Average Error: 0.4 → 0.3
Time: 25.9s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.im \cdot x.re + x.im \cdot x.re\right), x.re\right)\right)\]
\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.im \cdot x.re + x.im \cdot x.re\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r830330 = x_re;
        double r830331 = r830330 * r830330;
        double r830332 = x_im;
        double r830333 = r830332 * r830332;
        double r830334 = r830331 - r830333;
        double r830335 = r830334 * r830332;
        double r830336 = r830330 * r830332;
        double r830337 = r830332 * r830330;
        double r830338 = r830336 + r830337;
        double r830339 = r830338 * r830330;
        double r830340 = r830335 + r830339;
        return r830340;
}
↓
double f(double x_re, double x_im) {
        double r830341 = x_im;
        double r830342 = x_re;
        double r830343 = r830342 - r830341;
        double r830344 = r830341 * r830343;
        double r830345 = r830341 + r830342;
        double r830346 = r830344 * r830345;
        double r830347 = /*Error: no posit support in C */;
        double r830348 = r830341 * r830342;
        double r830349 = r830348 + r830348;
        double r830350 = /*Error: no posit support in C */;
        double r830351 = /*Error: no posit support in C */;
        return r830351;
}

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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 
(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)))