math.cube on complex, imaginary part

Average Error: 0.4 → 0.3

Time: 22.5s

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 r819810 = x_re;
        double r819811 = r819810 * r819810;
        double r819812 = x_im;
        double r819813 = r819812 * r819812;
        double r819814 = r819811 - r819813;
        double r819815 = r819814 * r819812;
        double r819816 = r819810 * r819812;
        double r819817 = r819812 * r819810;
        double r819818 = r819816 + r819817;
        double r819819 = r819818 * r819810;
        double r819820 = r819815 + r819819;
        return r819820;
}

↓

double f(double x_re, double x_im) {
        double r819821 = x_re;
        double r819822 = x_im;
        double r819823 = r819821 - r819822;
        double r819824 = r819822 + r819821;
        double r819825 = r819823 / r819824;
        double r819826 = r819822 * r819824;
        double r819827 = r819825 * r819826;
        double r819828 = r819827 * r819824;
        double r819829 = /*Error: no posit support in C */;
        double r819830 = r819822 * r819821;
        double r819831 = r819830 + r819830;
        double r819832 = /*Error: no posit support in C */;
        double r819833 = /*Error: no posit support in C */;
        return r819833;
}

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 
(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)))