Average Error: 1.1 → 1.1
Time: 11.2s
Precision: 64
\[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
↓
\[\frac{\frac{x.im \cdot y.re + x.re \cdot y.im}{\frac{x.im \cdot y.re + x.re \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}}{y.re \cdot y.re + y.im \cdot y.im}\]
\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 r1814872 = x_im;
        double r1814873 = y_re;
        double r1814874 = r1814872 * r1814873;
        double r1814875 = x_re;
        double r1814876 = y_im;
        double r1814877 = r1814875 * r1814876;
        double r1814878 = r1814874 - r1814877;
        double r1814879 = r1814873 * r1814873;
        double r1814880 = r1814876 * r1814876;
        double r1814881 = r1814879 + r1814880;
        double r1814882 = r1814878 / r1814881;
        return r1814882;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1814883 = x_im;
        double r1814884 = y_re;
        double r1814885 = r1814883 * r1814884;
        double r1814886 = x_re;
        double r1814887 = y_im;
        double r1814888 = r1814886 * r1814887;
        double r1814889 = r1814885 + r1814888;
        double r1814890 = r1814885 - r1814888;
        double r1814891 = r1814889 / r1814890;
        double r1814892 = r1814889 / r1814891;
        double r1814893 = r1814884 * r1814884;
        double r1814894 = r1814887 * r1814887;
        double r1814895 = r1814893 + r1814894;
        double r1814896 = r1814892 / r1814895;
        return r1814896;
}

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 
(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))))