Average Error: 0.4 → 0.3
Time: 18.3s
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 r564057 = x_re;
        double r564058 = r564057 * r564057;
        double r564059 = x_im;
        double r564060 = r564059 * r564059;
        double r564061 = r564058 - r564060;
        double r564062 = r564061 * r564057;
        double r564063 = r564057 * r564059;
        double r564064 = r564059 * r564057;
        double r564065 = r564063 + r564064;
        double r564066 = r564065 * r564059;
        double r564067 = r564062 - r564066;
        return r564067;
}
↓
double f(double x_re, double x_im) {
        double r564068 = x_re;
        double r564069 = x_im;
        double r564070 = r564069 + r564068;
        double r564071 = r564068 - r564069;
        double r564072 = r564070 * r564071;
        double r564073 = /*Error: no posit support in C */;
        double r564074 = r564069 + r564069;
        double r564075 = /*Error: no posit support in C */;
        double r564076 = /*Error: no posit support in C */;
        double r564077 = r564068 * r564076;
        return r564077;
}

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 2019154 
(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)))