Average Error: 32.9 → 8.6
Time: 33.0s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -1.8407009899098441 \cdot 10^{-06}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -2.9130712528480477 \cdot 10^{-164}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -4.2640896224449 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]
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 -1.8407009899098441 \cdot 10^{-06}:\\
\;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le -2.9130712528480477 \cdot 10^{-164}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le -4.2640896224449 \cdot 10^{-310}:\\
\;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1436384 = x_re;
        double r1436385 = r1436384 * r1436384;
        double r1436386 = x_im;
        double r1436387 = r1436386 * r1436386;
        double r1436388 = r1436385 + r1436387;
        double r1436389 = sqrt(r1436388);
        double r1436390 = log(r1436389);
        double r1436391 = y_re;
        double r1436392 = r1436390 * r1436391;
        double r1436393 = atan2(r1436386, r1436384);
        double r1436394 = y_im;
        double r1436395 = r1436393 * r1436394;
        double r1436396 = r1436392 - r1436395;
        double r1436397 = exp(r1436396);
        double r1436398 = r1436390 * r1436394;
        double r1436399 = r1436393 * r1436391;
        double r1436400 = r1436398 + r1436399;
        double r1436401 = cos(r1436400);
        double r1436402 = r1436397 * r1436401;
        return r1436402;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1436403 = x_re;
        double r1436404 = -1.8407009899098441e-06;
        bool r1436405 = r1436403 <= r1436404;
        double r1436406 = -r1436403;
        double r1436407 = log(r1436406);
        double r1436408 = y_re;
        double r1436409 = r1436407 * r1436408;
        double r1436410 = y_im;
        double r1436411 = x_im;
        double r1436412 = atan2(r1436411, r1436403);
        double r1436413 = r1436410 * r1436412;
        double r1436414 = r1436409 - r1436413;
        double r1436415 = exp(r1436414);
        double r1436416 = -2.9130712528480477e-164;
        bool r1436417 = r1436403 <= r1436416;
        double r1436418 = r1436403 * r1436403;
        double r1436419 = r1436411 * r1436411;
        double r1436420 = r1436418 + r1436419;
        double r1436421 = sqrt(r1436420);
        double r1436422 = log(r1436421);
        double r1436423 = r1436408 * r1436422;
        double r1436424 = r1436423 - r1436413;
        double r1436425 = exp(r1436424);
        double r1436426 = -4.2640896224449e-310;
        bool r1436427 = r1436403 <= r1436426;
        double r1436428 = log(r1436403);
        double r1436429 = r1436408 * r1436428;
        double r1436430 = r1436429 - r1436413;
        double r1436431 = exp(r1436430);
        double r1436432 = r1436427 ? r1436415 : r1436431;
        double r1436433 = r1436417 ? r1436425 : r1436432;
        double r1436434 = r1436405 ? r1436415 : r1436433;
        return r1436434;
}

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 < -1.8407009899098441e-06 or -2.9130712528480477e-164 < x.re < -4.2640896224449e-310

    1. Initial program 36.3

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

    2. Taylor expanded around 0 21.2

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

    3. Taylor expanded around -inf 4.1

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

    4. Simplified4.1

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

    if -1.8407009899098441e-06 < x.re < -2.9130712528480477e-164

    1. Initial program 15.9

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

    2. Taylor expanded around 0 8.3

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

    if -4.2640896224449e-310 < x.re

    1. Initial program 34.9

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

    2. Taylor expanded around 0 22.3

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

    3. Taylor expanded around inf 11.9

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -1.8407009899098441 \cdot 10^{-06}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -2.9130712528480477 \cdot 10^{-164}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -4.2640896224449 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))))