math.cube on complex, real part

Average Error: 0.4 → 0.4

Time: 11.7s

Precision: 64

Math

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

↓

\[x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) - \left(x.im + x.im\right) \cdot x.im\right)\]

C

double f(double x_re, double x_im) {
        double r519653 = x_re;
        double r519654 = r519653 * r519653;
        double r519655 = x_im;
        double r519656 = r519655 * r519655;
        double r519657 = r519654 - r519656;
        double r519658 = r519657 * r519653;
        double r519659 = r519653 * r519655;
        double r519660 = r519655 * r519653;
        double r519661 = r519659 + r519660;
        double r519662 = r519661 * r519655;
        double r519663 = r519658 - r519662;
        return r519663;
}

↓

double f(double x_re, double x_im) {
        double r519664 = x_re;
        double r519665 = x_im;
        double r519666 = r519664 - r519665;
        double r519667 = r519665 + r519664;
        double r519668 = r519666 * r519667;
        double r519669 = r519665 + r519665;
        double r519670 = r519669 * r519665;
        double r519671 = r519668 - r519670;
        double r519672 = r519664 * r519671;
        return r519672;
}

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. Simplified 0.4

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

4. Applied sub-neg 0.4

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

5. Applied distribute-rgt-in 0.4

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

6. Applied associate--l+ 0.4

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

8. Applied associate-+r- 0.4

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

9. Simplified 0.4

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

10. Final simplification 0.4

    \[\leadsto x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) - \left(x.im + x.im\right) \cdot x.im\right)\]

Reproduce

herbie shell --seed 2019128 
(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)))