Average Error: 0.4 → 0.4
Time: 23.5s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)\]
\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)
↓
\left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)
double f(double x_re, double x_im) {
        double r1825795 = x_re;
        double r1825796 = r1825795 * r1825795;
        double r1825797 = x_im;
        double r1825798 = r1825797 * r1825797;
        double r1825799 = r1825796 - r1825798;
        double r1825800 = r1825799 * r1825795;
        double r1825801 = r1825795 * r1825797;
        double r1825802 = r1825797 * r1825795;
        double r1825803 = r1825801 + r1825802;
        double r1825804 = r1825803 * r1825797;
        double r1825805 = r1825800 - r1825804;
        return r1825805;
}
↓
double f(double x_re, double x_im) {
        double r1825806 = x_re;
        double r1825807 = x_im;
        double r1825808 = r1825807 + r1825806;
        double r1825809 = r1825806 * r1825808;
        double r1825810 = r1825806 - r1825807;
        double r1825811 = r1825809 * r1825810;
        double r1825812 = r1825807 + r1825807;
        double r1825813 = r1825812 * r1825806;
        double r1825814 = r1825807 * r1825813;
        double r1825815 = r1825811 - r1825814;
        return r1825815;
}

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

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

  4. Applied associate-*r*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019125 +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)))