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 r3688603 = x_im;
        double r3688604 = y_re;
        double r3688605 = r3688603 * r3688604;
        double r3688606 = x_re;
        double r3688607 = y_im;
        double r3688608 = r3688606 * r3688607;
        double r3688609 = r3688605 - r3688608;
        double r3688610 = r3688604 * r3688604;
        double r3688611 = r3688607 * r3688607;
        double r3688612 = r3688610 + r3688611;
        double r3688613 = r3688609 / r3688612;
        return r3688613;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r3688614 = x_im;
        double r3688615 = y_re;
        double r3688616 = r3688614 * r3688615;
        double r3688617 = x_re;
        double r3688618 = y_im;
        double r3688619 = r3688617 * r3688618;
        double r3688620 = r3688616 - r3688619;
        double r3688621 = r3688615 * r3688615;
        double r3688622 = r3688618 * r3688618;
        double r3688623 = r3688621 + r3688622;
        double r3688624 = r3688620 / r3688623;
        return r3688624;
}

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 2019133 +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))))