Average Error: 0.4 → 0.4
Time: 33.1s
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 r4132591 = x_re;
        double r4132592 = r4132591 * r4132591;
        double r4132593 = x_im;
        double r4132594 = r4132593 * r4132593;
        double r4132595 = r4132592 - r4132594;
        double r4132596 = r4132595 * r4132591;
        double r4132597 = r4132591 * r4132593;
        double r4132598 = r4132593 * r4132591;
        double r4132599 = r4132597 + r4132598;
        double r4132600 = r4132599 * r4132593;
        double r4132601 = r4132596 - r4132600;
        return r4132601;
}
↓
double f(double x_re, double x_im) {
        double r4132602 = x_re;
        double r4132603 = x_im;
        double r4132604 = r4132602 - r4132603;
        double r4132605 = r4132603 + r4132602;
        double r4132606 = r4132604 * r4132605;
        double r4132607 = r4132603 + r4132603;
        double r4132608 = r4132607 * r4132603;
        double r4132609 = r4132606 - r4132608;
        double r4132610 = r4132602 * r4132609;
        return r4132610;
}

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 +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)))