Average Error: 0.4 → 0.3
Time: 18.6s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[x.re \cdot \left(\mathsf{qms}\left(\left(\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\right), \left(x.im + x.im\right), x.im\right)\right)\]
\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)
↓
x.re \cdot \left(\mathsf{qms}\left(\left(\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\right), \left(x.im + x.im\right), x.im\right)\right)
double f(double x_re, double x_im) {
        double r426964 = x_re;
        double r426965 = r426964 * r426964;
        double r426966 = x_im;
        double r426967 = r426966 * r426966;
        double r426968 = r426965 - r426967;
        double r426969 = r426968 * r426964;
        double r426970 = r426964 * r426966;
        double r426971 = r426966 * r426964;
        double r426972 = r426970 + r426971;
        double r426973 = r426972 * r426966;
        double r426974 = r426969 - r426973;
        return r426974;
}
↓
double f(double x_re, double x_im) {
        double r426975 = x_re;
        double r426976 = x_im;
        double r426977 = r426976 + r426975;
        double r426978 = r426975 - r426976;
        double r426979 = r426977 * r426978;
        double r426980 = /*Error: no posit support in C */;
        double r426981 = r426976 + r426976;
        double r426982 = /*Error: no posit support in C */;
        double r426983 = /*Error: no posit support in C */;
        double r426984 = r426975 * r426983;
        return r426984;
}

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-rgt-in0.4

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

  5. Applied associate--r+0.4

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

  6. Simplified0.4

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

  8. Applied introduce-quire0.4

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

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

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019154 +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)))