Average Error: 31.7 → 0.3
Time: 4.5s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\left(\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right) \cdot \log \left({\left(\mathsf{hypot}\left(im, re\right)\right)}^{\left(\sqrt[3]{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right)}\right)\]
\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(im, re\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 ((double) (((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) / ((double) log(10.0))));
}
↓
double code(double re, double im) {
	return ((double) (((double) (((double) (1.0 / ((double) sqrt(((double) log(10.0)))))) * ((double) cbrt(((double) (1.0 / ((double) sqrt(((double) log(10.0)))))))))) * ((double) log(((double) pow(((double) hypot(im, re)), ((double) (((double) cbrt(((double) (1.0 / ((double) sqrt(((double) log(10.0)))))))) * ((double) cbrt(((double) (1.0 / ((double) sqrt(((double) 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 31.7

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

  3. Applied +-commutative31.7

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

  4. Applied hypot-def0.6

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

  6. Applied add-sqr-sqrt0.6

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

  7. Applied pow10.6

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

  8. Applied log-pow0.6

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

  9. Applied times-frac0.5

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

  11. Applied add-log-exp0.5

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

  12. Simplified0.3

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

  14. Applied add-cube-cbrt0.3

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

  15. Applied pow-unpow0.6

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

  16. Applied log-pow0.6

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

  17. Applied associate-*r*0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))