Average Error: 32.9 → 3.6
Time: 10.1s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\left(\sqrt[3]{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \sqrt[3]{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \sqrt[3]{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right)}}\right) \cdot y.im\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
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 \sin \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^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\left(\sqrt[3]{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \sqrt[3]{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \sqrt[3]{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right)}}\right) \cdot y.im\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r15919 = x_re;
        double r15920 = r15919 * r15919;
        double r15921 = x_im;
        double r15922 = r15921 * r15921;
        double r15923 = r15920 + r15922;
        double r15924 = sqrt(r15923);
        double r15925 = log(r15924);
        double r15926 = y_re;
        double r15927 = r15925 * r15926;
        double r15928 = atan2(r15921, r15919);
        double r15929 = y_im;
        double r15930 = r15928 * r15929;
        double r15931 = r15927 - r15930;
        double r15932 = exp(r15931);
        double r15933 = r15925 * r15929;
        double r15934 = r15928 * r15926;
        double r15935 = r15933 + r15934;
        double r15936 = sin(r15935);
        double r15937 = r15932 * r15936;
        return r15937;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r15938 = x_re;
        double r15939 = x_im;
        double r15940 = hypot(r15938, r15939);
        double r15941 = log(r15940);
        double r15942 = y_re;
        double r15943 = r15941 * r15942;
        double r15944 = atan2(r15939, r15938);
        double r15945 = y_im;
        double r15946 = r15944 * r15945;
        double r15947 = r15943 - r15946;
        double r15948 = exp(r15947);
        double r15949 = 1.0;
        double r15950 = r15949 * r15940;
        double r15951 = log(r15950);
        double r15952 = cbrt(r15951);
        double r15953 = r15952 * r15952;
        double r15954 = cbrt(r15953);
        double r15955 = cbrt(r15952);
        double r15956 = r15954 * r15955;
        double r15957 = r15956 * r15945;
        double r15958 = r15953 * r15957;
        double r15959 = r15944 * r15942;
        double r15960 = r15958 + r15959;
        double r15961 = sin(r15960);
        double r15962 = r15948 * r15961;
        return r15962;
}

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 32.9

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

  3. Applied *-un-lft-identity32.9

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

  4. Applied sqrt-prod32.9

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

  5. Simplified32.9

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

  6. Simplified19.3

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

  8. Applied hypot-def3.3

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

  10. Applied add-cube-cbrt3.5

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

  11. Applied associate-*l*3.6

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

  13. Applied add-cube-cbrt3.6

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

  14. Applied cbrt-prod3.6

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

  15. Final simplification3.6

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))