math.cube on complex, real part

Average Error: 0.4 → 0.3

Time: 23.1s

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 r1964454 = x_re;
        double r1964455 = r1964454 * r1964454;
        double r1964456 = x_im;
        double r1964457 = r1964456 * r1964456;
        double r1964458 = r1964455 - r1964457;
        double r1964459 = r1964458 * r1964454;
        double r1964460 = r1964454 * r1964456;
        double r1964461 = r1964456 * r1964454;
        double r1964462 = r1964460 + r1964461;
        double r1964463 = r1964462 * r1964456;
        double r1964464 = r1964459 - r1964463;
        return r1964464;
}

↓

double f(double x_re, double x_im) {
        double r1964465 = x_re;
        double r1964466 = x_im;
        double r1964467 = r1964465 - r1964466;
        double r1964468 = r1964465 * r1964467;
        double r1964469 = r1964465 + r1964466;
        double r1964470 = r1964468 * r1964469;
        double r1964471 = /*Error: no posit support in C */;
        double r1964472 = r1964465 + r1964465;
        double r1964473 = r1964466 * r1964472;
        double r1964474 = /*Error: no posit support in C */;
        double r1964475 = /*Error: no posit support in C */;
        return r1964475;
}

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 +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)))