Average Error: 1.1 → 1.1
Time: 4.2s
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 r740738 = x_re;
        double r740739 = y_re;
        double r740740 = r740738 * r740739;
        double r740741 = x_im;
        double r740742 = y_im;
        double r740743 = r740741 * r740742;
        double r740744 = r740740 + r740743;
        double r740745 = r740739 * r740739;
        double r740746 = r740742 * r740742;
        double r740747 = r740745 + r740746;
        double r740748 = r740744 / r740747;
        return r740748;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r740749 = x_re;
        double r740750 = y_re;
        double r740751 = r740749 * r740750;
        double r740752 = x_im;
        double r740753 = y_im;
        double r740754 = r740752 * r740753;
        double r740755 = r740751 + r740754;
        double r740756 = r740750 * r740750;
        double r740757 = r740753 * r740753;
        double r740758 = r740756 + r740757;
        double r740759 = r740755 / r740758;
        return r740759;
}

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 2019104 +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))))