Average Error: 1.1 → 1.1
Time: 5.1s
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 r734884 = x_re;
        double r734885 = y_re;
        double r734886 = r734884 * r734885;
        double r734887 = x_im;
        double r734888 = y_im;
        double r734889 = r734887 * r734888;
        double r734890 = r734886 + r734889;
        double r734891 = r734885 * r734885;
        double r734892 = r734888 * r734888;
        double r734893 = r734891 + r734892;
        double r734894 = r734890 / r734893;
        return r734894;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r734895 = x_re;
        double r734896 = y_re;
        double r734897 = r734895 * r734896;
        double r734898 = x_im;
        double r734899 = y_im;
        double r734900 = r734898 * r734899;
        double r734901 = r734897 + r734900;
        double r734902 = r734896 * r734896;
        double r734903 = r734899 * r734899;
        double r734904 = r734902 + r734903;
        double r734905 = r734901 / r734904;
        return r734905;
}

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 2019107 +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))))