Average Error: 1.1 → 1.1
Time: 19.5s
Precision: 64
\[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
↓
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
\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{x.im \cdot y.re - x.re \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 r2242277 = x_im;
        double r2242278 = y_re;
        double r2242279 = r2242277 * r2242278;
        double r2242280 = x_re;
        double r2242281 = y_im;
        double r2242282 = r2242280 * r2242281;
        double r2242283 = r2242279 - r2242282;
        double r2242284 = r2242278 * r2242278;
        double r2242285 = r2242281 * r2242281;
        double r2242286 = r2242284 + r2242285;
        double r2242287 = r2242283 / r2242286;
        return r2242287;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2242288 = x_im;
        double r2242289 = y_re;
        double r2242290 = r2242288 * r2242289;
        double r2242291 = x_re;
        double r2242292 = y_im;
        double r2242293 = r2242291 * r2242292;
        double r2242294 = r2242290 - r2242293;
        double r2242295 = r2242289 * r2242289;
        double r2242296 = r2242292 * r2242292;
        double r2242297 = r2242295 + r2242296;
        double r2242298 = r2242294 / r2242297;
        return r2242298;
}

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. Final simplification1.1

    \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

Reproduce

herbie shell --seed 2019134 +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))))