Average Error: 0.4 → 0.4
Time: 10.6s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\]
\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(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)
double f(double x_re, double x_im) {
        double r1354354 = x_re;
        double r1354355 = r1354354 * r1354354;
        double r1354356 = x_im;
        double r1354357 = r1354356 * r1354356;
        double r1354358 = r1354355 - r1354357;
        double r1354359 = r1354358 * r1354356;
        double r1354360 = r1354354 * r1354356;
        double r1354361 = r1354356 * r1354354;
        double r1354362 = r1354360 + r1354361;
        double r1354363 = r1354362 * r1354354;
        double r1354364 = r1354359 + r1354363;
        return r1354364;
}
↓
double f(double x_re, double x_im) {
        double r1354365 = x_im;
        double r1354366 = x_re;
        double r1354367 = r1354365 + r1354366;
        double r1354368 = r1354366 - r1354365;
        double r1354369 = r1354367 * r1354368;
        double r1354370 = r1354366 + r1354366;
        double r1354371 = r1354366 * r1354370;
        double r1354372 = r1354369 + r1354371;
        double r1354373 = r1354365 * r1354372;
        return r1354373;
}

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

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(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)))