Average Error: 1.1 → 1.1
Time: 3.7s
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 r1657441 = x_re;
        double r1657442 = y_re;
        double r1657443 = r1657441 * r1657442;
        double r1657444 = x_im;
        double r1657445 = y_im;
        double r1657446 = r1657444 * r1657445;
        double r1657447 = r1657443 + r1657446;
        double r1657448 = r1657442 * r1657442;
        double r1657449 = r1657445 * r1657445;
        double r1657450 = r1657448 + r1657449;
        double r1657451 = r1657447 / r1657450;
        return r1657451;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1657452 = x_re;
        double r1657453 = y_re;
        double r1657454 = r1657452 * r1657453;
        double r1657455 = x_im;
        double r1657456 = y_im;
        double r1657457 = r1657455 * r1657456;
        double r1657458 = r1657454 + r1657457;
        double r1657459 = r1657453 * r1657453;
        double r1657460 = r1657456 * r1657456;
        double r1657461 = r1657459 + r1657460;
        double r1657462 = r1657458 / r1657461;
        return r1657462;
}

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 2019128 +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))))