Average Error: 33.2 → 24.2
Time: 42.2s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -5.907306680385344179981604616356048283101 \cdot 10^{-241}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{elif}\;x.re \le 1.808989908199788750558131282319198960687 \cdot 10^{-128}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}} \cdot \left(\sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}} \cdot \sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right)\\ \end{array}\]
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)
↓
\begin{array}{l}
\mathbf{if}\;x.re \le -5.907306680385344179981604616356048283101 \cdot 10^{-241}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\

\mathbf{elif}\;x.re \le 1.808989908199788750558131282319198960687 \cdot 10^{-128}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}} \cdot \left(\sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}} \cdot \sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\

\mathbf{else}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right)\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1490715 = x_re;
        double r1490716 = r1490715 * r1490715;
        double r1490717 = x_im;
        double r1490718 = r1490717 * r1490717;
        double r1490719 = r1490716 + r1490718;
        double r1490720 = sqrt(r1490719);
        double r1490721 = log(r1490720);
        double r1490722 = y_re;
        double r1490723 = r1490721 * r1490722;
        double r1490724 = atan2(r1490717, r1490715);
        double r1490725 = y_im;
        double r1490726 = r1490724 * r1490725;
        double r1490727 = r1490723 - r1490726;
        double r1490728 = exp(r1490727);
        double r1490729 = r1490721 * r1490725;
        double r1490730 = r1490724 * r1490722;
        double r1490731 = r1490729 + r1490730;
        double r1490732 = sin(r1490731);
        double r1490733 = r1490728 * r1490732;
        return r1490733;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1490734 = x_re;
        double r1490735 = -5.907306680385344e-241;
        bool r1490736 = r1490734 <= r1490735;
        double r1490737 = y_re;
        double r1490738 = r1490734 * r1490734;
        double r1490739 = x_im;
        double r1490740 = r1490739 * r1490739;
        double r1490741 = r1490738 + r1490740;
        double r1490742 = sqrt(r1490741);
        double r1490743 = log(r1490742);
        double r1490744 = r1490737 * r1490743;
        double r1490745 = atan2(r1490739, r1490734);
        double r1490746 = y_im;
        double r1490747 = r1490745 * r1490746;
        double r1490748 = r1490744 - r1490747;
        double r1490749 = exp(r1490748);
        double r1490750 = r1490745 * r1490737;
        double r1490751 = -r1490734;
        double r1490752 = log(r1490751);
        double r1490753 = r1490746 * r1490752;
        double r1490754 = r1490750 + r1490753;
        double r1490755 = sin(r1490754);
        double r1490756 = r1490749 * r1490755;
        double r1490757 = 1.8089899081997888e-128;
        bool r1490758 = r1490734 <= r1490757;
        double r1490759 = cbrt(r1490747);
        double r1490760 = r1490759 * r1490759;
        double r1490761 = r1490759 * r1490760;
        double r1490762 = r1490744 - r1490761;
        double r1490763 = exp(r1490762);
        double r1490764 = cbrt(r1490742);
        double r1490765 = r1490764 * r1490764;
        double r1490766 = r1490764 * r1490765;
        double r1490767 = log(r1490766);
        double r1490768 = r1490746 * r1490767;
        double r1490769 = r1490768 + r1490750;
        double r1490770 = sin(r1490769);
        double r1490771 = r1490763 * r1490770;
        double r1490772 = log(r1490734);
        double r1490773 = r1490772 * r1490746;
        double r1490774 = r1490750 + r1490773;
        double r1490775 = sin(r1490774);
        double r1490776 = r1490749 * r1490775;
        double r1490777 = r1490758 ? r1490771 : r1490776;
        double r1490778 = r1490736 ? r1490756 : r1490777;
        return r1490778;
}

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. Split input into 3 regimes

  2. if x.re < -5.907306680385344e-241

    1. Initial program 31.4

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

    2. Taylor expanded around -inf 20.2

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

    3. Simplified20.2

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

    if -5.907306680385344e-241 < x.re < 1.8089899081997888e-128

    1. Initial program 30.5

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

    3. Applied add-cube-cbrt30.5

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

    5. Applied add-cube-cbrt30.5

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

    if 1.8089899081997888e-128 < x.re

    1. Initial program 36.7

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

    2. Taylor expanded around inf 25.5

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

    3. Simplified25.5

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

  4. Final simplification24.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -5.907306680385344179981604616356048283101 \cdot 10^{-241}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{elif}\;x.re \le 1.808989908199788750558131282319198960687 \cdot 10^{-128}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}} \cdot \left(\sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}} \cdot \sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (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)))))