Average Error: 32.2 → 9.3
Time: 1.0m
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -8.138202357158243 \cdot 10^{-284}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le 7.402062416285907 \cdot 10^{-205}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]
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 -8.138202357158243 \cdot 10^{-284}:\\
\;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le 7.402062416285907 \cdot 10^{-205}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\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 r1589817 = x_re;
        double r1589818 = r1589817 * r1589817;
        double r1589819 = x_im;
        double r1589820 = r1589819 * r1589819;
        double r1589821 = r1589818 + r1589820;
        double r1589822 = sqrt(r1589821);
        double r1589823 = log(r1589822);
        double r1589824 = y_re;
        double r1589825 = r1589823 * r1589824;
        double r1589826 = atan2(r1589819, r1589817);
        double r1589827 = y_im;
        double r1589828 = r1589826 * r1589827;
        double r1589829 = r1589825 - r1589828;
        double r1589830 = exp(r1589829);
        double r1589831 = r1589823 * r1589827;
        double r1589832 = r1589826 * r1589824;
        double r1589833 = r1589831 + r1589832;
        double r1589834 = cos(r1589833);
        double r1589835 = r1589830 * r1589834;
        return r1589835;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1589836 = x_re;
        double r1589837 = -8.138202357158243e-284;
        bool r1589838 = r1589836 <= r1589837;
        double r1589839 = -r1589836;
        double r1589840 = log(r1589839);
        double r1589841 = y_re;
        double r1589842 = r1589840 * r1589841;
        double r1589843 = y_im;
        double r1589844 = x_im;
        double r1589845 = atan2(r1589844, r1589836);
        double r1589846 = r1589843 * r1589845;
        double r1589847 = r1589842 - r1589846;
        double r1589848 = exp(r1589847);
        double r1589849 = 7.402062416285907e-205;
        bool r1589850 = r1589836 <= r1589849;
        double r1589851 = r1589836 * r1589836;
        double r1589852 = r1589844 * r1589844;
        double r1589853 = r1589851 + r1589852;
        double r1589854 = sqrt(r1589853);
        double r1589855 = log(r1589854);
        double r1589856 = r1589841 * r1589855;
        double r1589857 = r1589856 - r1589846;
        double r1589858 = exp(r1589857);
        double r1589859 = log(r1589836);
        double r1589860 = r1589859 * r1589841;
        double r1589861 = r1589860 - r1589846;
        double r1589862 = exp(r1589861);
        double r1589863 = r1589850 ? r1589858 : r1589862;
        double r1589864 = r1589838 ? r1589848 : r1589863;
        return r1589864;
}

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 < -8.138202357158243e-284

    1. Initial program 31.7

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

    2. Taylor expanded around 0 18.0

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

    3. Taylor expanded around -inf 5.8

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

    4. Simplified5.8

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

    if -8.138202357158243e-284 < x.re < 7.402062416285907e-205

    1. Initial program 29.6

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

    2. Taylor expanded around 0 15.9

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

    if 7.402062416285907e-205 < x.re

    1. Initial program 33.5

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

    2. Taylor expanded around 0 21.7

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

    3. Taylor expanded around inf 11.6

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -8.138202357158243 \cdot 10^{-284}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le 7.402062416285907 \cdot 10^{-205}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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)))))