Average Error: 0.4 → 0.4
Time: 29.7s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) - \left(x.im + x.im\right) \cdot x.im\right)\]
\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)
↓
x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) - \left(x.im + x.im\right) \cdot x.im\right)
double f(double x_re, double x_im) {
        double r4133163 = x_re;
        double r4133164 = r4133163 * r4133163;
        double r4133165 = x_im;
        double r4133166 = r4133165 * r4133165;
        double r4133167 = r4133164 - r4133166;
        double r4133168 = r4133167 * r4133163;
        double r4133169 = r4133163 * r4133165;
        double r4133170 = r4133165 * r4133163;
        double r4133171 = r4133169 + r4133170;
        double r4133172 = r4133171 * r4133165;
        double r4133173 = r4133168 - r4133172;
        return r4133173;
}
↓
double f(double x_re, double x_im) {
        double r4133174 = x_re;
        double r4133175 = x_im;
        double r4133176 = r4133174 - r4133175;
        double r4133177 = r4133175 + r4133174;
        double r4133178 = r4133176 * r4133177;
        double r4133179 = r4133175 + r4133175;
        double r4133180 = r4133179 * r4133175;
        double r4133181 = r4133178 - r4133180;
        double r4133182 = r4133174 * r4133181;
        return r4133182;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

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

  2. Simplified0.4

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

  3. Final simplification0.4

    \[\leadsto x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) - \left(x.im + x.im\right) \cdot x.im\right)\]

Reproduce

herbie shell --seed 2019130 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  (-.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.re) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.im)))