Average Error: 33.6 → 22.6
Time: 1.1m
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -1.91205330543206 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log x.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]
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 \sin \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 -1.91205330543206 \cdot 10^{-310}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log x.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \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 r1590176 = x_re;
        double r1590177 = r1590176 * r1590176;
        double r1590178 = x_im;
        double r1590179 = r1590178 * r1590178;
        double r1590180 = r1590177 + r1590179;
        double r1590181 = sqrt(r1590180);
        double r1590182 = log(r1590181);
        double r1590183 = y_re;
        double r1590184 = r1590182 * r1590183;
        double r1590185 = atan2(r1590178, r1590176);
        double r1590186 = y_im;
        double r1590187 = r1590185 * r1590186;
        double r1590188 = r1590184 - r1590187;
        double r1590189 = exp(r1590188);
        double r1590190 = r1590182 * r1590186;
        double r1590191 = r1590185 * r1590183;
        double r1590192 = r1590190 + r1590191;
        double r1590193 = sin(r1590192);
        double r1590194 = r1590189 * r1590193;
        return r1590194;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1590195 = x_re;
        double r1590196 = -1.91205330543206e-310;
        bool r1590197 = r1590195 <= r1590196;
        double r1590198 = y_re;
        double r1590199 = r1590195 * r1590195;
        double r1590200 = x_im;
        double r1590201 = r1590200 * r1590200;
        double r1590202 = r1590199 + r1590201;
        double r1590203 = sqrt(r1590202);
        double r1590204 = log(r1590203);
        double r1590205 = r1590198 * r1590204;
        double r1590206 = atan2(r1590200, r1590195);
        double r1590207 = y_im;
        double r1590208 = r1590206 * r1590207;
        double r1590209 = r1590205 - r1590208;
        double r1590210 = exp(r1590209);
        double r1590211 = r1590206 * r1590198;
        double r1590212 = -r1590195;
        double r1590213 = log(r1590212);
        double r1590214 = r1590207 * r1590213;
        double r1590215 = r1590211 + r1590214;
        double r1590216 = sin(r1590215);
        double r1590217 = r1590210 * r1590216;
        double r1590218 = log(r1590195);
        double r1590219 = r1590207 * r1590218;
        double r1590220 = r1590211 + r1590219;
        double r1590221 = sin(r1590220);
        double r1590222 = r1590221 * r1590210;
        double r1590223 = r1590197 ? r1590217 : r1590222;
        return r1590223;
}

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 < -1.91205330543206e-310

    1. Initial program 31.8

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

    2. Taylor expanded around -inf 20.3

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

    3. Simplified20.3

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

    if -1.91205330543206e-310 < x.re

    1. Initial program 35.3

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

    2. Taylor expanded around inf 24.7

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

  4. Final simplification22.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -1.91205330543206 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log x.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))