Average Error: 1.1 → 1.1
Time: 13.4s
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 r2778117 = x_im;
        double r2778118 = y_re;
        double r2778119 = r2778117 * r2778118;
        double r2778120 = x_re;
        double r2778121 = y_im;
        double r2778122 = r2778120 * r2778121;
        double r2778123 = r2778119 - r2778122;
        double r2778124 = r2778118 * r2778118;
        double r2778125 = r2778121 * r2778121;
        double r2778126 = r2778124 + r2778125;
        double r2778127 = r2778123 / r2778126;
        return r2778127;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2778128 = x_im;
        double r2778129 = y_re;
        double r2778130 = r2778128 * r2778129;
        double r2778131 = x_re;
        double r2778132 = y_im;
        double r2778133 = r2778131 * r2778132;
        double r2778134 = r2778130 + r2778133;
        double r2778135 = r2778130 - r2778133;
        double r2778136 = r2778134 / r2778135;
        double r2778137 = r2778134 / r2778136;
        double r2778138 = r2778129 * r2778129;
        double r2778139 = r2778132 * r2778132;
        double r2778140 = r2778138 + r2778139;
        double r2778141 = r2778137 / r2778140;
        return r2778141;
}

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-squares1.9

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

  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 2019132 +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))))