

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

    8. Applied add-log-exp_binary6454.0

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

    9. Simplified54.0

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

    11. Applied add-sqr-sqrt_binary6454.1

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

    12. Applied add-cube-cbrt_binary6454.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{0.5}}}{\sqrt{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\log 10}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

    13. Applied times-frac_binary6454.0

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

    14. Applied associate-*l*_binary6454.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)}\]

    15. Simplified54.0

      \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \color{blue}{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)}\]

    16. Taylor expanded around -inf 9.6

      \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\color{blue}{-2 \cdot \log \left(\frac{-1}{re}\right)}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\]

    17. Simplified9.6

      \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\color{blue}{\log \left(\frac{-1}{re}\right) \cdot -2}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\]

    if -4.68589793687304724e105 < re < 1.6162243242536524e94

    1. Initial program 21.8

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

    3. Applied add-sqr-sqrt_binary6421.8

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

    4. Applied pow1/2_binary6421.8

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

    5. Applied log-pow_binary6421.8

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

    6. Applied times-frac_binary6421.7

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

    8. Applied add-log-exp_binary6421.7

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

    9. Simplified21.6

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

    if 1.6162243242536524e94 < re

    1. Initial program 50.2

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

    3. Applied add-sqr-sqrt_binary6450.2

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

    4. Applied pow1/2_binary6450.2

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

    5. Applied log-pow_binary6450.2

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

    6. Applied times-frac_binary6450.2

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

    8. Applied add-log-exp_binary6450.2

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

    9. Simplified50.1

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

    11. Applied add-sqr-sqrt_binary6450.3

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

    12. Applied add-cube-cbrt_binary6450.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{0.5}}}{\sqrt{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\log 10}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

    13. Applied times-frac_binary6450.1

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

    14. Applied associate-*l*_binary6450.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)}\]

    15. Simplified50.1

      \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \color{blue}{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)}\]

    16. Taylor expanded around inf 9.3

      \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\color{blue}{-2 \cdot \log \left(\frac{1}{re}\right)}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\]

    17. Simplified9.3

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.685897936873047 \cdot 10^{+105}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(\frac{-1}{re}\right) \cdot -2}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;re \leq 1.6162243242536524 \cdot 10^{+94}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{2 \cdot \log re}{\sqrt{\log 10}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))