Average Error: 1.1 → 1.1
Time: 3.8s
Precision: 64
\[\frac{\left(\frac{\left(x.re \cdot y.re\right)}{\left(x.im \cdot y.im\right)}\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
↓
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
\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)}
↓
\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 r1272784 = x_re;
        double r1272785 = y_re;
        double r1272786 = r1272784 * r1272785;
        double r1272787 = x_im;
        double r1272788 = y_im;
        double r1272789 = r1272787 * r1272788;
        double r1272790 = r1272786 + r1272789;
        double r1272791 = r1272785 * r1272785;
        double r1272792 = r1272788 * r1272788;
        double r1272793 = r1272791 + r1272792;
        double r1272794 = r1272790 / r1272793;
        return r1272794;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1272795 = x_re;
        double r1272796 = y_re;
        double r1272797 = r1272795 * r1272796;
        double r1272798 = x_im;
        double r1272799 = y_im;
        double r1272800 = r1272798 * r1272799;
        double r1272801 = r1272797 + r1272800;
        double r1272802 = r1272796 * r1272796;
        double r1272803 = r1272799 * r1272799;
        double r1272804 = r1272802 + r1272803;
        double r1272805 = r1272801 / r1272804;
        return r1272805;
}

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

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

  2. 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 2019124 +o rules:numerics
(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))))