Average Error: 1.1 → 1.1
Time: 1.1m
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 r1128286 = x_im;
        double r1128287 = y_re;
        double r1128288 = r1128286 * r1128287;
        double r1128289 = x_re;
        double r1128290 = y_im;
        double r1128291 = r1128289 * r1128290;
        double r1128292 = r1128288 - r1128291;
        double r1128293 = r1128287 * r1128287;
        double r1128294 = r1128290 * r1128290;
        double r1128295 = r1128293 + r1128294;
        double r1128296 = r1128292 / r1128295;
        return r1128296;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1128297 = x_im;
        double r1128298 = y_re;
        double r1128299 = r1128297 * r1128298;
        double r1128300 = /*Error: no posit support in C */;
        double r1128301 = x_re;
        double r1128302 = y_im;
        double r1128303 = /*Error: no posit support in C */;
        double r1128304 = /*Error: no posit support in C */;
        double r1128305 = r1128298 * r1128298;
        double r1128306 = r1128302 * r1128302;
        double r1128307 = r1128305 + r1128306;
        double r1128308 = r1128304 / r1128307;
        return r1128308;
}

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 +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))))