Average Error: 31.4 → 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 r1002254 = x_re;
        double r1002255 = r1002254 * r1002254;
        double r1002256 = x_im;
        double r1002257 = r1002256 * r1002256;
        double r1002258 = r1002255 + r1002257;
        double r1002259 = sqrt(r1002258);
        double r1002260 = log(r1002259);
        double r1002261 = y_re;
        double r1002262 = r1002260 * r1002261;
        double r1002263 = atan2(r1002256, r1002254);
        double r1002264 = y_im;
        double r1002265 = r1002263 * r1002264;
        double r1002266 = r1002262 - r1002265;
        double r1002267 = exp(r1002266);
        double r1002268 = r1002260 * r1002264;
        double r1002269 = r1002263 * r1002261;
        double r1002270 = r1002268 + r1002269;
        double r1002271 = sin(r1002270);
        double r1002272 = r1002267 * r1002271;
        return r1002272;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1002273 = y_im;
        double r1002274 = x_re;
        double r1002275 = x_im;
        double r1002276 = hypot(r1002274, r1002275);
        double r1002277 = log(r1002276);
        double r1002278 = atan2(r1002275, r1002274);
        double r1002279 = y_re;
        double r1002280 = r1002278 * r1002279;
        double r1002281 = fma(r1002273, r1002277, r1002280);
        double r1002282 = sin(r1002281);
        double r1002283 = r1002278 * r1002273;
        double r1002284 = r1002279 * r1002277;
        double r1002285 = r1002283 - r1002284;
        double r1002286 = exp(r1002285);
        double r1002287 = r1002282 / r1002286;
        return r1002287;
}

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 31.4

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

  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 2019130 +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)))))