Average Error: 1.1 → 1.1
Time: 1.2m
Precision: 64
\[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
↓
\[\frac{\left(\mathsf{qms}\left(\left(\left(x.im \cdot y.re\right)\right), x.re, y.im\right)\right)}{y.re \cdot y.re + y.im \cdot y.im}\]
\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(\mathsf{qms}\left(\left(\left(x.im \cdot y.re\right)\right), x.re, y.im\right)\right)}{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 r947843 = x_im;
        double r947844 = y_re;
        double r947845 = r947843 * r947844;
        double r947846 = x_re;
        double r947847 = y_im;
        double r947848 = r947846 * r947847;
        double r947849 = r947845 - r947848;
        double r947850 = r947844 * r947844;
        double r947851 = r947847 * r947847;
        double r947852 = r947850 + r947851;
        double r947853 = r947849 / r947852;
        return r947853;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r947854 = x_im;
        double r947855 = y_re;
        double r947856 = r947854 * r947855;
        double r947857 = /*Error: no posit support in C */;
        double r947858 = x_re;
        double r947859 = y_im;
        double r947860 = /*Error: no posit support in C */;
        double r947861 = /*Error: no posit support in C */;
        double r947862 = r947855 * r947855;
        double r947863 = r947859 * r947859;
        double r947864 = r947862 + r947863;
        double r947865 = r947861 / r947864;
        return r947865;
}

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

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

  4. Applied insert-quire-fdp-sub1.1

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

  5. Final simplification1.1

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

Reproduce

herbie shell --seed 2019153 
(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))))