Average Error: 33.6 → 9.9
Time: 1.8m
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -5.429027010745199184876313702270882432857 \cdot 10^{-203}:\\ \;\;\;\;e^{\left(-y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \le -1.575802734573826837157530749461839110785 \cdot 10^{-254}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \le -3.566576781276111012185154762547093345551 \cdot 10^{-310}:\\ \;\;\;\;e^{\left(-y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]
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 -5.429027010745199184876313702270882432857 \cdot 10^{-203}:\\
\;\;\;\;e^{\left(-y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\

\mathbf{elif}\;x.re \le -1.575802734573826837157530749461839110785 \cdot 10^{-254}:\\
\;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\

\mathbf{elif}\;x.re \le -3.566576781276111012185154762547093345551 \cdot 10^{-310}:\\
\;\;\;\;e^{\left(-y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r21623 = x_re;
        double r21624 = r21623 * r21623;
        double r21625 = x_im;
        double r21626 = r21625 * r21625;
        double r21627 = r21624 + r21626;
        double r21628 = sqrt(r21627);
        double r21629 = log(r21628);
        double r21630 = y_re;
        double r21631 = r21629 * r21630;
        double r21632 = atan2(r21625, r21623);
        double r21633 = y_im;
        double r21634 = r21632 * r21633;
        double r21635 = r21631 - r21634;
        double r21636 = exp(r21635);
        double r21637 = r21629 * r21633;
        double r21638 = r21632 * r21630;
        double r21639 = r21637 + r21638;
        double r21640 = cos(r21639);
        double r21641 = r21636 * r21640;
        return r21641;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r21642 = x_re;
        double r21643 = -5.429027010745199e-203;
        bool r21644 = r21642 <= r21643;
        double r21645 = y_re;
        double r21646 = -1.0;
        double r21647 = r21646 / r21642;
        double r21648 = log(r21647);
        double r21649 = r21645 * r21648;
        double r21650 = -r21649;
        double r21651 = x_im;
        double r21652 = atan2(r21651, r21642);
        double r21653 = y_im;
        double r21654 = r21652 * r21653;
        double r21655 = r21650 - r21654;
        double r21656 = exp(r21655);
        double r21657 = -1.5758027345738268e-254;
        bool r21658 = r21642 <= r21657;
        double r21659 = -r21651;
        double r21660 = log(r21659);
        double r21661 = r21660 * r21645;
        double r21662 = r21661 - r21654;
        double r21663 = exp(r21662);
        double r21664 = -3.5665767812761e-310;
        bool r21665 = r21642 <= r21664;
        double r21666 = log(r21642);
        double r21667 = r21666 * r21645;
        double r21668 = r21667 - r21654;
        double r21669 = exp(r21668);
        double r21670 = r21665 ? r21656 : r21669;
        double r21671 = r21658 ? r21663 : r21670;
        double r21672 = r21644 ? r21656 : r21671;
        return r21672;
}

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 3 regimes

  2. if x.re < -5.429027010745199e-203 or -1.5758027345738268e-254 < x.re < -3.5665767812761e-310

    1. Initial program 31.9

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

    2. Taylor expanded around 0 17.2

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

    3. Taylor expanded around -inf 5.1

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

    4. Simplified5.1

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

    if -5.429027010745199e-203 < x.re < -1.5758027345738268e-254

    1. Initial program 32.6

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

    2. Taylor expanded around 0 15.6

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

    4. Applied add-cbrt-cube21.4

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

    5. Simplified21.4

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

    6. Taylor expanded around -inf 36.8

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

    7. Simplified36.8

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

    if -3.5665767812761e-310 < x.re

    1. Initial program 35.2

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

    2. Taylor expanded around 0 21.4

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

    3. Taylor expanded around inf 11.7

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -5.429027010745199184876313702270882432857 \cdot 10^{-203}:\\ \;\;\;\;e^{\left(-y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \le -1.575802734573826837157530749461839110785 \cdot 10^{-254}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \le -3.566576781276111012185154762547093345551 \cdot 10^{-310}:\\ \;\;\;\;e^{\left(-y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))))