Average Error: 1.1 → 1.1
Time: 16.6s
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 r1332463 = x_im;
        double r1332464 = y_re;
        double r1332465 = r1332463 * r1332464;
        double r1332466 = x_re;
        double r1332467 = y_im;
        double r1332468 = r1332466 * r1332467;
        double r1332469 = r1332465 - r1332468;
        double r1332470 = r1332464 * r1332464;
        double r1332471 = r1332467 * r1332467;
        double r1332472 = r1332470 + r1332471;
        double r1332473 = r1332469 / r1332472;
        return r1332473;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1332474 = x_im;
        double r1332475 = y_re;
        double r1332476 = r1332474 * r1332475;
        double r1332477 = x_re;
        double r1332478 = y_im;
        double r1332479 = r1332477 * r1332478;
        double r1332480 = r1332476 - r1332479;
        double r1332481 = r1332475 * r1332475;
        double r1332482 = r1332478 * r1332478;
        double r1332483 = r1332481 + r1332482;
        double r1332484 = r1332480 / r1332483;
        return r1332484;
}

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 2019130 +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))))