Average Error: 1.1 → 1.0
Time: 2.2m
Precision: 64
\[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
↓
\[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\left(\mathsf{qma}\left(\left(\left(y.re \cdot y.re\right)\right), y.im, y.im\right)\right)\right)}\]
\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}
↓
\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\left(\mathsf{qma}\left(\left(\left(y.re \cdot y.re\right)\right), y.im, y.im\right)\right)\right)}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2445745 = x_im;
        double r2445746 = y_re;
        double r2445747 = r2445745 * r2445746;
        double r2445748 = x_re;
        double r2445749 = y_im;
        double r2445750 = r2445748 * r2445749;
        double r2445751 = r2445747 - r2445750;
        double r2445752 = r2445746 * r2445746;
        double r2445753 = r2445749 * r2445749;
        double r2445754 = r2445752 + r2445753;
        double r2445755 = r2445751 / r2445754;
        return r2445755;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2445756 = x_im;
        double r2445757 = y_re;
        double r2445758 = r2445756 * r2445757;
        double r2445759 = x_re;
        double r2445760 = y_im;
        double r2445761 = r2445759 * r2445760;
        double r2445762 = r2445758 - r2445761;
        double r2445763 = r2445757 * r2445757;
        double r2445764 = /*Error: no posit support in C */;
        double r2445765 = /*Error: no posit support in C */;
        double r2445766 = /*Error: no posit support in C */;
        double r2445767 = r2445762 / r2445766;
        return r2445767;
}

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

    \[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
  2. Using strategy rm

  3. Applied introduce-quire1.1

    \[\leadsto \frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\color{blue}{\left(\left(\left(y.re \cdot y.re\right)\right)\right)}}{\left(y.im \cdot y.im\right)}\right)}\]

  4. Applied insert-quire-fdp-add1.0

    \[\leadsto \frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\left(y.re \cdot y.re\right)\right), y.im, y.im\right)\right)\right)}}\]

  5. Final simplification1.0

    \[\leadsto \frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\left(\mathsf{qma}\left(\left(\left(y.re \cdot y.re\right)\right), y.im, y.im\right)\right)\right)}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/.p16 (-.p16 (*.p16 x.im y.re) (*.p16 x.re y.im)) (+.p16 (*.p16 y.re y.re) (*.p16 y.im y.im))))