Average Error: 1.1 → 1.1
Time: 10.6s
Precision: 64
\[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
↓
\[\frac{\frac{x.im \cdot y.re + x.re \cdot y.im}{\frac{x.im \cdot y.re + x.re \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}}{y.re \cdot y.re + y.im \cdot y.im}\]
\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{\frac{x.im \cdot y.re + x.re \cdot y.im}{\frac{x.im \cdot y.re + x.re \cdot y.im}{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 r1084055 = x_im;
        double r1084056 = y_re;
        double r1084057 = r1084055 * r1084056;
        double r1084058 = x_re;
        double r1084059 = y_im;
        double r1084060 = r1084058 * r1084059;
        double r1084061 = r1084057 - r1084060;
        double r1084062 = r1084056 * r1084056;
        double r1084063 = r1084059 * r1084059;
        double r1084064 = r1084062 + r1084063;
        double r1084065 = r1084061 / r1084064;
        return r1084065;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1084066 = x_im;
        double r1084067 = y_re;
        double r1084068 = r1084066 * r1084067;
        double r1084069 = x_re;
        double r1084070 = y_im;
        double r1084071 = r1084069 * r1084070;
        double r1084072 = r1084068 + r1084071;
        double r1084073 = r1084068 - r1084071;
        double r1084074 = r1084072 / r1084073;
        double r1084075 = r1084072 / r1084074;
        double r1084076 = r1084067 * r1084067;
        double r1084077 = r1084070 * r1084070;
        double r1084078 = r1084076 + r1084077;
        double r1084079 = r1084075 / r1084078;
        return r1084079;
}

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. Using strategy rm

  3. Applied p16-flip--2.1

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\left(\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right)\right) - \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right)\right)\right)}{\left(\frac{\left(x.im \cdot y.re\right)}{\left(x.re \cdot y.im\right)}\right)}\right)}}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
  4. Using strategy rm

  5. Applied difference-of-squares2.0

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(x.im \cdot y.re\right)}{\left(x.re \cdot y.im\right)}\right) \cdot \left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)\right)}}{\left(\frac{\left(x.im \cdot y.re\right)}{\left(x.re \cdot y.im\right)}\right)}\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]

  6. Applied associate-/l*1.1

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(x.im \cdot y.re\right)}{\left(x.re \cdot y.im\right)}\right)}{\left(\frac{\left(\frac{\left(x.im \cdot y.re\right)}{\left(x.re \cdot y.im\right)}\right)}{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}\right)}\right)}}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]

  7. Final simplification1.1

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

Reproduce

herbie shell --seed 2019119 +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))))