Average Error: 0.4 → 0.3
Time: 14.1s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\mathsf{qms}\left(\left(\left(x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\right)\right), \left(x.im \cdot \left(x.re + x.re\right)\right), x.im\right)\right)\]
\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.im + x.re\right)\right)\right)\right), \left(x.im \cdot \left(x.re + x.re\right)\right), x.im\right)\right)
double f(double x_re, double x_im) {
        double r625846 = x_re;
        double r625847 = r625846 * r625846;
        double r625848 = x_im;
        double r625849 = r625848 * r625848;
        double r625850 = r625847 - r625849;
        double r625851 = r625850 * r625846;
        double r625852 = r625846 * r625848;
        double r625853 = r625848 * r625846;
        double r625854 = r625852 + r625853;
        double r625855 = r625854 * r625848;
        double r625856 = r625851 - r625855;
        return r625856;
}
↓
double f(double x_re, double x_im) {
        double r625857 = x_re;
        double r625858 = x_im;
        double r625859 = r625857 - r625858;
        double r625860 = r625858 + r625857;
        double r625861 = r625859 * r625860;
        double r625862 = r625857 * r625861;
        double r625863 = /*Error: no posit support in C */;
        double r625864 = r625857 + r625857;
        double r625865 = r625858 * r625864;
        double r625866 = /*Error: no posit support in C */;
        double r625867 = /*Error: no posit support in C */;
        return r625867;
}

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

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

  7. Applied distribute-lft-out0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019156 
(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)))