Average Error: 0.4 → 0.4
Time: 20.3s
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 r628571 = x_re;
        double r628572 = r628571 * r628571;
        double r628573 = x_im;
        double r628574 = r628573 * r628573;
        double r628575 = r628572 - r628574;
        double r628576 = r628575 * r628571;
        double r628577 = r628571 * r628573;
        double r628578 = r628573 * r628571;
        double r628579 = r628577 + r628578;
        double r628580 = r628579 * r628573;
        double r628581 = r628576 - r628580;
        return r628581;
}
↓
double f(double x_re, double x_im) {
        double r628582 = x_re;
        double r628583 = x_im;
        double r628584 = r628582 + r628583;
        double r628585 = r628582 - r628583;
        double r628586 = r628585 * r628582;
        double r628587 = r628584 * r628586;
        double r628588 = r628582 * r628583;
        double r628589 = r628583 * r628582;
        double r628590 = r628588 + r628589;
        double r628591 = r628590 * r628583;
        double r628592 = r628587 - r628591;
        return r628592;
}

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 2019153 +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)))