Average Error: 0.4 → 0.4
Time: 24.4s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\mathsf{qms}\left(\left(\left(\left(x.re \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(\left(x.im \cdot x.im\right) \cdot \left(x.re + x.re\right)\right), 1.0\right)\right)\]
\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 r1843758 = x_re;
        double r1843759 = r1843758 * r1843758;
        double r1843760 = x_im;
        double r1843761 = r1843760 * r1843760;
        double r1843762 = r1843759 - r1843761;
        double r1843763 = r1843762 * r1843758;
        double r1843764 = r1843758 * r1843760;
        double r1843765 = r1843760 * r1843758;
        double r1843766 = r1843764 + r1843765;
        double r1843767 = r1843766 * r1843760;
        double r1843768 = r1843763 - r1843767;
        return r1843768;
}
↓
double f(double x_re, double x_im) {
        double r1843769 = x_re;
        double r1843770 = x_im;
        double r1843771 = r1843769 - r1843770;
        double r1843772 = r1843769 * r1843771;
        double r1843773 = r1843770 + r1843769;
        double r1843774 = r1843772 * r1843773;
        double r1843775 = /*Error: no posit support in C */;
        double r1843776 = r1843770 * r1843770;
        double r1843777 = r1843769 + r1843769;
        double r1843778 = r1843776 * r1843777;
        double r1843779 = 1.0;
        double r1843780 = /*Error: no posit support in C */;
        double r1843781 = /*Error: no posit support in C */;
        return r1843781;
}

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 
(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)))