math.cube on complex, real part

Average Error: 0.4 → 0.3

Time: 26.0s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r1984610 = x_re;
        double r1984611 = r1984610 * r1984610;
        double r1984612 = x_im;
        double r1984613 = r1984612 * r1984612;
        double r1984614 = r1984611 - r1984613;
        double r1984615 = r1984614 * r1984610;
        double r1984616 = r1984610 * r1984612;
        double r1984617 = r1984612 * r1984610;
        double r1984618 = r1984616 + r1984617;
        double r1984619 = r1984618 * r1984612;
        double r1984620 = r1984615 - r1984619;
        return r1984620;
}

↓

double f(double x_re, double x_im) {
        double r1984621 = x_re;
        double r1984622 = x_im;
        double r1984623 = r1984622 + r1984621;
        double r1984624 = r1984621 * r1984623;
        double r1984625 = r1984621 - r1984622;
        double r1984626 = r1984624 * r1984625;
        double r1984627 = /*Error: no posit support in C */;
        double r1984628 = /*Error: no posit support in C */;
        double r1984629 = /*Error: no posit support in C */;
        double r1984630 = r1984621 + r1984621;
        double r1984631 = r1984630 * r1984622;
        double r1984632 = /*Error: no posit support in C */;
        double r1984633 = /*Error: no posit support in C */;
        return r1984633;
}

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-lft-in 0.3

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

10. Applied distribute-rgt-in 0.4

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

12. Applied introduce-quire 0.4

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019162 +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)))