Average Error: 33.9 → 18.2
Time: 2.5m
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -9.243103292273186243733064370902117084868 \cdot 10^{-70}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(-x.re\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \le 3.033060010744536784370034757295084712292 \cdot 10^{-310}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(-x.re\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log x.re \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array}\]
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 -9.243103292273186243733064370902117084868 \cdot 10^{-70}:\\
\;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(-x.re\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\

\mathbf{elif}\;x.re \le 3.033060010744536784370034757295084712292 \cdot 10^{-310}:\\
\;\;\;\;\sin \left(y.im \cdot \log \left(-x.re\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot 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^{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}} \cdot \sin \left(\log x.re \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1399953 = x_re;
        double r1399954 = r1399953 * r1399953;
        double r1399955 = x_im;
        double r1399956 = r1399955 * r1399955;
        double r1399957 = r1399954 + r1399956;
        double r1399958 = sqrt(r1399957);
        double r1399959 = log(r1399958);
        double r1399960 = y_re;
        double r1399961 = r1399959 * r1399960;
        double r1399962 = atan2(r1399955, r1399953);
        double r1399963 = y_im;
        double r1399964 = r1399962 * r1399963;
        double r1399965 = r1399961 - r1399964;
        double r1399966 = exp(r1399965);
        double r1399967 = r1399959 * r1399963;
        double r1399968 = r1399962 * r1399960;
        double r1399969 = r1399967 + r1399968;
        double r1399970 = sin(r1399969);
        double r1399971 = r1399966 * r1399970;
        return r1399971;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1399972 = x_re;
        double r1399973 = -9.243103292273186e-70;
        bool r1399974 = r1399972 <= r1399973;
        double r1399975 = y_re;
        double r1399976 = -r1399972;
        double r1399977 = log(r1399976);
        double r1399978 = r1399975 * r1399977;
        double r1399979 = y_im;
        double r1399980 = x_im;
        double r1399981 = atan2(r1399980, r1399972);
        double r1399982 = r1399979 * r1399981;
        double r1399983 = r1399978 - r1399982;
        double r1399984 = exp(r1399983);
        double r1399985 = r1399979 * r1399977;
        double r1399986 = r1399975 * r1399981;
        double r1399987 = r1399985 + r1399986;
        double r1399988 = sin(r1399987);
        double r1399989 = r1399984 * r1399988;
        double r1399990 = 3.03306001074454e-310;
        bool r1399991 = r1399972 <= r1399990;
        double r1399992 = r1399980 * r1399980;
        double r1399993 = r1399972 * r1399972;
        double r1399994 = r1399992 + r1399993;
        double r1399995 = sqrt(r1399994);
        double r1399996 = log(r1399995);
        double r1399997 = r1399975 * r1399996;
        double r1399998 = r1399997 - r1399982;
        double r1399999 = exp(r1399998);
        double r1400000 = r1399988 * r1399999;
        double r1400001 = log(r1399972);
        double r1400002 = r1400001 * r1399979;
        double r1400003 = r1400002 + r1399986;
        double r1400004 = sin(r1400003);
        double r1400005 = r1399999 * r1400004;
        double r1400006 = r1399991 ? r1400000 : r1400005;
        double r1400007 = r1399974 ? r1399989 : r1400006;
        return r1400007;
}

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 < -9.243103292273186e-70

    1. Initial program 36.5

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

    2. Taylor expanded around -inf 20.7

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

    3. Simplified20.7

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

    4. Taylor expanded around -inf 5.8

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

    5. Simplified5.8

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

    if -9.243103292273186e-70 < x.re < 3.03306001074454e-310

    1. Initial program 27.0

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

    2. Taylor expanded around -inf 20.9

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

    3. Simplified20.9

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

    if 3.03306001074454e-310 < x.re

    1. Initial program 34.9

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

    2. Taylor expanded around inf 24.3

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

    3. Simplified24.3

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

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -9.243103292273186243733064370902117084868 \cdot 10^{-70}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(-x.re\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \le 3.033060010744536784370034757295084712292 \cdot 10^{-310}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(-x.re\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log x.re \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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)))))