Average Error: 0.4 → 0.3
Time: 26.4s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\mathsf{qms}\left(\left(\left(x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(\frac{x.im}{x.re}\right)\right)\right)\right), \left(x.im \cdot \left(\frac{x.re}{x.re}\right)\right), x.im\right)\right)\]
\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(\frac{x.im}{x.re}\right)\right)\right)\right), \left(x.im \cdot \left(\frac{x.re}{x.re}\right)\right), x.im\right)\right)
double f(double x_re, double x_im) {
        double r1779854 = x_re;
        double r1779855 = r1779854 * r1779854;
        double r1779856 = x_im;
        double r1779857 = r1779856 * r1779856;
        double r1779858 = r1779855 - r1779857;
        double r1779859 = r1779858 * r1779854;
        double r1779860 = r1779854 * r1779856;
        double r1779861 = r1779856 * r1779854;
        double r1779862 = r1779860 + r1779861;
        double r1779863 = r1779862 * r1779856;
        double r1779864 = r1779859 - r1779863;
        return r1779864;
}
↓
double f(double x_re, double x_im) {
        double r1779865 = x_re;
        double r1779866 = x_im;
        double r1779867 = r1779865 - r1779866;
        double r1779868 = r1779866 + r1779865;
        double r1779869 = r1779867 * r1779868;
        double r1779870 = r1779865 * r1779869;
        double r1779871 = /*Error: no posit support in C */;
        double r1779872 = r1779865 + r1779865;
        double r1779873 = r1779866 * r1779872;
        double r1779874 = /*Error: no posit support in C */;
        double r1779875 = /*Error: no posit support in C */;
        return r1779875;
}

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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 +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)))