Average Error: 0.4 → 0.4
Time: 20.3s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[x.im \cdot \left(\frac{\left(\left(x.re - x.im\right) \cdot \left(\frac{x.im}{x.re}\right)\right)}{\left(\left(\frac{x.re}{x.re}\right) \cdot x.re\right)}\right)\]
\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)}
↓
x.im \cdot \left(\frac{\left(\left(x.re - x.im\right) \cdot \left(\frac{x.im}{x.re}\right)\right)}{\left(\left(\frac{x.re}{x.re}\right) \cdot x.re\right)}\right)
double f(double x_re, double x_im) {
        double r2036862 = x_re;
        double r2036863 = r2036862 * r2036862;
        double r2036864 = x_im;
        double r2036865 = r2036864 * r2036864;
        double r2036866 = r2036863 - r2036865;
        double r2036867 = r2036866 * r2036864;
        double r2036868 = r2036862 * r2036864;
        double r2036869 = r2036864 * r2036862;
        double r2036870 = r2036868 + r2036869;
        double r2036871 = r2036870 * r2036862;
        double r2036872 = r2036867 + r2036871;
        return r2036872;
}
↓
double f(double x_re, double x_im) {
        double r2036873 = x_im;
        double r2036874 = x_re;
        double r2036875 = r2036874 - r2036873;
        double r2036876 = r2036873 + r2036874;
        double r2036877 = r2036875 * r2036876;
        double r2036878 = r2036874 + r2036874;
        double r2036879 = r2036878 * r2036874;
        double r2036880 = r2036877 + r2036879;
        double r2036881 = r2036873 * r2036880;
        return r2036881;
}

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. Simplified0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 
(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)))