Average Error: 33.3 → 22.1
Time: 8.4s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -5.434160650439028118361824534749858467457 \cdot 10^{-309}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\\ \end{array}\]
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 -5.434160650439028118361824534749858467457 \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 r15374 = x_re;
        double r15375 = r15374 * r15374;
        double r15376 = x_im;
        double r15377 = r15376 * r15376;
        double r15378 = r15375 + r15377;
        double r15379 = sqrt(r15378);
        double r15380 = log(r15379);
        double r15381 = y_re;
        double r15382 = r15380 * r15381;
        double r15383 = atan2(r15376, r15374);
        double r15384 = y_im;
        double r15385 = r15383 * r15384;
        double r15386 = r15382 - r15385;
        double r15387 = exp(r15386);
        double r15388 = r15380 * r15384;
        double r15389 = r15383 * r15381;
        double r15390 = r15388 + r15389;
        double r15391 = sin(r15390);
        double r15392 = r15387 * r15391;
        return r15392;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r15393 = x_re;
        double r15394 = -5.43416065043903e-309;
        bool r15395 = r15393 <= r15394;
        double r15396 = r15393 * r15393;
        double r15397 = x_im;
        double r15398 = r15397 * r15397;
        double r15399 = r15396 + r15398;
        double r15400 = sqrt(r15399);
        double r15401 = log(r15400);
        double r15402 = y_re;
        double r15403 = r15401 * r15402;
        double r15404 = atan2(r15397, r15393);
        double r15405 = y_im;
        double r15406 = r15404 * r15405;
        double r15407 = r15403 - r15406;
        double r15408 = exp(r15407);
        double r15409 = r15404 * r15402;
        double r15410 = -1.0;
        double r15411 = r15410 / r15393;
        double r15412 = log(r15411);
        double r15413 = r15405 * r15412;
        double r15414 = r15409 - r15413;
        double r15415 = sin(r15414);
        double r15416 = r15408 * r15415;
        double r15417 = 1.0;
        double r15418 = r15417 / r15393;
        double r15419 = log(r15418);
        double r15420 = r15405 * r15419;
        double r15421 = r15409 - r15420;
        double r15422 = sin(r15421);
        double r15423 = r15408 * r15422;
        double r15424 = r15395 ? r15416 : r15423;
        return r15424;
}

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 < -5.43416065043903e-309

    1. Initial program 31.6

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

    2. Taylor expanded around -inf 19.4

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

    if -5.43416065043903e-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. Using strategy rm

    3. Applied add-cube-cbrt35.0

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

    4. Applied associate-*l*35.0

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

    5. Taylor expanded around inf 24.8

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -5.434160650439028118361824534749858467457 \cdot 10^{-309}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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)))))