Average Error: 30.9 → 0.2
Time: 4.0m
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 r652329 = x_re;
        double r652330 = r652329 * r652329;
        double r652331 = x_im;
        double r652332 = r652331 * r652331;
        double r652333 = r652330 + r652332;
        double r652334 = sqrt(r652333);
        double r652335 = log(r652334);
        double r652336 = y_re;
        double r652337 = r652335 * r652336;
        double r652338 = atan2(r652331, r652329);
        double r652339 = y_im;
        double r652340 = r652338 * r652339;
        double r652341 = r652337 - r652340;
        double r652342 = exp(r652341);
        double r652343 = r652335 * r652339;
        double r652344 = r652338 * r652336;
        double r652345 = r652343 + r652344;
        double r652346 = sin(r652345);
        double r652347 = r652342 * r652346;
        return r652347;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r652348 = y_im;
        double r652349 = x_re;
        double r652350 = x_im;
        double r652351 = hypot(r652349, r652350);
        double r652352 = log(r652351);
        double r652353 = atan2(r652350, r652349);
        double r652354 = y_re;
        double r652355 = r652353 * r652354;
        double r652356 = fma(r652348, r652352, r652355);
        double r652357 = sin(r652356);
        double r652358 = r652353 * r652348;
        double r652359 = r652354 * r652352;
        double r652360 = r652358 - r652359;
        double r652361 = exp(r652360);
        double r652362 = r652357 / r652361;
        return r652362;
}

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 30.9

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

  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 2019128 +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)))))