Average Error: 0.4 → 0.4
Time: 28.0s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\mathsf{qms}\left(\left(\left(\left(x.re \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(\left(x.im \cdot x.im\right) \cdot \left(x.re + x.re\right)\right), 1.0\right)\right)\]
\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(\left(x.im \cdot x.im\right) \cdot \left(x.re + x.re\right)\right), 1.0\right)\right)
double f(double x_re, double x_im) {
        double r1792413 = x_re;
        double r1792414 = r1792413 * r1792413;
        double r1792415 = x_im;
        double r1792416 = r1792415 * r1792415;
        double r1792417 = r1792414 - r1792416;
        double r1792418 = r1792417 * r1792413;
        double r1792419 = r1792413 * r1792415;
        double r1792420 = r1792415 * r1792413;
        double r1792421 = r1792419 + r1792420;
        double r1792422 = r1792421 * r1792415;
        double r1792423 = r1792418 - r1792422;
        return r1792423;
}
↓
double f(double x_re, double x_im) {
        double r1792424 = x_re;
        double r1792425 = x_im;
        double r1792426 = r1792424 - r1792425;
        double r1792427 = r1792424 * r1792426;
        double r1792428 = r1792425 + r1792424;
        double r1792429 = r1792427 * r1792428;
        double r1792430 = /*Error: no posit support in C */;
        double r1792431 = r1792425 * r1792425;
        double r1792432 = r1792424 + r1792424;
        double r1792433 = r1792431 * r1792432;
        double r1792434 = 1.0;
        double r1792435 = /*Error: no posit support in C */;
        double r1792436 = /*Error: no posit support in C */;
        return r1792436;
}

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-quire0.4

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

  4. Applied insert-quire-sub0.4

    \[\leadsto \color{blue}{\left(\mathsf{qms}\left(\left(\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right)\right), \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right), \left(1.0\right)\right)\right)}\]

  5. Simplified0.4

    \[\leadsto \color{blue}{\left(\mathsf{qms}\left(\left(\left(\left(x.re \cdot \left(x.re - x.im\right)\right) \cdot \left(\frac{x.im}{x.re}\right)\right)\right), \left(\left(x.im \cdot x.im\right) \cdot \left(\frac{x.re}{x.re}\right)\right), \left(1.0\right)\right)\right)}\]

  6. Final simplification0.4

    \[\leadsto \left(\mathsf{qms}\left(\left(\left(\left(x.re \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(\left(x.im \cdot x.im\right) \cdot \left(x.re + x.re\right)\right), 1.0\right)\right)\]

Reproduce

herbie shell --seed 2019163 +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)))