math.cube on complex, imaginary part

Average Error: 0.4 → 0.3

Time: 16.8s

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 r753820 = x_re;
        double r753821 = r753820 * r753820;
        double r753822 = x_im;
        double r753823 = r753822 * r753822;
        double r753824 = r753821 - r753823;
        double r753825 = r753824 * r753822;
        double r753826 = r753820 * r753822;
        double r753827 = r753822 * r753820;
        double r753828 = r753826 + r753827;
        double r753829 = r753828 * r753820;
        double r753830 = r753825 + r753829;
        return r753830;
}

↓

double f(double x_re, double x_im) {
        double r753831 = x_re;
        double r753832 = x_im;
        double r753833 = r753831 - r753832;
        double r753834 = r753832 + r753831;
        double r753835 = r753833 / r753834;
        double r753836 = r753832 * r753834;
        double r753837 = r753835 * r753836;
        double r753838 = r753837 * r753834;
        double r753839 = /*Error: no posit support in C */;
        double r753840 = r753832 * r753831;
        double r753841 = r753840 + r753840;
        double r753842 = /*Error: no posit support in C */;
        double r753843 = /*Error: no posit support in C */;
        return r753843;
}

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)))