Average Error: 0.4 → 0.3
Time: 48.1s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot \left(x.im + x.im\right)\right), x.re\right)\right)\]
\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.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot \left(x.im + x.im\right)\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r4226274 = x_re;
        double r4226275 = r4226274 * r4226274;
        double r4226276 = x_im;
        double r4226277 = r4226276 * r4226276;
        double r4226278 = r4226275 - r4226277;
        double r4226279 = r4226278 * r4226276;
        double r4226280 = r4226274 * r4226276;
        double r4226281 = r4226276 * r4226274;
        double r4226282 = r4226280 + r4226281;
        double r4226283 = r4226282 * r4226274;
        double r4226284 = r4226279 + r4226283;
        return r4226284;
}
↓
double f(double x_re, double x_im) {
        double r4226285 = x_im;
        double r4226286 = x_re;
        double r4226287 = r4226286 - r4226285;
        double r4226288 = r4226285 * r4226287;
        double r4226289 = r4226285 + r4226286;
        double r4226290 = r4226288 * r4226289;
        double r4226291 = /*Error: no posit support in C */;
        double r4226292 = r4226285 + r4226285;
        double r4226293 = r4226286 * r4226292;
        double r4226294 = /*Error: no posit support in C */;
        double r4226295 = /*Error: no posit support in C */;
        return r4226295;
}

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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019142 +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)))