Average Error: 0.4 → 0.4
Time: 17.8s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[x.im \cdot \left(\left(x.re \cdot x.re + x.re \cdot x.re\right) + \left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\]
\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}
↓
x.im \cdot \left(\left(x.re \cdot x.re + x.re \cdot x.re\right) + \left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)
double f(double x_re, double x_im) {
        double r2843567 = x_re;
        double r2843568 = r2843567 * r2843567;
        double r2843569 = x_im;
        double r2843570 = r2843569 * r2843569;
        double r2843571 = r2843568 - r2843570;
        double r2843572 = r2843571 * r2843569;
        double r2843573 = r2843567 * r2843569;
        double r2843574 = r2843569 * r2843567;
        double r2843575 = r2843573 + r2843574;
        double r2843576 = r2843575 * r2843567;
        double r2843577 = r2843572 + r2843576;
        return r2843577;
}
↓
double f(double x_re, double x_im) {
        double r2843578 = x_im;
        double r2843579 = x_re;
        double r2843580 = r2843579 * r2843579;
        double r2843581 = r2843580 + r2843580;
        double r2843582 = r2843579 - r2843578;
        double r2843583 = r2843578 + r2843579;
        double r2843584 = r2843582 * r2843583;
        double r2843585 = r2843581 + r2843584;
        double r2843586 = r2843578 * r2843585;
        return r2843586;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

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

  2. Simplified0.4

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

  4. Applied distribute-rgt-in0.4

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

  5. Applied associate-+l+0.4

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

  7. Applied associate-+r+0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.im) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.re)))