Average Error: 1.1 → 1.1
Time: 5.3s
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 r1801541 = x_re;
        double r1801542 = y_re;
        double r1801543 = r1801541 * r1801542;
        double r1801544 = x_im;
        double r1801545 = y_im;
        double r1801546 = r1801544 * r1801545;
        double r1801547 = r1801543 + r1801546;
        double r1801548 = r1801542 * r1801542;
        double r1801549 = r1801545 * r1801545;
        double r1801550 = r1801548 + r1801549;
        double r1801551 = r1801547 / r1801550;
        return r1801551;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1801552 = x_re;
        double r1801553 = y_re;
        double r1801554 = r1801552 * r1801553;
        double r1801555 = x_im;
        double r1801556 = y_im;
        double r1801557 = r1801555 * r1801556;
        double r1801558 = r1801554 + r1801557;
        double r1801559 = r1801553 * r1801553;
        double r1801560 = r1801556 * r1801556;
        double r1801561 = r1801559 + r1801560;
        double r1801562 = r1801558 / r1801561;
        return r1801562;
}

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 2019134 
(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))))