Average Error: 0.4 → 0.3
Time: 24.1s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[x.re \cdot \left(\mathsf{qms}\left(\left(\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\right), x.im, \left(x.im + x.im\right)\right)\right)\]
\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)
↓
x.re \cdot \left(\mathsf{qms}\left(\left(\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\right), x.im, \left(x.im + x.im\right)\right)\right)
double f(double x_re, double x_im) {
        double r693914 = x_re;
        double r693915 = r693914 * r693914;
        double r693916 = x_im;
        double r693917 = r693916 * r693916;
        double r693918 = r693915 - r693917;
        double r693919 = r693918 * r693914;
        double r693920 = r693914 * r693916;
        double r693921 = r693916 * r693914;
        double r693922 = r693920 + r693921;
        double r693923 = r693922 * r693916;
        double r693924 = r693919 - r693923;
        return r693924;
}
↓
double f(double x_re, double x_im) {
        double r693925 = x_re;
        double r693926 = x_im;
        double r693927 = r693926 + r693925;
        double r693928 = r693925 - r693926;
        double r693929 = r693927 * r693928;
        double r693930 = /*Error: no posit support in C */;
        double r693931 = r693926 + r693926;
        double r693932 = /*Error: no posit support in C */;
        double r693933 = /*Error: no posit support in C */;
        double r693934 = r693925 * r693933;
        return r693934;
}

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. Simplified0.4

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

  4. Applied distribute-lft-in0.4

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

  5. Applied associate--r+0.4

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

  6. Simplified0.4

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

  8. Applied introduce-quire0.4

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

  9. Applied insert-quire-fdp-sub0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019151 +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)))