Average Error: 1.1 → 1.1
Time: 3.8s
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 r1843280 = x_re;
        double r1843281 = y_re;
        double r1843282 = r1843280 * r1843281;
        double r1843283 = x_im;
        double r1843284 = y_im;
        double r1843285 = r1843283 * r1843284;
        double r1843286 = r1843282 + r1843285;
        double r1843287 = r1843281 * r1843281;
        double r1843288 = r1843284 * r1843284;
        double r1843289 = r1843287 + r1843288;
        double r1843290 = r1843286 / r1843289;
        return r1843290;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1843291 = x_re;
        double r1843292 = y_re;
        double r1843293 = r1843291 * r1843292;
        double r1843294 = x_im;
        double r1843295 = y_im;
        double r1843296 = r1843294 * r1843295;
        double r1843297 = r1843293 + r1843296;
        double r1843298 = r1843292 * r1843292;
        double r1843299 = r1843295 * r1843295;
        double r1843300 = r1843298 + r1843299;
        double r1843301 = r1843297 / r1843300;
        return r1843301;
}

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 2019125 
(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))))