Average Error: 31.9 → 0.3
Time: 5.2s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\left(\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right) \cdot \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right)}\right)\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\left(\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right) \cdot \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right)}\right)
double code(double re, double im) {
	return (log(sqrt(((re * re) + (im * im)))) / log(10.0));
}
↓
double code(double re, double im) {
	return (((1.0 / sqrt(log(10.0))) * cbrt((1.0 / sqrt(log(10.0))))) * log(pow(hypot(re, im), (cbrt((1.0 / sqrt(log(10.0)))) * cbrt((1.0 / sqrt(log(10.0))))))));
}

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. Initial program 31.9

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

  3. Applied *-un-lft-identity31.9

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

  4. Applied sqrt-prod31.9

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

  5. Simplified31.9

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

  6. Simplified0.6

    \[\leadsto \frac{\log \left(1 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)}{\log 10}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.6

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

  9. Applied pow10.6

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

  10. Applied pow10.6

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

  11. Applied pow-prod-down0.6

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

  12. Applied log-pow0.6

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

  13. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}}\]
  14. Using strategy rm

  15. Applied add-log-exp0.6

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\log \left(e^{\frac{\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}}\right)}\]

  16. Simplified0.3

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

  18. Applied add-cube-cbrt0.3

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

  19. Applied pow-unpow0.6

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

  20. Applied log-pow0.6

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

  21. Applied associate-*r*0.3

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

  22. Final simplification0.3

    \[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right) \cdot \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right)}\right)\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))