Average Error: 33.8 → 9.3
Time: 7.1s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -1.99303880436604133 \cdot 10^{-80}:\\ \;\;\;\;e^{-\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)} \cdot 1\\ \mathbf{elif}\;x.re \le 2.986375179024046 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 8.899433436760234 \cdot 10^{-153}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 5.92453119316243509 \cdot 10^{-30}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]
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 -1.99303880436604133 \cdot 10^{-80}:\\
\;\;\;\;e^{-\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)} \cdot 1\\

\mathbf{elif}\;x.re \le 2.986375179024046 \cdot 10^{-310}:\\
\;\;\;\;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 1\\

\mathbf{elif}\;x.re \le 8.899433436760234 \cdot 10^{-153}:\\
\;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\

\mathbf{elif}\;x.re \le 5.92453119316243509 \cdot 10^{-30}:\\
\;\;\;\;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 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 r14402 = x_re;
        double r14403 = r14402 * r14402;
        double r14404 = x_im;
        double r14405 = r14404 * r14404;
        double r14406 = r14403 + r14405;
        double r14407 = sqrt(r14406);
        double r14408 = log(r14407);
        double r14409 = y_re;
        double r14410 = r14408 * r14409;
        double r14411 = atan2(r14404, r14402);
        double r14412 = y_im;
        double r14413 = r14411 * r14412;
        double r14414 = r14410 - r14413;
        double r14415 = exp(r14414);
        double r14416 = r14408 * r14412;
        double r14417 = r14411 * r14409;
        double r14418 = r14416 + r14417;
        double r14419 = cos(r14418);
        double r14420 = r14415 * r14419;
        return r14420;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r14421 = x_re;
        double r14422 = -1.9930388043660413e-80;
        bool r14423 = r14421 <= r14422;
        double r14424 = x_im;
        double r14425 = atan2(r14424, r14421);
        double r14426 = y_im;
        double r14427 = r14425 * r14426;
        double r14428 = y_re;
        double r14429 = -1.0;
        double r14430 = r14429 / r14421;
        double r14431 = log(r14430);
        double r14432 = r14428 * r14431;
        double r14433 = r14427 + r14432;
        double r14434 = -r14433;
        double r14435 = exp(r14434);
        double r14436 = 1.0;
        double r14437 = r14435 * r14436;
        double r14438 = 2.98637517902405e-310;
        bool r14439 = r14421 <= r14438;
        double r14440 = r14421 * r14421;
        double r14441 = r14424 * r14424;
        double r14442 = r14440 + r14441;
        double r14443 = sqrt(r14442);
        double r14444 = log(r14443);
        double r14445 = r14444 * r14428;
        double r14446 = r14445 - r14427;
        double r14447 = exp(r14446);
        double r14448 = r14447 * r14436;
        double r14449 = 8.899433436760234e-153;
        bool r14450 = r14421 <= r14449;
        double r14451 = log(r14421);
        double r14452 = r14451 * r14428;
        double r14453 = r14452 - r14427;
        double r14454 = exp(r14453);
        double r14455 = r14454 * r14436;
        double r14456 = 5.924531193162435e-30;
        bool r14457 = r14421 <= r14456;
        double r14458 = r14457 ? r14448 : r14455;
        double r14459 = r14450 ? r14455 : r14458;
        double r14460 = r14439 ? r14448 : r14459;
        double r14461 = r14423 ? r14437 : r14460;
        return r14461;
}

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 < -1.9930388043660413e-80

    1. Initial program 35.3

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

    2. Taylor expanded around 0 19.6

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

    3. Taylor expanded around -inf 2.9

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

    if -1.9930388043660413e-80 < x.re < 2.98637517902405e-310 or 8.899433436760234e-153 < x.re < 5.924531193162435e-30

    1. Initial program 23.4

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

    2. Taylor expanded around 0 13.2

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

    if 2.98637517902405e-310 < x.re < 8.899433436760234e-153 or 5.924531193162435e-30 < x.re

    1. Initial program 39.3

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

    2. Taylor expanded around 0 24.3

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

    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 simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -1.99303880436604133 \cdot 10^{-80}:\\ \;\;\;\;e^{-\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)} \cdot 1\\ \mathbf{elif}\;x.re \le 2.986375179024046 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 8.899433436760234 \cdot 10^{-153}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 5.92453119316243509 \cdot 10^{-30}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))))