Average Error: 1.1 → 1.1
Time: 4.7s
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 r845158 = x_re;
        double r845159 = y_re;
        double r845160 = r845158 * r845159;
        double r845161 = x_im;
        double r845162 = y_im;
        double r845163 = r845161 * r845162;
        double r845164 = r845160 + r845163;
        double r845165 = r845159 * r845159;
        double r845166 = r845162 * r845162;
        double r845167 = r845165 + r845166;
        double r845168 = r845164 / r845167;
        return r845168;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r845169 = x_re;
        double r845170 = y_re;
        double r845171 = r845169 * r845170;
        double r845172 = x_im;
        double r845173 = y_im;
        double r845174 = r845172 * r845173;
        double r845175 = r845171 + r845174;
        double r845176 = r845170 * r845170;
        double r845177 = r845173 * r845173;
        double r845178 = r845176 + r845177;
        double r845179 = r845175 / r845178;
        return r845179;
}

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 2019119 +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))))