Average Error: 0.4 → 0.3
Time: 24.9s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\mathsf{qms}\left(\left(\left(\left(\left(\left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)\right)\right)\right)\right), \left(\left(x.re + x.re\right) \cdot x.im\right), x.im\right)\right)\]
\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(\mathsf{qms}\left(\left(\left(\left(\left(\left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)\right)\right)\right)\right), \left(\left(x.re + x.re\right) \cdot x.im\right), x.im\right)\right)
double f(double x_re, double x_im) {
        double r1984596 = x_re;
        double r1984597 = r1984596 * r1984596;
        double r1984598 = x_im;
        double r1984599 = r1984598 * r1984598;
        double r1984600 = r1984597 - r1984599;
        double r1984601 = r1984600 * r1984596;
        double r1984602 = r1984596 * r1984598;
        double r1984603 = r1984598 * r1984596;
        double r1984604 = r1984602 + r1984603;
        double r1984605 = r1984604 * r1984598;
        double r1984606 = r1984601 - r1984605;
        return r1984606;
}
↓
double f(double x_re, double x_im) {
        double r1984607 = x_re;
        double r1984608 = x_im;
        double r1984609 = r1984608 + r1984607;
        double r1984610 = r1984607 * r1984609;
        double r1984611 = r1984607 - r1984608;
        double r1984612 = r1984610 * r1984611;
        double r1984613 = /*Error: no posit support in C */;
        double r1984614 = /*Error: no posit support in C */;
        double r1984615 = /*Error: no posit support in C */;
        double r1984616 = r1984607 + r1984607;
        double r1984617 = r1984616 * r1984608;
        double r1984618 = /*Error: no posit support in C */;
        double r1984619 = /*Error: no posit support in C */;
        return r1984619;
}

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 introduce-quire0.4

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

  4. Applied insert-quire-fdp-sub0.3

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

  5. Simplified0.3

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

  7. Applied associate-*l*0.3

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

  9. Applied distribute-lft-in0.3

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

  10. Applied distribute-rgt-in0.4

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

  12. Applied introduce-quire0.4

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019162 
(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)))