math.cube on complex, real part

Average Error: 0.4 → 0.3

Time: 26.1s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r4117586 = x_re;
        double r4117587 = r4117586 * r4117586;
        double r4117588 = x_im;
        double r4117589 = r4117588 * r4117588;
        double r4117590 = r4117587 - r4117589;
        double r4117591 = r4117590 * r4117586;
        double r4117592 = r4117586 * r4117588;
        double r4117593 = r4117588 * r4117586;
        double r4117594 = r4117592 + r4117593;
        double r4117595 = r4117594 * r4117588;
        double r4117596 = r4117591 - r4117595;
        return r4117596;
}

↓

double f(double x_re, double x_im) {
        double r4117597 = x_re;
        double r4117598 = x_im;
        double r4117599 = r4117597 - r4117598;
        double r4117600 = r4117597 * r4117599;
        double r4117601 = r4117598 + r4117597;
        double r4117602 = r4117600 * r4117601;
        double r4117603 = /*Error: no posit support in C */;
        double r4117604 = r4117598 + r4117598;
        double r4117605 = r4117597 * r4117604;
        double r4117606 = /*Error: no posit support in C */;
        double r4117607 = /*Error: no posit support in C */;
        return r4117607;
}

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

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

7. Applied distribute-lft-out 0.3

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019152 +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)))