Average Error: 30.8 → 0.2
Time: 3.8m
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)}}\]
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)
↓
\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)}}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r362313 = x_re;
        double r362314 = r362313 * r362313;
        double r362315 = x_im;
        double r362316 = r362315 * r362315;
        double r362317 = r362314 + r362316;
        double r362318 = sqrt(r362317);
        double r362319 = log(r362318);
        double r362320 = y_re;
        double r362321 = r362319 * r362320;
        double r362322 = atan2(r362315, r362313);
        double r362323 = y_im;
        double r362324 = r362322 * r362323;
        double r362325 = r362321 - r362324;
        double r362326 = exp(r362325);
        double r362327 = r362319 * r362323;
        double r362328 = r362322 * r362320;
        double r362329 = r362327 + r362328;
        double r362330 = sin(r362329);
        double r362331 = r362326 * r362330;
        return r362331;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r362332 = y_im;
        double r362333 = x_re;
        double r362334 = x_im;
        double r362335 = hypot(r362333, r362334);
        double r362336 = log(r362335);
        double r362337 = atan2(r362334, r362333);
        double r362338 = y_re;
        double r362339 = r362337 * r362338;
        double r362340 = fma(r362332, r362336, r362339);
        double r362341 = sin(r362340);
        double r362342 = r362337 * r362332;
        double r362343 = r362338 * r362336;
        double r362344 = r362342 - r362343;
        double r362345 = exp(r362344);
        double r362346 = r362341 / r362345;
        return r362346;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 30.8

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

  2. Simplified0.2

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

  3. Final simplification0.2

    \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)}}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(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)))))