Average Error: 0.4 → 0.3
Time: 18.0s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot x.re + x.im \cdot \left(-x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.im \cdot \left(x.re + x.re\right)\right), x.re\right)\right)\]
\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}
↓
\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot x.re + x.im \cdot \left(-x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.im \cdot \left(x.re + x.re\right)\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r510777 = x_re;
        double r510778 = r510777 * r510777;
        double r510779 = x_im;
        double r510780 = r510779 * r510779;
        double r510781 = r510778 - r510780;
        double r510782 = r510781 * r510779;
        double r510783 = r510777 * r510779;
        double r510784 = r510779 * r510777;
        double r510785 = r510783 + r510784;
        double r510786 = r510785 * r510777;
        double r510787 = r510782 + r510786;
        return r510787;
}
↓
double f(double x_re, double x_im) {
        double r510788 = x_im;
        double r510789 = x_re;
        double r510790 = r510788 * r510789;
        double r510791 = -r510788;
        double r510792 = r510788 * r510791;
        double r510793 = r510790 + r510792;
        double r510794 = r510788 + r510789;
        double r510795 = r510793 * r510794;
        double r510796 = /*Error: no posit support in C */;
        double r510797 = r510789 + r510789;
        double r510798 = r510788 * r510797;
        double r510799 = /*Error: no posit support in C */;
        double r510800 = /*Error: no posit support in C */;
        return r510800;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

    \[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
  2. Using strategy rm

  3. Applied introduce-quire0.4

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

  4. Applied insert-quire-fdp-add0.3

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

  5. Simplified0.3

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

  7. Applied distribute-lft-out0.3

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

  9. Applied sub-neg0.3

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

  10. Applied distribute-lft-in0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019155 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.im) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.re)))