math.cube on complex, real part

Average Error: 0.4 → 0.3

Time: 1.1m

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r2578530 = x_re;
        double r2578531 = r2578530 * r2578530;
        double r2578532 = x_im;
        double r2578533 = r2578532 * r2578532;
        double r2578534 = r2578531 - r2578533;
        double r2578535 = r2578534 * r2578530;
        double r2578536 = r2578530 * r2578532;
        double r2578537 = r2578532 * r2578530;
        double r2578538 = r2578536 + r2578537;
        double r2578539 = r2578538 * r2578532;
        double r2578540 = r2578535 - r2578539;
        return r2578540;
}

↓

double f(double x_re, double x_im) {
        double r2578541 = x_re;
        double r2578542 = x_im;
        double r2578543 = r2578541 - r2578542;
        double r2578544 = r2578542 + r2578541;
        double r2578545 = r2578543 * r2578544;
        double r2578546 = r2578541 * r2578545;
        double r2578547 = /*Error: no posit support in C */;
        double r2578548 = r2578541 + r2578541;
        double r2578549 = r2578542 * r2578548;
        double r2578550 = /*Error: no posit support in C */;
        double r2578551 = /*Error: no posit support in C */;
        return r2578551;
}

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

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

7. Applied distribute-rgt-in 0.3

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

9. Applied distribute-lft-out 0.3

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

10. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019142 +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)))