Average Error: 33.4 → 3.9
Time: 30.8s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sqrt[3]{y.re}\right)\right)\right)\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\]
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{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sqrt[3]{y.re}\right)\right)\right)\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 r1078814 = x_re;
        double r1078815 = r1078814 * r1078814;
        double r1078816 = x_im;
        double r1078817 = r1078816 * r1078816;
        double r1078818 = r1078815 + r1078817;
        double r1078819 = sqrt(r1078818);
        double r1078820 = log(r1078819);
        double r1078821 = y_re;
        double r1078822 = r1078820 * r1078821;
        double r1078823 = atan2(r1078816, r1078814);
        double r1078824 = y_im;
        double r1078825 = r1078823 * r1078824;
        double r1078826 = r1078822 - r1078825;
        double r1078827 = exp(r1078826);
        double r1078828 = r1078820 * r1078824;
        double r1078829 = r1078823 * r1078821;
        double r1078830 = r1078828 + r1078829;
        double r1078831 = sin(r1078830);
        double r1078832 = r1078827 * r1078831;
        return r1078832;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1078833 = y_im;
        double r1078834 = x_re;
        double r1078835 = x_im;
        double r1078836 = hypot(r1078834, r1078835);
        double r1078837 = log(r1078836);
        double r1078838 = y_re;
        double r1078839 = cbrt(r1078838);
        double r1078840 = r1078839 * r1078839;
        double r1078841 = atan2(r1078835, r1078834);
        double r1078842 = r1078840 * r1078841;
        double r1078843 = r1078842 * r1078839;
        double r1078844 = fma(r1078833, r1078837, r1078843);
        double r1078845 = sin(r1078844);
        double r1078846 = log1p(r1078845);
        double r1078847 = expm1(r1078846);
        double r1078848 = expm1(r1078847);
        double r1078849 = log1p(r1078848);
        double r1078850 = r1078841 * r1078833;
        double r1078851 = r1078838 * r1078837;
        double r1078852 = r1078850 - r1078851;
        double r1078853 = exp(r1078852);
        double r1078854 = r1078849 / r1078853;
        return r1078854;
}

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 33.4

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

  2. Simplified3.7

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

  4. Applied log1p-expm1-u3.7

    \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\]
  5. Using strategy rm

  6. Applied expm1-log1p-u3.7

    \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt3.9

    \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \sqrt[3]{y.re}\right)}\right)\right)\right)\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\]

  9. Applied associate-*r*3.9

    \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right)\right) \cdot \sqrt[3]{y.re}}\right)\right)\right)\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\]

  10. Final simplification3.9

    \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sqrt[3]{y.re}\right)\right)\right)\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\]

Reproduce

herbie shell --seed 2019169 +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)))))