Average Error: 0.4 → 0.4
Time: 34.9s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
\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)
↓
\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im
double f(double x_re, double x_im) {
        double r4086737 = x_re;
        double r4086738 = r4086737 * r4086737;
        double r4086739 = x_im;
        double r4086740 = r4086739 * r4086739;
        double r4086741 = r4086738 - r4086740;
        double r4086742 = r4086741 * r4086737;
        double r4086743 = r4086737 * r4086739;
        double r4086744 = r4086739 * r4086737;
        double r4086745 = r4086743 + r4086744;
        double r4086746 = r4086745 * r4086739;
        double r4086747 = r4086742 - r4086746;
        return r4086747;
}
↓
double f(double x_re, double x_im) {
        double r4086748 = x_re;
        double r4086749 = x_im;
        double r4086750 = r4086748 + r4086749;
        double r4086751 = r4086748 - r4086749;
        double r4086752 = r4086751 * r4086748;
        double r4086753 = r4086750 * r4086752;
        double r4086754 = r4086748 * r4086749;
        double r4086755 = r4086749 * r4086748;
        double r4086756 = r4086754 + r4086755;
        double r4086757 = r4086756 * r4086749;
        double r4086758 = r4086753 - r4086757;
        return r4086758;
}

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. Using strategy rm

  3. Applied difference-of-squares0.4

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

  4. Applied associate-*l*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(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)))