Average Error: 0.4 → 0.3
Time: 24.7s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\mathsf{qms}\left(\left(\left(x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.im\right)\right)\]
\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(x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.im\right)\right)
double f(double x_re, double x_im) {
        double r667317 = x_re;
        double r667318 = r667317 * r667317;
        double r667319 = x_im;
        double r667320 = r667319 * r667319;
        double r667321 = r667318 - r667320;
        double r667322 = r667321 * r667317;
        double r667323 = r667317 * r667319;
        double r667324 = r667319 * r667317;
        double r667325 = r667323 + r667324;
        double r667326 = r667325 * r667319;
        double r667327 = r667322 - r667326;
        return r667327;
}
↓
double f(double x_re, double x_im) {
        double r667328 = x_re;
        double r667329 = x_im;
        double r667330 = r667328 + r667329;
        double r667331 = r667328 - r667329;
        double r667332 = r667330 * r667331;
        double r667333 = r667328 * r667332;
        double r667334 = /*Error: no posit support in C */;
        double r667335 = r667328 * r667329;
        double r667336 = r667335 + r667335;
        double r667337 = /*Error: no posit support in C */;
        double r667338 = /*Error: no posit support in C */;
        return r667338;
}

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-fdp-sub0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 
(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)))