Average Error: 1.1 → 1.1
Time: 4.4s
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 r229822 = x_re;
        double r229823 = y_re;
        double r229824 = r229822 * r229823;
        double r229825 = x_im;
        double r229826 = y_im;
        double r229827 = r229825 * r229826;
        double r229828 = r229824 + r229827;
        double r229829 = r229823 * r229823;
        double r229830 = r229826 * r229826;
        double r229831 = r229829 + r229830;
        double r229832 = r229828 / r229831;
        return r229832;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r229833 = x_re;
        double r229834 = y_re;
        double r229835 = r229833 * r229834;
        double r229836 = x_im;
        double r229837 = y_im;
        double r229838 = r229836 * r229837;
        double r229839 = r229835 + r229838;
        double r229840 = r229834 * r229834;
        double r229841 = r229837 * r229837;
        double r229842 = r229840 + r229841;
        double r229843 = r229839 / r229842;
        return r229843;
}

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 2019132 +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))))