math.cube on complex, imaginary part

Average Error: 0.4 → 0.3

Time: 15.9s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r917860 = x_re;
        double r917861 = r917860 * r917860;
        double r917862 = x_im;
        double r917863 = r917862 * r917862;
        double r917864 = r917861 - r917863;
        double r917865 = r917864 * r917862;
        double r917866 = r917860 * r917862;
        double r917867 = r917862 * r917860;
        double r917868 = r917866 + r917867;
        double r917869 = r917868 * r917860;
        double r917870 = r917865 + r917869;
        return r917870;
}

↓

double f(double x_re, double x_im) {
        double r917871 = x_re;
        double r917872 = x_im;
        double r917873 = r917871 - r917872;
        double r917874 = r917872 + r917871;
        double r917875 = r917873 / r917874;
        double r917876 = r917872 * r917874;
        double r917877 = r917875 * r917876;
        double r917878 = r917877 * r917874;
        double r917879 = /*Error: no posit support in C */;
        double r917880 = r917872 * r917871;
        double r917881 = r917880 + r917880;
        double r917882 = /*Error: no posit support in C */;
        double r917883 = /*Error: no posit support in C */;
        return r917883;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

1. Initial program 0.4

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

3. Applied introduce-quire 0.4

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

4. Applied insert-quire-fdp-add 0.3

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

5. Simplified 0.3

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

7. Applied p16-flip-- 0.3

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

8. Applied associate-*r/ 0.3

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

9. Simplified 0.3

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

11. Applied *p16-rgt-identity-expand 0.3

    \[\leadsto \left(\mathsf{qma}\left(\left(\left(\left(\frac{\left(\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(\frac{x.im}{x.re}\right)\right)\right)}{\color{blue}{\left(\left(\frac{x.re}{x.im}\right) \cdot \left(1.0\right)\right)}}\right) \cdot \left(\frac{x.im}{x.re}\right)\right)\right), \left(\frac{\left(x.im \cdot x.re\right)}{\left(x.im \cdot x.re\right)}\right), x.re\right)\right)\]

12. Applied p16-times-frac 0.3

    \[\leadsto \left(\mathsf{qma}\left(\left(\left(\color{blue}{\left(\left(\frac{\left(x.re - x.im\right)}{\left(\frac{x.re}{x.im}\right)}\right) \cdot \left(\frac{\left(x.im \cdot \left(\frac{x.im}{x.re}\right)\right)}{\left(1.0\right)}\right)\right)} \cdot \left(\frac{x.im}{x.re}\right)\right)\right), \left(\frac{\left(x.im \cdot x.re\right)}{\left(x.im \cdot x.re\right)}\right), x.re\right)\right)\]

13. Simplified 0.3

    \[\leadsto \left(\mathsf{qma}\left(\left(\left(\left(\color{blue}{\left(\frac{\left(x.re - x.im\right)}{\left(\frac{x.im}{x.re}\right)}\right)} \cdot \left(\frac{\left(x.im \cdot \left(\frac{x.im}{x.re}\right)\right)}{\left(1.0\right)}\right)\right) \cdot \left(\frac{x.im}{x.re}\right)\right)\right), \left(\frac{\left(x.im \cdot x.re\right)}{\left(x.im \cdot x.re\right)}\right), x.re\right)\right)\]

14. Simplified 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.im) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.re)))