Average Error: 0.4 → 0.4
Time: 26.4s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\mathsf{qms}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot x.re + \left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right), \left(\left(x.re + x.re\right) \cdot x.im\right), x.im\right)\right)\]
\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(x.im \cdot \left(x.re - x.im\right)\right) \cdot x.re + \left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\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 r1939564 = x_re;
        double r1939565 = r1939564 * r1939564;
        double r1939566 = x_im;
        double r1939567 = r1939566 * r1939566;
        double r1939568 = r1939565 - r1939567;
        double r1939569 = r1939568 * r1939564;
        double r1939570 = r1939564 * r1939566;
        double r1939571 = r1939566 * r1939564;
        double r1939572 = r1939570 + r1939571;
        double r1939573 = r1939572 * r1939566;
        double r1939574 = r1939569 - r1939573;
        return r1939574;
}
↓
double f(double x_re, double x_im) {
        double r1939575 = x_im;
        double r1939576 = x_re;
        double r1939577 = r1939576 - r1939575;
        double r1939578 = r1939575 * r1939577;
        double r1939579 = r1939578 * r1939576;
        double r1939580 = r1939576 * r1939576;
        double r1939581 = r1939577 * r1939580;
        double r1939582 = r1939579 + r1939581;
        double r1939583 = /*Error: no posit support in C */;
        double r1939584 = r1939576 + r1939576;
        double r1939585 = r1939584 * r1939575;
        double r1939586 = /*Error: no posit support in C */;
        double r1939587 = /*Error: no posit support in C */;
        return r1939587;
}

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-rgt-in0.3

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

  10. Applied distribute-rgt-in0.4

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

  12. Applied +p16-rgt-identity-expand0.4

    \[\leadsto \left(\mathsf{qms}\left(\left(\left(\frac{\color{blue}{\left(\frac{\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot x.re\right)}{\left(0.0\right)}\right)}}{\left(\left(x.re \cdot \left(x.re - x.im\right)\right) \cdot x.re\right)}\right)\right), \left(\left(\frac{x.re}{x.re}\right) \cdot x.im\right), x.im\right)\right)\]

  13. Applied associate-+l+0.4

    \[\leadsto \left(\mathsf{qms}\left(\left(\color{blue}{\left(\frac{\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot x.re\right)}{\left(\frac{\left(0.0\right)}{\left(\left(x.re \cdot \left(x.re - x.im\right)\right) \cdot x.re\right)}\right)}\right)}\right), \left(\left(\frac{x.re}{x.re}\right) \cdot x.im\right), x.im\right)\right)\]

  14. Simplified0.4

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

  15. Final simplification0.4

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

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)))