Average Error: 33.4 → 22.7
Time: 32.9s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -5.121219671519223731301371692327955218506 \cdot 10^{-309}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]
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 -5.121219671519223731301371692327955218506 \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(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r26429 = x_re;
        double r26430 = r26429 * r26429;
        double r26431 = x_im;
        double r26432 = r26431 * r26431;
        double r26433 = r26430 + r26432;
        double r26434 = sqrt(r26433);
        double r26435 = log(r26434);
        double r26436 = y_re;
        double r26437 = r26435 * r26436;
        double r26438 = atan2(r26431, r26429);
        double r26439 = y_im;
        double r26440 = r26438 * r26439;
        double r26441 = r26437 - r26440;
        double r26442 = exp(r26441);
        double r26443 = r26435 * r26439;
        double r26444 = r26438 * r26436;
        double r26445 = r26443 + r26444;
        double r26446 = sin(r26445);
        double r26447 = r26442 * r26446;
        return r26447;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r26448 = x_re;
        double r26449 = -5.121219671519224e-309;
        bool r26450 = r26448 <= r26449;
        double r26451 = r26448 * r26448;
        double r26452 = x_im;
        double r26453 = r26452 * r26452;
        double r26454 = r26451 + r26453;
        double r26455 = sqrt(r26454);
        double r26456 = log(r26455);
        double r26457 = y_re;
        double r26458 = r26456 * r26457;
        double r26459 = atan2(r26452, r26448);
        double r26460 = y_im;
        double r26461 = r26459 * r26460;
        double r26462 = r26458 - r26461;
        double r26463 = exp(r26462);
        double r26464 = r26459 * r26457;
        double r26465 = -1.0;
        double r26466 = r26465 / r26448;
        double r26467 = log(r26466);
        double r26468 = r26460 * r26467;
        double r26469 = r26464 - r26468;
        double r26470 = sin(r26469);
        double r26471 = r26463 * r26470;
        double r26472 = log(r26448);
        double r26473 = r26472 * r26460;
        double r26474 = r26473 + r26464;
        double r26475 = sin(r26474);
        double r26476 = r26463 * r26475;
        double r26477 = r26450 ? r26471 : r26476;
        return r26477;
}

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 < -5.121219671519224e-309

    1. Initial program 32.2

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

    2. Taylor expanded around -inf 20.9

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

    if -5.121219671519224e-309 < x.re

    1. Initial program 34.7

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

    2. Taylor expanded around inf 24.4

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

    3. Simplified24.4

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

  4. Final simplification22.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -5.121219671519223731301371692327955218506 \cdot 10^{-309}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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)))))