Average Error: 33.7 → 22.0
Time: 32.0s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -3.566576781276111012185154762547093345551 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]
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 -3.566576781276111012185154762547093345551 \cdot 10^{-310}:\\
\;\;\;\;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(-x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r24777 = x_re;
        double r24778 = r24777 * r24777;
        double r24779 = x_im;
        double r24780 = r24779 * r24779;
        double r24781 = r24778 + r24780;
        double r24782 = sqrt(r24781);
        double r24783 = log(r24782);
        double r24784 = y_re;
        double r24785 = r24783 * r24784;
        double r24786 = atan2(r24779, r24777);
        double r24787 = y_im;
        double r24788 = r24786 * r24787;
        double r24789 = r24785 - r24788;
        double r24790 = exp(r24789);
        double r24791 = r24783 * r24787;
        double r24792 = r24786 * r24784;
        double r24793 = r24791 + r24792;
        double r24794 = sin(r24793);
        double r24795 = r24790 * r24794;
        return r24795;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r24796 = x_re;
        double r24797 = -3.5665767812761e-310;
        bool r24798 = r24796 <= r24797;
        double r24799 = r24796 * r24796;
        double r24800 = x_im;
        double r24801 = r24800 * r24800;
        double r24802 = r24799 + r24801;
        double r24803 = sqrt(r24802);
        double r24804 = log(r24803);
        double r24805 = y_re;
        double r24806 = r24804 * r24805;
        double r24807 = atan2(r24800, r24796);
        double r24808 = y_im;
        double r24809 = r24807 * r24808;
        double r24810 = r24806 - r24809;
        double r24811 = exp(r24810);
        double r24812 = -r24796;
        double r24813 = log(r24812);
        double r24814 = r24813 * r24808;
        double r24815 = r24807 * r24805;
        double r24816 = r24814 + r24815;
        double r24817 = sin(r24816);
        double r24818 = r24811 * r24817;
        double r24819 = log(r24796);
        double r24820 = r24819 * r24808;
        double r24821 = r24820 + r24815;
        double r24822 = sin(r24821);
        double r24823 = r24811 * r24822;
        double r24824 = r24798 ? r24818 : r24823;
        return r24824;
}

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 2 regimes

  2. if x.re < -3.5665767812761e-310

    1. Initial program 32.0

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

    2. Taylor expanded around -inf 19.7

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

    3. Simplified19.7

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

    if -3.5665767812761e-310 < x.re

    1. Initial program 35.3

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

    2. Taylor expanded around inf 24.2

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{x.re} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -3.566576781276111012185154762547093345551 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))))