Average Error: 1.1 → 1.1
Time: 4.3s
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 r1331739 = x_re;
        double r1331740 = y_re;
        double r1331741 = r1331739 * r1331740;
        double r1331742 = x_im;
        double r1331743 = y_im;
        double r1331744 = r1331742 * r1331743;
        double r1331745 = r1331741 + r1331744;
        double r1331746 = r1331740 * r1331740;
        double r1331747 = r1331743 * r1331743;
        double r1331748 = r1331746 + r1331747;
        double r1331749 = r1331745 / r1331748;
        return r1331749;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1331750 = x_re;
        double r1331751 = y_re;
        double r1331752 = r1331750 * r1331751;
        double r1331753 = x_im;
        double r1331754 = y_im;
        double r1331755 = r1331753 * r1331754;
        double r1331756 = r1331752 + r1331755;
        double r1331757 = r1331751 * r1331751;
        double r1331758 = r1331754 * r1331754;
        double r1331759 = r1331757 + r1331758;
        double r1331760 = r1331756 / r1331759;
        return r1331760;
}

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 2019128 
(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))))