Average Error: 1.1 → 1.1
Time: 9.3s
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 r493986 = x_im;
        double r493987 = y_re;
        double r493988 = r493986 * r493987;
        double r493989 = x_re;
        double r493990 = y_im;
        double r493991 = r493989 * r493990;
        double r493992 = r493988 - r493991;
        double r493993 = r493987 * r493987;
        double r493994 = r493990 * r493990;
        double r493995 = r493993 + r493994;
        double r493996 = r493992 / r493995;
        return r493996;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r493997 = x_im;
        double r493998 = y_re;
        double r493999 = r493997 * r493998;
        double r494000 = x_re;
        double r494001 = y_im;
        double r494002 = r494000 * r494001;
        double r494003 = r493999 - r494002;
        double r494004 = r493998 * r493998;
        double r494005 = r494001 * r494001;
        double r494006 = r494004 + r494005;
        double r494007 = r494003 / r494006;
        return r494007;
}

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 2019120 
(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))))