Average Error: 32.8 → 4.2
Time: 8.5s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y.re}}}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]
e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)
↓
e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y.re}}}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re))));
}
↓
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp(((((cbrt(y_46_re) * cbrt(y_46_re)) * (1.0 * log(hypot(x_46_re, x_46_im)))) * (cbrt((cbrt(y_46_re) * cbrt(y_46_re))) * (cbrt(cbrt((cbrt(y_46_re) * cbrt(y_46_re)))) * cbrt(cbrt(cbrt(y_46_re)))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.8

    \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

  2. Taylor expanded around 0 19.5

    \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt19.5

    \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \sqrt[3]{y.re}\right)} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

  5. Applied associate-*r*19.5

    \[\leadsto e^{\color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot \left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right)\right) \cdot \sqrt[3]{y.re}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

  6. Simplified4.2

    \[\leadsto e^{\color{blue}{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right)} \cdot \sqrt[3]{y.re} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]
  7. Using strategy rm

  8. Applied add-cube-cbrt4.2

    \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \sqrt[3]{y.re}}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

  9. Applied cbrt-prod4.2

    \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}} \cdot \sqrt[3]{\sqrt[3]{y.re}}\right)} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]
  10. Using strategy rm

  11. Applied add-cube-cbrt4.2

    \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}} \cdot \sqrt[3]{\sqrt[3]{\color{blue}{\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \sqrt[3]{y.re}}}}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

  12. Applied cbrt-prod4.2

    \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}} \cdot \sqrt[3]{\color{blue}{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}} \cdot \sqrt[3]{\sqrt[3]{y.re}}}}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

  13. Applied cbrt-prod4.2

    \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y.re}}}\right)}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

  14. Final simplification4.2

    \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y.re}}}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))