Average Error: 33.8 → 10.5
Time: 2.4m
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -8.732803126791036021627347672393718855249 \cdot 10^{-70}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le 1.359755228258073250742917907568916334331 \cdot 10^{-34}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]
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 \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)
↓
\begin{array}{l}
\mathbf{if}\;x.re \le -8.732803126791036021627347672393718855249 \cdot 10^{-70}:\\
\;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le 1.359755228258073250742917907568916334331 \cdot 10^{-34}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{else}:\\
\;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1420728 = x_re;
        double r1420729 = r1420728 * r1420728;
        double r1420730 = x_im;
        double r1420731 = r1420730 * r1420730;
        double r1420732 = r1420729 + r1420731;
        double r1420733 = sqrt(r1420732);
        double r1420734 = log(r1420733);
        double r1420735 = y_re;
        double r1420736 = r1420734 * r1420735;
        double r1420737 = atan2(r1420730, r1420728);
        double r1420738 = y_im;
        double r1420739 = r1420737 * r1420738;
        double r1420740 = r1420736 - r1420739;
        double r1420741 = exp(r1420740);
        double r1420742 = r1420734 * r1420738;
        double r1420743 = r1420737 * r1420735;
        double r1420744 = r1420742 + r1420743;
        double r1420745 = cos(r1420744);
        double r1420746 = r1420741 * r1420745;
        return r1420746;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1420747 = x_re;
        double r1420748 = -8.732803126791036e-70;
        bool r1420749 = r1420747 <= r1420748;
        double r1420750 = y_re;
        double r1420751 = -r1420747;
        double r1420752 = log(r1420751);
        double r1420753 = r1420750 * r1420752;
        double r1420754 = y_im;
        double r1420755 = x_im;
        double r1420756 = atan2(r1420755, r1420747);
        double r1420757 = r1420754 * r1420756;
        double r1420758 = r1420753 - r1420757;
        double r1420759 = exp(r1420758);
        double r1420760 = 1.3597552282580733e-34;
        bool r1420761 = r1420747 <= r1420760;
        double r1420762 = r1420755 * r1420755;
        double r1420763 = r1420747 * r1420747;
        double r1420764 = r1420762 + r1420763;
        double r1420765 = sqrt(r1420764);
        double r1420766 = log(r1420765);
        double r1420767 = r1420750 * r1420766;
        double r1420768 = r1420767 - r1420757;
        double r1420769 = exp(r1420768);
        double r1420770 = log(r1420747);
        double r1420771 = r1420770 * r1420750;
        double r1420772 = r1420771 - r1420757;
        double r1420773 = exp(r1420772);
        double r1420774 = r1420761 ? r1420769 : r1420773;
        double r1420775 = r1420749 ? r1420759 : r1420774;
        return r1420775;
}

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 < -8.732803126791036e-70

    1. Initial program 36.4

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

    2. Taylor expanded around 0 20.1

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

    3. Taylor expanded around -inf 3.0

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

    4. Simplified3.0

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

    if -8.732803126791036e-70 < x.re < 1.3597552282580733e-34

    1. Initial program 26.2

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

    2. Taylor expanded around 0 15.1

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

    if 1.3597552282580733e-34 < x.re

    1. Initial program 41.7

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

    2. Taylor expanded around 0 26.6

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

    3. Taylor expanded around inf 11.4

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

    4. Simplified11.4

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -8.732803126791036021627347672393718855249 \cdot 10^{-70}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le 1.359755228258073250742917907568916334331 \cdot 10^{-34}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (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)))))