Average Error: 33.5 → 8.5
Time: 21.2s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le 2.694637325986234635023828792620054235417 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]
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 2.694637325986234635023828792620054235417 \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 r21422 = x_re;
        double r21423 = r21422 * r21422;
        double r21424 = x_im;
        double r21425 = r21424 * r21424;
        double r21426 = r21423 + r21425;
        double r21427 = sqrt(r21426);
        double r21428 = log(r21427);
        double r21429 = y_re;
        double r21430 = r21428 * r21429;
        double r21431 = atan2(r21424, r21422);
        double r21432 = y_im;
        double r21433 = r21431 * r21432;
        double r21434 = r21430 - r21433;
        double r21435 = exp(r21434);
        double r21436 = r21428 * r21432;
        double r21437 = r21431 * r21429;
        double r21438 = r21436 + r21437;
        double r21439 = cos(r21438);
        double r21440 = r21435 * r21439;
        return r21440;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r21441 = x_re;
        double r21442 = 2.69463732598623e-310;
        bool r21443 = r21441 <= r21442;
        double r21444 = -1.0;
        double r21445 = r21444 * r21441;
        double r21446 = log(r21445);
        double r21447 = y_re;
        double r21448 = r21446 * r21447;
        double r21449 = x_im;
        double r21450 = atan2(r21449, r21441);
        double r21451 = y_im;
        double r21452 = r21450 * r21451;
        double r21453 = r21448 - r21452;
        double r21454 = exp(r21453);
        double r21455 = 1.0;
        double r21456 = r21454 * r21455;
        double r21457 = log(r21441);
        double r21458 = r21457 * r21447;
        double r21459 = r21458 - r21452;
        double r21460 = exp(r21459);
        double r21461 = r21460 * r21455;
        double r21462 = r21443 ? r21456 : r21461;
        return r21462;
}

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 < 2.69463732598623e-310

    1. Initial program 32.5

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

    2. Taylor expanded around 0 18.1

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

    3. Taylor expanded around -inf 6.0

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

    if 2.69463732598623e-310 < x.re

    1. Initial program 34.5

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

    2. Taylor expanded around 0 21.5

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

    3. Taylor expanded around inf 10.9

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le 2.694637325986234635023828792620054235417 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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)))))