Average Error: 0.4 → 0.4
Time: 17.5s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) + \left(x.re + x.re\right) \cdot x.re\right)\]
\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(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) + \left(x.re + x.re\right) \cdot x.re\right)
double f(double x_re, double x_im) {
        double r2977224 = x_re;
        double r2977225 = r2977224 * r2977224;
        double r2977226 = x_im;
        double r2977227 = r2977226 * r2977226;
        double r2977228 = r2977225 - r2977227;
        double r2977229 = r2977228 * r2977226;
        double r2977230 = r2977224 * r2977226;
        double r2977231 = r2977226 * r2977224;
        double r2977232 = r2977230 + r2977231;
        double r2977233 = r2977232 * r2977224;
        double r2977234 = r2977229 + r2977233;
        return r2977234;
}
↓
double f(double x_re, double x_im) {
        double r2977235 = x_im;
        double r2977236 = x_re;
        double r2977237 = r2977236 - r2977235;
        double r2977238 = r2977235 + r2977236;
        double r2977239 = r2977237 * r2977238;
        double r2977240 = r2977236 + r2977236;
        double r2977241 = r2977240 * r2977236;
        double r2977242 = r2977239 + r2977241;
        double r2977243 = r2977235 * r2977242;
        return r2977243;
}

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

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

Reproduce

herbie shell --seed 2019134 
(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)))