Average Error: 0.4 → 0.4
Time: 10.4s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
\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 + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
double f(double x_re, double x_im) {
        double r1409276 = x_re;
        double r1409277 = r1409276 * r1409276;
        double r1409278 = x_im;
        double r1409279 = r1409278 * r1409278;
        double r1409280 = r1409277 - r1409279;
        double r1409281 = r1409280 * r1409278;
        double r1409282 = r1409276 * r1409278;
        double r1409283 = r1409278 * r1409276;
        double r1409284 = r1409282 + r1409283;
        double r1409285 = r1409284 * r1409276;
        double r1409286 = r1409281 + r1409285;
        return r1409286;
}
↓
double f(double x_re, double x_im) {
        double r1409287 = x_re;
        double r1409288 = x_im;
        double r1409289 = r1409287 + r1409288;
        double r1409290 = r1409287 - r1409288;
        double r1409291 = r1409290 * r1409288;
        double r1409292 = r1409289 * r1409291;
        double r1409293 = r1409287 * r1409288;
        double r1409294 = r1409288 * r1409287;
        double r1409295 = r1409293 + r1409294;
        double r1409296 = r1409295 * r1409287;
        double r1409297 = r1409292 + r1409296;
        return r1409297;
}

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. Using strategy rm

  3. Applied difference-of-squares0.4

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

  4. Applied associate-*l*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019107 +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)))