Average Error: 33.0 → 22.1
Time: 32.0s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -5.189251590720128421046416075251281086703 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]
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 -5.189251590720128421046416075251281086703 \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 r2169015 = x_re;
        double r2169016 = r2169015 * r2169015;
        double r2169017 = x_im;
        double r2169018 = r2169017 * r2169017;
        double r2169019 = r2169016 + r2169018;
        double r2169020 = sqrt(r2169019);
        double r2169021 = log(r2169020);
        double r2169022 = y_re;
        double r2169023 = r2169021 * r2169022;
        double r2169024 = atan2(r2169017, r2169015);
        double r2169025 = y_im;
        double r2169026 = r2169024 * r2169025;
        double r2169027 = r2169023 - r2169026;
        double r2169028 = exp(r2169027);
        double r2169029 = r2169021 * r2169025;
        double r2169030 = r2169024 * r2169022;
        double r2169031 = r2169029 + r2169030;
        double r2169032 = sin(r2169031);
        double r2169033 = r2169028 * r2169032;
        return r2169033;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2169034 = x_re;
        double r2169035 = -5.18925159072013e-310;
        bool r2169036 = r2169034 <= r2169035;
        double r2169037 = r2169034 * r2169034;
        double r2169038 = x_im;
        double r2169039 = r2169038 * r2169038;
        double r2169040 = r2169037 + r2169039;
        double r2169041 = sqrt(r2169040);
        double r2169042 = log(r2169041);
        double r2169043 = y_re;
        double r2169044 = r2169042 * r2169043;
        double r2169045 = atan2(r2169038, r2169034);
        double r2169046 = y_im;
        double r2169047 = r2169045 * r2169046;
        double r2169048 = r2169044 - r2169047;
        double r2169049 = exp(r2169048);
        double r2169050 = -r2169034;
        double r2169051 = log(r2169050);
        double r2169052 = r2169051 * r2169046;
        double r2169053 = r2169045 * r2169043;
        double r2169054 = r2169052 + r2169053;
        double r2169055 = sin(r2169054);
        double r2169056 = r2169049 * r2169055;
        double r2169057 = log(r2169034);
        double r2169058 = r2169057 * r2169046;
        double r2169059 = r2169058 + r2169053;
        double r2169060 = sin(r2169059);
        double r2169061 = r2169049 * r2169060;
        double r2169062 = r2169036 ? r2169056 : r2169061;
        return r2169062;
}

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 < -5.18925159072013e-310

    1. Initial program 31.7

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

    2. Taylor expanded around -inf 20.8

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

    3. Simplified20.8

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

    if -5.18925159072013e-310 < x.re

    1. Initial program 34.3

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

    2. Taylor expanded around inf 23.4

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -5.189251590720128421046416075251281086703 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))))