Average Error: 1.1 → 1.1
Time: 5.2s
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 r1590578 = x_re;
        double r1590579 = y_re;
        double r1590580 = r1590578 * r1590579;
        double r1590581 = x_im;
        double r1590582 = y_im;
        double r1590583 = r1590581 * r1590582;
        double r1590584 = r1590580 + r1590583;
        double r1590585 = r1590579 * r1590579;
        double r1590586 = r1590582 * r1590582;
        double r1590587 = r1590585 + r1590586;
        double r1590588 = r1590584 / r1590587;
        return r1590588;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1590589 = x_re;
        double r1590590 = y_re;
        double r1590591 = r1590589 * r1590590;
        double r1590592 = x_im;
        double r1590593 = y_im;
        double r1590594 = r1590592 * r1590593;
        double r1590595 = r1590591 + r1590594;
        double r1590596 = r1590590 * r1590590;
        double r1590597 = r1590593 * r1590593;
        double r1590598 = r1590596 + r1590597;
        double r1590599 = r1590595 / r1590598;
        return r1590599;
}

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 2019130 
(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))))