Average Error: 0.4 → 0.4
Time: 9.8s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right) - x.im \cdot x.im\right) \cdot x.im\]
\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)}
↓
\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right) - x.im \cdot x.im\right) \cdot x.im
double f(double x_re, double x_im) {
        double r1566848 = x_re;
        double r1566849 = r1566848 * r1566848;
        double r1566850 = x_im;
        double r1566851 = r1566850 * r1566850;
        double r1566852 = r1566849 - r1566851;
        double r1566853 = r1566852 * r1566850;
        double r1566854 = r1566848 * r1566850;
        double r1566855 = r1566850 * r1566848;
        double r1566856 = r1566854 + r1566855;
        double r1566857 = r1566856 * r1566848;
        double r1566858 = r1566853 + r1566857;
        return r1566858;
}
↓
double f(double x_re, double x_im) {
        double r1566859 = x_re;
        double r1566860 = r1566859 + r1566859;
        double r1566861 = r1566860 + r1566859;
        double r1566862 = r1566859 * r1566861;
        double r1566863 = x_im;
        double r1566864 = r1566863 * r1566863;
        double r1566865 = r1566862 - r1566864;
        double r1566866 = r1566865 * r1566863;
        return r1566866;
}

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. Simplified0.3

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

  3. Simplified0.4

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

  4. Final simplification0.4

    \[\leadsto \left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right) - x.im \cdot x.im\right) \cdot x.im\]

Reproduce

herbie shell --seed 2019104 
(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)))