Average Error: 0.4 → 0.3
Time: 24.7s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\mathsf{qms}\left(\left(\left(\left(x.re \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.im \cdot \left(x.re + x.re\right)\right), x.im\right)\right)\]
\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)
↓
\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(x.im \cdot \left(x.re + x.re\right)\right), x.im\right)\right)
double f(double x_re, double x_im) {
        double r1626030 = x_re;
        double r1626031 = r1626030 * r1626030;
        double r1626032 = x_im;
        double r1626033 = r1626032 * r1626032;
        double r1626034 = r1626031 - r1626033;
        double r1626035 = r1626034 * r1626030;
        double r1626036 = r1626030 * r1626032;
        double r1626037 = r1626032 * r1626030;
        double r1626038 = r1626036 + r1626037;
        double r1626039 = r1626038 * r1626032;
        double r1626040 = r1626035 - r1626039;
        return r1626040;
}
↓
double f(double x_re, double x_im) {
        double r1626041 = x_re;
        double r1626042 = x_im;
        double r1626043 = r1626041 - r1626042;
        double r1626044 = r1626041 * r1626043;
        double r1626045 = r1626042 + r1626041;
        double r1626046 = r1626044 * r1626045;
        double r1626047 = /*Error: no posit support in C */;
        double r1626048 = r1626041 + r1626041;
        double r1626049 = r1626042 * r1626048;
        double r1626050 = /*Error: no posit support in C */;
        double r1626051 = /*Error: no posit support in C */;
        return r1626051;
}

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-fdp-sub0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019165 
(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)))