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

↓

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

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1100728 = x_re;
        double r1100729 = y_re;
        double r1100730 = r1100728 * r1100729;
        double r1100731 = x_im;
        double r1100732 = y_im;
        double r1100733 = r1100731 * r1100732;
        double r1100734 = r1100730 + r1100733;
        double r1100735 = r1100729 * r1100729;
        double r1100736 = r1100732 * r1100732;
        double r1100737 = r1100735 + r1100736;
        double r1100738 = r1100734 / r1100737;
        return r1100738;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1100739 = x_re;
        double r1100740 = y_re;
        double r1100741 = r1100739 * r1100740;
        double r1100742 = x_im;
        double r1100743 = y_im;
        double r1100744 = r1100742 * r1100743;
        double r1100745 = r1100741 + r1100744;
        double r1100746 = r1100740 * r1100740;
        double r1100747 = r1100743 * r1100743;
        double r1100748 = r1100746 + r1100747;
        double r1100749 = r1100745 / r1100748;
        return r1100749;
}

\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}

↓

\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}

Derivation

Initial program 1.1
\[\frac{\left(\frac{\left(x.re \cdot y.re\right)}{\left(x.im \cdot y.im\right)}\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
Final simplification1.1
\[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

Reproduce

herbie shell --seed 2019101 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  (/.p16 (+.p16 (*.p16 x.re y.re) (*.p16 x.im y.im)) (+.p16 (*.p16 y.re y.re) (*.p16 y.im y.im))))

Error

Derivation

Reproduce