Average Error: 33.0 → 11.7
Time: 26.1s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -1.43693106990943942236142186325654954095 \cdot 10^{-21}:\\ \;\;\;\;\frac{{\left(\frac{-1}{x.re}\right)}^{\left(-y.re\right)}}{\sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\\ \mathbf{elif}\;x.re \le -3.862547000878270908215764314997251341808 \cdot 10^{-82}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{3}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \le -2.203136696212098368153966931412365744688 \cdot 10^{-306}:\\ \;\;\;\;\frac{{\left(\frac{-1}{x.re}\right)}^{\left(-y.re\right)}}{\sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\\ \mathbf{elif}\;x.re \le 1.111286203230868220787610421722651910977 \cdot 10^{-97} \lor \neg \left(x.re \le 0.1107239265288796903341506094875512644649\right):\\ \;\;\;\;e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{3}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]
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 -1.43693106990943942236142186325654954095 \cdot 10^{-21}:\\
\;\;\;\;\frac{{\left(\frac{-1}{x.re}\right)}^{\left(-y.re\right)}}{\sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\\

\mathbf{elif}\;x.re \le -3.862547000878270908215764314997251341808 \cdot 10^{-82}:\\
\;\;\;\;e^{\log \left(\sqrt[3]{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{3}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\

\mathbf{elif}\;x.re \le -2.203136696212098368153966931412365744688 \cdot 10^{-306}:\\
\;\;\;\;\frac{{\left(\frac{-1}{x.re}\right)}^{\left(-y.re\right)}}{\sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\\

\mathbf{elif}\;x.re \le 1.111286203230868220787610421722651910977 \cdot 10^{-97} \lor \neg \left(x.re \le 0.1107239265288796903341506094875512644649\right):\\
\;\;\;\;e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left(\sqrt[3]{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{3}}\right) \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 r24529 = x_re;
        double r24530 = r24529 * r24529;
        double r24531 = x_im;
        double r24532 = r24531 * r24531;
        double r24533 = r24530 + r24532;
        double r24534 = sqrt(r24533);
        double r24535 = log(r24534);
        double r24536 = y_re;
        double r24537 = r24535 * r24536;
        double r24538 = atan2(r24531, r24529);
        double r24539 = y_im;
        double r24540 = r24538 * r24539;
        double r24541 = r24537 - r24540;
        double r24542 = exp(r24541);
        double r24543 = r24535 * r24539;
        double r24544 = r24538 * r24536;
        double r24545 = r24543 + r24544;
        double r24546 = cos(r24545);
        double r24547 = r24542 * r24546;
        return r24547;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r24548 = x_re;
        double r24549 = -1.4369310699094394e-21;
        bool r24550 = r24548 <= r24549;
        double r24551 = -1.0;
        double r24552 = r24551 / r24548;
        double r24553 = y_re;
        double r24554 = -r24553;
        double r24555 = pow(r24552, r24554);
        double r24556 = y_im;
        double r24557 = exp(r24556);
        double r24558 = x_im;
        double r24559 = atan2(r24558, r24548);
        double r24560 = pow(r24557, r24559);
        double r24561 = sqrt(r24560);
        double r24562 = r24561 * r24561;
        double r24563 = r24555 / r24562;
        double r24564 = -3.862547000878271e-82;
        bool r24565 = r24548 <= r24564;
        double r24566 = r24548 * r24548;
        double r24567 = r24558 * r24558;
        double r24568 = r24566 + r24567;
        double r24569 = sqrt(r24568);
        double r24570 = 3.0;
        double r24571 = pow(r24569, r24570);
        double r24572 = cbrt(r24571);
        double r24573 = log(r24572);
        double r24574 = r24573 * r24553;
        double r24575 = r24559 * r24556;
        double r24576 = r24574 - r24575;
        double r24577 = exp(r24576);
        double r24578 = -2.2031366962120984e-306;
        bool r24579 = r24548 <= r24578;
        double r24580 = 1.1112862032308682e-97;
        bool r24581 = r24548 <= r24580;
        double r24582 = 0.11072392652887969;
        bool r24583 = r24548 <= r24582;
        double r24584 = !r24583;
        bool r24585 = r24581 || r24584;
        double r24586 = log(r24548);
        double r24587 = r24553 * r24586;
        double r24588 = r24587 - r24575;
        double r24589 = exp(r24588);
        double r24590 = r24585 ? r24589 : r24577;
        double r24591 = r24579 ? r24563 : r24590;
        double r24592 = r24565 ? r24577 : r24591;
        double r24593 = r24550 ? r24563 : r24592;
        return r24593;
}

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 < -1.4369310699094394e-21 or -3.862547000878271e-82 < x.re < -2.2031366962120984e-306

    1. Initial program 33.5

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

    2. Taylor expanded around 0 18.5

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

    3. Taylor expanded around -inf 5.5

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

    4. Simplified11.0

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

    6. Applied add-sqr-sqrt11.0

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

    7. Simplified11.0

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

    8. Simplified11.0

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

    if -1.4369310699094394e-21 < x.re < -3.862547000878271e-82 or 1.1112862032308682e-97 < x.re < 0.11072392652887969

    1. Initial program 17.7

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

    2. Taylor expanded around 0 11.6

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

    4. Applied add-cbrt-cube14.7

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

    5. Simplified14.7

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

    if -2.2031366962120984e-306 < x.re < 1.1112862032308682e-97 or 0.11072392652887969 < x.re

    1. Initial program 36.8

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

    2. Taylor expanded around 0 22.6

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

    3. Taylor expanded around inf 11.5

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -1.43693106990943942236142186325654954095 \cdot 10^{-21}:\\ \;\;\;\;\frac{{\left(\frac{-1}{x.re}\right)}^{\left(-y.re\right)}}{\sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\\ \mathbf{elif}\;x.re \le -3.862547000878270908215764314997251341808 \cdot 10^{-82}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{3}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \le -2.203136696212098368153966931412365744688 \cdot 10^{-306}:\\ \;\;\;\;\frac{{\left(\frac{-1}{x.re}\right)}^{\left(-y.re\right)}}{\sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sqrt{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}\\ \mathbf{elif}\;x.re \le 1.111286203230868220787610421722651910977 \cdot 10^{-97} \lor \neg \left(x.re \le 0.1107239265288796903341506094875512644649\right):\\ \;\;\;\;e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{3}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (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)))))