Average Error: 0.4 → 0.3
Time: 44.8s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot \left(x.im + x.im\right)\right), x.re\right)\right)\]
\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)}
↓
\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot \left(x.im + x.im\right)\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r1647712 = x_re;
        double r1647713 = r1647712 * r1647712;
        double r1647714 = x_im;
        double r1647715 = r1647714 * r1647714;
        double r1647716 = r1647713 - r1647715;
        double r1647717 = r1647716 * r1647714;
        double r1647718 = r1647712 * r1647714;
        double r1647719 = r1647714 * r1647712;
        double r1647720 = r1647718 + r1647719;
        double r1647721 = r1647720 * r1647712;
        double r1647722 = r1647717 + r1647721;
        return r1647722;
}
↓
double f(double x_re, double x_im) {
        double r1647723 = x_im;
        double r1647724 = x_re;
        double r1647725 = r1647724 - r1647723;
        double r1647726 = r1647723 * r1647725;
        double r1647727 = r1647723 + r1647724;
        double r1647728 = r1647726 * r1647727;
        double r1647729 = /*Error: no posit support in C */;
        double r1647730 = r1647723 + r1647723;
        double r1647731 = r1647724 * r1647730;
        double r1647732 = /*Error: no posit support in C */;
        double r1647733 = /*Error: no posit support in C */;
        return r1647733;
}

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. Using strategy rm

  3. Applied introduce-quire0.4

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

  4. Applied insert-quire-fdp-add0.3

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

  5. Simplified0.3

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

  7. Applied associate-*r*0.3

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

  8. Final simplification0.3

    \[\leadsto \left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot \left(x.im + x.im\right)\right), x.re\right)\right)\]

Reproduce

herbie shell --seed 2019149 +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)))