Average Error: 1.1 → 1.1
Time: 13.9s
Precision: 64
\[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
↓
\[\frac{\frac{x.im \cdot y.re + x.re \cdot y.im}{\frac{x.im \cdot y.re + x.re \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}}{y.re \cdot y.re + y.im \cdot y.im}\]
\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}
↓
\frac{\frac{x.im \cdot y.re + x.re \cdot y.im}{\frac{x.im \cdot y.re + x.re \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}}{y.re \cdot y.re + y.im \cdot y.im}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2924844 = x_im;
        double r2924845 = y_re;
        double r2924846 = r2924844 * r2924845;
        double r2924847 = x_re;
        double r2924848 = y_im;
        double r2924849 = r2924847 * r2924848;
        double r2924850 = r2924846 - r2924849;
        double r2924851 = r2924845 * r2924845;
        double r2924852 = r2924848 * r2924848;
        double r2924853 = r2924851 + r2924852;
        double r2924854 = r2924850 / r2924853;
        return r2924854;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2924855 = x_im;
        double r2924856 = y_re;
        double r2924857 = r2924855 * r2924856;
        double r2924858 = x_re;
        double r2924859 = y_im;
        double r2924860 = r2924858 * r2924859;
        double r2924861 = r2924857 + r2924860;
        double r2924862 = r2924857 - r2924860;
        double r2924863 = r2924861 / r2924862;
        double r2924864 = r2924861 / r2924863;
        double r2924865 = r2924856 * r2924856;
        double r2924866 = r2924859 * r2924859;
        double r2924867 = r2924865 + r2924866;
        double r2924868 = r2924864 / r2924867;
        return r2924868;
}

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 1.1

    \[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
  2. Using strategy rm

  3. Applied p16-flip--2.0

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\left(\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right)\right) - \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right)\right)\right)}{\left(\frac{\left(x.im \cdot y.re\right)}{\left(x.re \cdot y.im\right)}\right)}\right)}}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
  4. Using strategy rm

  5. Applied difference-of-squares1.9

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(x.im \cdot y.re\right)}{\left(x.re \cdot y.im\right)}\right) \cdot \left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)\right)}}{\left(\frac{\left(x.im \cdot y.re\right)}{\left(x.re \cdot y.im\right)}\right)}\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]

  6. Applied associate-/l*1.1

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(x.im \cdot y.re\right)}{\left(x.re \cdot y.im\right)}\right)}{\left(\frac{\left(\frac{\left(x.im \cdot y.re\right)}{\left(x.re \cdot y.im\right)}\right)}{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}\right)}\right)}}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]

  7. Final simplification1.1

    \[\leadsto \frac{\frac{x.im \cdot y.re + x.re \cdot y.im}{\frac{x.im \cdot y.re + x.re \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}}{y.re \cdot y.re + y.im \cdot y.im}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/.p16 (-.p16 (*.p16 x.im y.re) (*.p16 x.re y.im)) (+.p16 (*.p16 y.re y.re) (*.p16 y.im y.im))))