Average Error: 33.4 → 22.4
Time: 9.9s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -6.2360011819834022 \cdot 10^{-309}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\\ \end{array}\]
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 -6.2360011819834022 \cdot 10^{-309}:\\
\;\;\;\;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(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r17368 = x_re;
        double r17369 = r17368 * r17368;
        double r17370 = x_im;
        double r17371 = r17370 * r17370;
        double r17372 = r17369 + r17371;
        double r17373 = sqrt(r17372);
        double r17374 = log(r17373);
        double r17375 = y_re;
        double r17376 = r17374 * r17375;
        double r17377 = atan2(r17370, r17368);
        double r17378 = y_im;
        double r17379 = r17377 * r17378;
        double r17380 = r17376 - r17379;
        double r17381 = exp(r17380);
        double r17382 = r17374 * r17378;
        double r17383 = r17377 * r17375;
        double r17384 = r17382 + r17383;
        double r17385 = sin(r17384);
        double r17386 = r17381 * r17385;
        return r17386;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r17387 = x_re;
        double r17388 = -6.2360011819834e-309;
        bool r17389 = r17387 <= r17388;
        double r17390 = r17387 * r17387;
        double r17391 = x_im;
        double r17392 = r17391 * r17391;
        double r17393 = r17390 + r17392;
        double r17394 = sqrt(r17393);
        double r17395 = log(r17394);
        double r17396 = y_re;
        double r17397 = r17395 * r17396;
        double r17398 = atan2(r17391, r17387);
        double r17399 = y_im;
        double r17400 = r17398 * r17399;
        double r17401 = r17397 - r17400;
        double r17402 = exp(r17401);
        double r17403 = r17398 * r17396;
        double r17404 = -1.0;
        double r17405 = r17404 / r17387;
        double r17406 = log(r17405);
        double r17407 = r17399 * r17406;
        double r17408 = r17403 - r17407;
        double r17409 = sin(r17408);
        double r17410 = r17402 * r17409;
        double r17411 = 1.0;
        double r17412 = r17411 / r17387;
        double r17413 = log(r17412);
        double r17414 = r17399 * r17413;
        double r17415 = r17403 - r17414;
        double r17416 = sin(r17415);
        double r17417 = r17402 * r17416;
        double r17418 = r17389 ? r17410 : r17417;
        return r17418;
}

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 < -6.2360011819834e-309

    1. Initial program 31.8

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

    2. Taylor expanded around -inf 19.9

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

    if -6.2360011819834e-309 < 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 24.7

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -6.2360011819834022 \cdot 10^{-309}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (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)))))