Average Error: 33.1 → 8.6
Time: 7.5s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le 7.7747768769685 \cdot 10^{-312}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]
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 7.7747768769685 \cdot 10^{-312}:\\
\;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \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 r15436 = x_re;
        double r15437 = r15436 * r15436;
        double r15438 = x_im;
        double r15439 = r15438 * r15438;
        double r15440 = r15437 + r15439;
        double r15441 = sqrt(r15440);
        double r15442 = log(r15441);
        double r15443 = y_re;
        double r15444 = r15442 * r15443;
        double r15445 = atan2(r15438, r15436);
        double r15446 = y_im;
        double r15447 = r15445 * r15446;
        double r15448 = r15444 - r15447;
        double r15449 = exp(r15448);
        double r15450 = r15442 * r15446;
        double r15451 = r15445 * r15443;
        double r15452 = r15450 + r15451;
        double r15453 = cos(r15452);
        double r15454 = r15449 * r15453;
        return r15454;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r15455 = x_re;
        double r15456 = 7.7747768769685e-312;
        bool r15457 = r15455 <= r15456;
        double r15458 = -1.0;
        double r15459 = y_re;
        double r15460 = r15458 / r15455;
        double r15461 = log(r15460);
        double r15462 = r15459 * r15461;
        double r15463 = r15458 * r15462;
        double r15464 = x_im;
        double r15465 = atan2(r15464, r15455);
        double r15466 = y_im;
        double r15467 = r15465 * r15466;
        double r15468 = r15463 - r15467;
        double r15469 = exp(r15468);
        double r15470 = 1.0;
        double r15471 = r15469 * r15470;
        double r15472 = log(r15455);
        double r15473 = r15472 * r15459;
        double r15474 = r15473 - r15467;
        double r15475 = exp(r15474);
        double r15476 = r15475 * r15470;
        double r15477 = r15457 ? r15471 : r15476;
        return r15477;
}

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 < 7.7747768769685e-312

    1. Initial program 31.2

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

    2. Taylor expanded around 0 17.1

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

    3. Taylor expanded around -inf 5.7

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

    if 7.7747768769685e-312 < x.re

    1. Initial program 34.8

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

    2. Taylor expanded around 0 21.6

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

    3. Taylor expanded around inf 11.2

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le 7.7747768769685 \cdot 10^{-312}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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)))))