math.cube on complex, real part

Average Error: 0.4 → 0.3

Time: 29.8s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r1976770 = x_re;
        double r1976771 = r1976770 * r1976770;
        double r1976772 = x_im;
        double r1976773 = r1976772 * r1976772;
        double r1976774 = r1976771 - r1976773;
        double r1976775 = r1976774 * r1976770;
        double r1976776 = r1976770 * r1976772;
        double r1976777 = r1976772 * r1976770;
        double r1976778 = r1976776 + r1976777;
        double r1976779 = r1976778 * r1976772;
        double r1976780 = r1976775 - r1976779;
        return r1976780;
}

↓

double f(double x_re, double x_im) {
        double r1976781 = x_re;
        double r1976782 = x_im;
        double r1976783 = r1976781 + r1976782;
        double r1976784 = r1976781 * r1976783;
        double r1976785 = r1976781 - r1976782;
        double r1976786 = r1976784 * r1976785;
        double r1976787 = /*Error: no posit support in C */;
        double r1976788 = /*Error: no posit support in C */;
        double r1976789 = /*Error: no posit support in C */;
        double r1976790 = r1976781 + r1976781;
        double r1976791 = r1976782 * r1976790;
        double r1976792 = /*Error: no posit support in C */;
        double r1976793 = /*Error: no posit support in C */;
        return r1976793;
}

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

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

7. Applied distribute-rgt-in 0.3

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

8. Applied distribute-rgt-in 0.4

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

9. Simplified 0.4

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

11. Applied introduce-quire 0.4

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019158 +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)))