Average Error: 1.1 → 1.1
Time: 10.2s
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 r1563409 = x_im;
        double r1563410 = y_re;
        double r1563411 = r1563409 * r1563410;
        double r1563412 = x_re;
        double r1563413 = y_im;
        double r1563414 = r1563412 * r1563413;
        double r1563415 = r1563411 - r1563414;
        double r1563416 = r1563410 * r1563410;
        double r1563417 = r1563413 * r1563413;
        double r1563418 = r1563416 + r1563417;
        double r1563419 = r1563415 / r1563418;
        return r1563419;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1563420 = x_im;
        double r1563421 = y_re;
        double r1563422 = r1563420 * r1563421;
        double r1563423 = x_re;
        double r1563424 = y_im;
        double r1563425 = r1563423 * r1563424;
        double r1563426 = r1563422 - r1563425;
        double r1563427 = r1563421 * r1563421;
        double r1563428 = r1563424 * r1563424;
        double r1563429 = r1563427 + r1563428;
        double r1563430 = r1563426 / r1563429;
        return r1563430;
}

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 2019112 +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))))