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 r442184 = x_re;
        double r442185 = y_re;
        double r442186 = r442184 * r442185;
        double r442187 = x_im;
        double r442188 = y_im;
        double r442189 = r442187 * r442188;
        double r442190 = r442186 + r442189;
        double r442191 = r442185 * r442185;
        double r442192 = r442188 * r442188;
        double r442193 = r442191 + r442192;
        double r442194 = r442190 / r442193;
        return r442194;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r442195 = x_re;
        double r442196 = y_re;
        double r442197 = r442195 * r442196;
        double r442198 = x_im;
        double r442199 = y_im;
        double r442200 = r442198 * r442199;
        double r442201 = r442197 + r442200;
        double r442202 = r442196 * r442196;
        double r442203 = r442199 * r442199;
        double r442204 = r442202 + r442203;
        double r442205 = r442201 / r442204;
        return r442205;
}

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 2019104 
(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))))