Average Error: 0.4 → 0.3
Time: 20.7s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.im\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.re\right)\right)\]
\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)}
↓
\left(\mathsf{qma}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.im\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r611356 = x_re;
        double r611357 = r611356 * r611356;
        double r611358 = x_im;
        double r611359 = r611358 * r611358;
        double r611360 = r611357 - r611359;
        double r611361 = r611360 * r611358;
        double r611362 = r611356 * r611358;
        double r611363 = r611358 * r611356;
        double r611364 = r611362 + r611363;
        double r611365 = r611364 * r611356;
        double r611366 = r611361 + r611365;
        return r611366;
}
↓
double f(double x_re, double x_im) {
        double r611367 = x_re;
        double r611368 = x_im;
        double r611369 = r611367 - r611368;
        double r611370 = r611368 + r611367;
        double r611371 = r611370 * r611368;
        double r611372 = r611369 * r611371;
        double r611373 = /*Error: no posit support in C */;
        double r611374 = r611367 * r611368;
        double r611375 = r611374 + r611374;
        double r611376 = /*Error: no posit support in C */;
        double r611377 = /*Error: no posit support in C */;
        return r611377;
}

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 introduce-quire0.4

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

  4. Applied insert-quire-fdp-add0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

    \[\leadsto \left(\mathsf{qma}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.im\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.re\right)\right)\]

Reproduce

herbie shell --seed 2019154 +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)))