Average Error: 30.6 → 0.1
Time: 3.9m
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)}}\]
double f(double x_re, double x_im, double y_re, double y_im) {
        double r361920 = x_re;
        double r361921 = r361920 * r361920;
        double r361922 = x_im;
        double r361923 = r361922 * r361922;
        double r361924 = r361921 + r361923;
        double r361925 = sqrt(r361924);
        double r361926 = log(r361925);
        double r361927 = y_re;
        double r361928 = r361926 * r361927;
        double r361929 = atan2(r361922, r361920);
        double r361930 = y_im;
        double r361931 = r361929 * r361930;
        double r361932 = r361928 - r361931;
        double r361933 = exp(r361932);
        double r361934 = r361926 * r361930;
        double r361935 = r361929 * r361927;
        double r361936 = r361934 + r361935;
        double r361937 = sin(r361936);
        double r361938 = r361933 * r361937;
        return r361938;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r361939 = y_im;
        double r361940 = x_re;
        double r361941 = x_im;
        double r361942 = hypot(r361940, r361941);
        double r361943 = log(r361942);
        double r361944 = atan2(r361941, r361940);
        double r361945 = y_re;
        double r361946 = r361944 * r361945;
        double r361947 = fma(r361939, r361943, r361946);
        double r361948 = sin(r361947);
        double r361949 = r361944 * r361939;
        double r361950 = r361945 * r361943;
        double r361951 = r361949 - r361950;
        double r361952 = exp(r361951);
        double r361953 = r361948 / r361952;
        return r361953;
}


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)}}

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 30.6

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019102 +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)))))