Average Error: 0.4 → 0.3
Time: 20.1s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.im\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.re\right)\right)\]
\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 r416600 = x_re;
        double r416601 = r416600 * r416600;
        double r416602 = x_im;
        double r416603 = r416602 * r416602;
        double r416604 = r416601 - r416603;
        double r416605 = r416604 * r416602;
        double r416606 = r416600 * r416602;
        double r416607 = r416602 * r416600;
        double r416608 = r416606 + r416607;
        double r416609 = r416608 * r416600;
        double r416610 = r416605 + r416609;
        return r416610;
}
↓
double f(double x_re, double x_im) {
        double r416611 = x_re;
        double r416612 = x_im;
        double r416613 = r416611 - r416612;
        double r416614 = r416612 + r416611;
        double r416615 = r416614 * r416612;
        double r416616 = r416613 * r416615;
        double r416617 = /*Error: no posit support in C */;
        double r416618 = r416611 * r416612;
        double r416619 = r416618 + r416618;
        double r416620 = /*Error: no posit support in C */;
        double r416621 = /*Error: no posit support in C */;
        return r416621;
}

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 
(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)))