Average Error: 33.9 → 9.1
Time: 7.1s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -1.340045005672196694155106662820309757076 \cdot 10^{-23}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -2.997344712807687659670489329095943021743 \cdot 10^{-193}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 1.945722595037461298301930831329184038453 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]
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 -1.340045005672196694155106662820309757076 \cdot 10^{-23}:\\
\;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\

\mathbf{elif}\;x.re \le -2.997344712807687659670489329095943021743 \cdot 10^{-193}:\\
\;\;\;\;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 1\\

\mathbf{elif}\;x.re \le 1.945722595037461298301930831329184038453 \cdot 10^{-310}:\\
\;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r10880 = x_re;
        double r10881 = r10880 * r10880;
        double r10882 = x_im;
        double r10883 = r10882 * r10882;
        double r10884 = r10881 + r10883;
        double r10885 = sqrt(r10884);
        double r10886 = log(r10885);
        double r10887 = y_re;
        double r10888 = r10886 * r10887;
        double r10889 = atan2(r10882, r10880);
        double r10890 = y_im;
        double r10891 = r10889 * r10890;
        double r10892 = r10888 - r10891;
        double r10893 = exp(r10892);
        double r10894 = r10886 * r10890;
        double r10895 = r10889 * r10887;
        double r10896 = r10894 + r10895;
        double r10897 = cos(r10896);
        double r10898 = r10893 * r10897;
        return r10898;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r10899 = x_re;
        double r10900 = -1.3400450056721967e-23;
        bool r10901 = r10899 <= r10900;
        double r10902 = -1.0;
        double r10903 = r10902 * r10899;
        double r10904 = log(r10903);
        double r10905 = y_re;
        double r10906 = r10904 * r10905;
        double r10907 = x_im;
        double r10908 = atan2(r10907, r10899);
        double r10909 = y_im;
        double r10910 = r10908 * r10909;
        double r10911 = r10906 - r10910;
        double r10912 = exp(r10911);
        double r10913 = 1.0;
        double r10914 = r10912 * r10913;
        double r10915 = -2.9973447128076877e-193;
        bool r10916 = r10899 <= r10915;
        double r10917 = r10899 * r10899;
        double r10918 = r10907 * r10907;
        double r10919 = r10917 + r10918;
        double r10920 = sqrt(r10919);
        double r10921 = log(r10920);
        double r10922 = r10921 * r10905;
        double r10923 = r10922 - r10910;
        double r10924 = exp(r10923);
        double r10925 = r10924 * r10913;
        double r10926 = 1.94572259503746e-310;
        bool r10927 = r10899 <= r10926;
        double r10928 = log(r10899);
        double r10929 = r10928 * r10905;
        double r10930 = r10929 - r10910;
        double r10931 = exp(r10930);
        double r10932 = r10931 * r10913;
        double r10933 = r10927 ? r10914 : r10932;
        double r10934 = r10916 ? r10925 : r10933;
        double r10935 = r10901 ? r10914 : r10934;
        return r10935;
}

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 < -1.3400450056721967e-23 or -2.9973447128076877e-193 < x.re < 1.94572259503746e-310

    1. Initial program 37.0

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

    2. Taylor expanded around 0 21.1

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

    3. Taylor expanded around -inf 4.3

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

    if -1.3400450056721967e-23 < x.re < -2.9973447128076877e-193

    1. Initial program 20.4

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

    2. Taylor expanded around 0 11.1

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

    if 1.94572259503746e-310 < x.re

    1. Initial program 35.0

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

    2. Taylor expanded around 0 22.7

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

    3. Taylor expanded around inf 11.9

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -1.340045005672196694155106662820309757076 \cdot 10^{-23}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -2.997344712807687659670489329095943021743 \cdot 10^{-193}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 1.945722595037461298301930831329184038453 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (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)))))