Average Error: 0.4 → 0.4
Time: 18.6s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(x.re \cdot \left(x.re + \left(x.re + x.re\right)\right) - x.im \cdot x.im\right) \cdot x.im\]
\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(x.re + \left(x.re + x.re\right)\right) - x.im \cdot x.im\right) \cdot x.im
double f(double x_re, double x_im) {
        double r2548230 = x_re;
        double r2548231 = r2548230 * r2548230;
        double r2548232 = x_im;
        double r2548233 = r2548232 * r2548232;
        double r2548234 = r2548231 - r2548233;
        double r2548235 = r2548234 * r2548232;
        double r2548236 = r2548230 * r2548232;
        double r2548237 = r2548232 * r2548230;
        double r2548238 = r2548236 + r2548237;
        double r2548239 = r2548238 * r2548230;
        double r2548240 = r2548235 + r2548239;
        return r2548240;
}
↓
double f(double x_re, double x_im) {
        double r2548241 = x_re;
        double r2548242 = r2548241 + r2548241;
        double r2548243 = r2548241 + r2548242;
        double r2548244 = r2548241 * r2548243;
        double r2548245 = x_im;
        double r2548246 = r2548245 * r2548245;
        double r2548247 = r2548244 - r2548246;
        double r2548248 = r2548247 * r2548245;
        return r2548248;
}

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. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019133 
(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)))