Average Error: 33.2 → 4.0
Time: 7.9s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot {\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{4}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}\]
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 \sqrt[3]{y.re} - \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot {\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{4}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r16587 = x_re;
        double r16588 = r16587 * r16587;
        double r16589 = x_im;
        double r16590 = r16589 * r16589;
        double r16591 = r16588 + r16590;
        double r16592 = sqrt(r16591);
        double r16593 = log(r16592);
        double r16594 = y_re;
        double r16595 = r16593 * r16594;
        double r16596 = atan2(r16589, r16587);
        double r16597 = y_im;
        double r16598 = r16596 * r16597;
        double r16599 = r16595 - r16598;
        double r16600 = exp(r16599);
        double r16601 = r16593 * r16597;
        double r16602 = r16596 * r16594;
        double r16603 = r16601 + r16602;
        double r16604 = cos(r16603);
        double r16605 = r16600 * r16604;
        return r16605;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r16606 = y_re;
        double r16607 = cbrt(r16606);
        double r16608 = r16607 * r16607;
        double r16609 = 1.0;
        double r16610 = x_re;
        double r16611 = x_im;
        double r16612 = hypot(r16610, r16611);
        double r16613 = log(r16612);
        double r16614 = r16609 * r16613;
        double r16615 = r16608 * r16614;
        double r16616 = r16615 * r16607;
        double r16617 = atan2(r16611, r16610);
        double r16618 = cbrt(r16617);
        double r16619 = r16618 * r16618;
        double r16620 = cbrt(r16619);
        double r16621 = cbrt(r16618);
        double r16622 = 4.0;
        double r16623 = pow(r16621, r16622);
        double r16624 = r16620 * r16623;
        double r16625 = y_im;
        double r16626 = r16618 * r16625;
        double r16627 = r16624 * r16626;
        double r16628 = r16616 - r16627;
        double r16629 = exp(r16628);
        return r16629;
}

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 33.2

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

  2. Taylor expanded around 0 19.9

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

  4. Applied add-cube-cbrt19.9

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

  5. Applied associate-*r*19.9

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

  6. Simplified4.0

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

  8. Applied add-cube-cbrt4.0

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

  9. Applied associate-*l*4.0

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

  11. Applied add-cube-cbrt4.0

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

  12. Applied cbrt-prod4.0

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

  13. Applied associate-*l*4.0

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

  14. Simplified4.0

    \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{4}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)} \cdot 1\]

  15. Final simplification4.0

    \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot {\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{4}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}\]

Reproduce

herbie shell --seed 2020059 +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)))))