math.cube on complex, real part

Average Error: 0.4 → 0.4

Time: 29.1s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r2064728 = x_re;
        double r2064729 = r2064728 * r2064728;
        double r2064730 = x_im;
        double r2064731 = r2064730 * r2064730;
        double r2064732 = r2064729 - r2064731;
        double r2064733 = r2064732 * r2064728;
        double r2064734 = r2064728 * r2064730;
        double r2064735 = r2064730 * r2064728;
        double r2064736 = r2064734 + r2064735;
        double r2064737 = r2064736 * r2064730;
        double r2064738 = r2064733 - r2064737;
        return r2064738;
}

↓

double f(double x_re, double x_im) {
        double r2064739 = x_re;
        double r2064740 = x_im;
        double r2064741 = r2064739 - r2064740;
        double r2064742 = r2064740 * r2064741;
        double r2064743 = r2064739 * r2064742;
        double r2064744 = r2064739 * r2064739;
        double r2064745 = r2064741 * r2064744;
        double r2064746 = r2064743 + r2064745;
        double r2064747 = /*Error: no posit support in C */;
        double r2064748 = r2064739 + r2064739;
        double r2064749 = r2064748 * r2064740;
        double r2064750 = /*Error: no posit support in C */;
        double r2064751 = /*Error: no posit support in C */;
        return r2064751;
}

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-quire 0.4

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

4. Applied insert-quire-fdp-sub 0.3

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

5. Simplified 0.3

   \[\leadsto \color{blue}{\left(\mathsf{qms}\left(\left(\left(\left(x.re \cdot \left(x.re - x.im\right)\right) \cdot \left(\frac{x.im}{x.re}\right)\right)\right), \left(\left(\frac{x.re}{x.re}\right) \cdot x.im\right), x.im\right)\right)}\]
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-in 0.3

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

10. Applied distribute-lft-in 0.4

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

12. Applied +p16-rgt-identity-expand 0.4

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

13. Applied distribute-lft-in 0.4

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

14. Applied distribute-lft-in 0.4

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

15. Applied associate-+l+ 0.4

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

16. Simplified 0.4

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

17. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019162 +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)))