Average Error: 32.2 → 9.3
Time: 32.1s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le 4.11803379523495 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]
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 4.11803379523495 \cdot 10^{-310}:\\
\;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - 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 r1448582 = x_re;
        double r1448583 = r1448582 * r1448582;
        double r1448584 = x_im;
        double r1448585 = r1448584 * r1448584;
        double r1448586 = r1448583 + r1448585;
        double r1448587 = sqrt(r1448586);
        double r1448588 = log(r1448587);
        double r1448589 = y_re;
        double r1448590 = r1448588 * r1448589;
        double r1448591 = atan2(r1448584, r1448582);
        double r1448592 = y_im;
        double r1448593 = r1448591 * r1448592;
        double r1448594 = r1448590 - r1448593;
        double r1448595 = exp(r1448594);
        double r1448596 = r1448588 * r1448592;
        double r1448597 = r1448591 * r1448589;
        double r1448598 = r1448596 + r1448597;
        double r1448599 = cos(r1448598);
        double r1448600 = r1448595 * r1448599;
        return r1448600;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1448601 = x_re;
        double r1448602 = 4.11803379523495e-310;
        bool r1448603 = r1448601 <= r1448602;
        double r1448604 = -r1448601;
        double r1448605 = log(r1448604);
        double r1448606 = y_re;
        double r1448607 = r1448605 * r1448606;
        double r1448608 = y_im;
        double r1448609 = x_im;
        double r1448610 = atan2(r1448609, r1448601);
        double r1448611 = r1448608 * r1448610;
        double r1448612 = r1448607 - r1448611;
        double r1448613 = exp(r1448612);
        double r1448614 = log(r1448601);
        double r1448615 = r1448614 * r1448606;
        double r1448616 = r1448615 - r1448611;
        double r1448617 = exp(r1448616);
        double r1448618 = r1448603 ? r1448613 : r1448617;
        return r1448618;
}

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 < 4.11803379523495e-310

    1. Initial program 30.2

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

    2. Taylor expanded around 0 16.7

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

    3. Taylor expanded around -inf 6.3

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

    4. Simplified6.3

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

    if 4.11803379523495e-310 < x.re

    1. Initial program 34.2

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

    2. Taylor expanded around 0 22.1

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

    3. Taylor expanded around inf 12.1

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

    4. Simplified12.1

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le 4.11803379523495 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))))