Average Error: 1.1 → 1.1
Time: 4.1s
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 r499623 = x_re;
        double r499624 = y_re;
        double r499625 = r499623 * r499624;
        double r499626 = x_im;
        double r499627 = y_im;
        double r499628 = r499626 * r499627;
        double r499629 = r499625 + r499628;
        double r499630 = r499624 * r499624;
        double r499631 = r499627 * r499627;
        double r499632 = r499630 + r499631;
        double r499633 = r499629 / r499632;
        return r499633;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r499634 = x_re;
        double r499635 = y_re;
        double r499636 = r499634 * r499635;
        double r499637 = x_im;
        double r499638 = y_im;
        double r499639 = r499637 * r499638;
        double r499640 = r499636 + r499639;
        double r499641 = r499635 * r499635;
        double r499642 = r499638 * r499638;
        double r499643 = r499641 + r499642;
        double r499644 = r499640 / r499643;
        return r499644;
}

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 2019112 +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))))