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 r1743440 = x_re;
        double r1743441 = y_re;
        double r1743442 = r1743440 * r1743441;
        double r1743443 = x_im;
        double r1743444 = y_im;
        double r1743445 = r1743443 * r1743444;
        double r1743446 = r1743442 + r1743445;
        double r1743447 = r1743441 * r1743441;
        double r1743448 = r1743444 * r1743444;
        double r1743449 = r1743447 + r1743448;
        double r1743450 = r1743446 / r1743449;
        return r1743450;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1743451 = x_re;
        double r1743452 = y_re;
        double r1743453 = r1743451 * r1743452;
        double r1743454 = x_im;
        double r1743455 = y_im;
        double r1743456 = r1743454 * r1743455;
        double r1743457 = r1743453 + r1743456;
        double r1743458 = r1743452 * r1743452;
        double r1743459 = r1743455 * r1743455;
        double r1743460 = r1743458 + r1743459;
        double r1743461 = r1743457 / r1743460;
        return r1743461;
}

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 2019135 +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))))