math.cube on complex, real part

Average Error: 0.4 → 0.3

Time: 23.4s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r1964582 = x_re;
        double r1964583 = r1964582 * r1964582;
        double r1964584 = x_im;
        double r1964585 = r1964584 * r1964584;
        double r1964586 = r1964583 - r1964585;
        double r1964587 = r1964586 * r1964582;
        double r1964588 = r1964582 * r1964584;
        double r1964589 = r1964584 * r1964582;
        double r1964590 = r1964588 + r1964589;
        double r1964591 = r1964590 * r1964584;
        double r1964592 = r1964587 - r1964591;
        return r1964592;
}

↓

double f(double x_re, double x_im) {
        double r1964593 = x_re;
        double r1964594 = x_im;
        double r1964595 = r1964593 - r1964594;
        double r1964596 = r1964593 * r1964595;
        double r1964597 = r1964593 + r1964594;
        double r1964598 = r1964596 * r1964597;
        double r1964599 = /*Error: no posit support in C */;
        double r1964600 = r1964593 + r1964593;
        double r1964601 = r1964594 * r1964600;
        double r1964602 = /*Error: no posit support in C */;
        double r1964603 = /*Error: no posit support in C */;
        return r1964603;
}

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

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

7. Applied associate-*r* 0.3

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019168 
(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)))