Average Error: 31.8 → 0.2
Time: 3.8m
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\frac{\sin \left(\mathsf{fma}\left(y.im, \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right), \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\]
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(\mathsf{fma}\left(y.im, \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right), \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r362372 = x_re;
        double r362373 = r362372 * r362372;
        double r362374 = x_im;
        double r362375 = r362374 * r362374;
        double r362376 = r362373 + r362375;
        double r362377 = sqrt(r362376);
        double r362378 = log(r362377);
        double r362379 = y_re;
        double r362380 = r362378 * r362379;
        double r362381 = atan2(r362374, r362372);
        double r362382 = y_im;
        double r362383 = r362381 * r362382;
        double r362384 = r362380 - r362383;
        double r362385 = exp(r362384);
        double r362386 = r362378 * r362382;
        double r362387 = r362381 * r362379;
        double r362388 = r362386 + r362387;
        double r362389 = sin(r362388);
        double r362390 = r362385 * r362389;
        return r362390;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r362391 = y_im;
        double r362392 = x_re;
        double r362393 = x_im;
        double r362394 = hypot(r362392, r362393);
        double r362395 = log(r362394);
        double r362396 = atan2(r362393, r362392);
        double r362397 = y_re;
        double r362398 = r362396 * r362397;
        double r362399 = fma(r362391, r362395, r362398);
        double r362400 = sin(r362399);
        double r362401 = r362396 * r362391;
        double r362402 = r362397 * r362395;
        double r362403 = r362401 - r362402;
        double r362404 = exp(r362403);
        double r362405 = r362400 / r362404;
        return r362405;
}

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 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. Simplified0.2

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

  3. Final simplification0.2

    \[\leadsto \frac{\sin \left(\mathsf{fma}\left(y.im, \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right), \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\]

Reproduce

herbie shell --seed 2019121 +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)))))