Average Error: 0.4 → 0.3
Time: 24.6s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\mathsf{qms}\left(\left(\left(\left(x.re \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot \left(x.im + x.im\right)\right), x.im\right)\right)\]
\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)
↓
\left(\mathsf{qms}\left(\left(\left(\left(x.re \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.im\right)\right)
double f(double x_re, double x_im) {
        double r4117548 = x_re;
        double r4117549 = r4117548 * r4117548;
        double r4117550 = x_im;
        double r4117551 = r4117550 * r4117550;
        double r4117552 = r4117549 - r4117551;
        double r4117553 = r4117552 * r4117548;
        double r4117554 = r4117548 * r4117550;
        double r4117555 = r4117550 * r4117548;
        double r4117556 = r4117554 + r4117555;
        double r4117557 = r4117556 * r4117550;
        double r4117558 = r4117553 - r4117557;
        return r4117558;
}
↓
double f(double x_re, double x_im) {
        double r4117559 = x_re;
        double r4117560 = x_im;
        double r4117561 = r4117559 - r4117560;
        double r4117562 = r4117559 * r4117561;
        double r4117563 = r4117560 + r4117559;
        double r4117564 = r4117562 * r4117563;
        double r4117565 = /*Error: no posit support in C */;
        double r4117566 = r4117560 + r4117560;
        double r4117567 = r4117559 * r4117566;
        double r4117568 = /*Error: no posit support in C */;
        double r4117569 = /*Error: no posit support in C */;
        return r4117569;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

    \[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
  2. Using strategy rm

  3. Applied introduce-quire0.4

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

  4. Applied insert-quire-fdp-sub0.3

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

  5. Simplified0.3

    \[\leadsto \color{blue}{\left(\mathsf{qms}\left(\left(\left(\left(x.re \cdot \left(x.re - x.im\right)\right) \cdot \left(\frac{x.im}{x.re}\right)\right)\right), \left(\frac{\left(x.re \cdot x.im\right)}{\left(x.re \cdot x.im\right)}\right), x.im\right)\right)}\]
  6. Using strategy rm

  7. Applied distribute-lft-out0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  (-.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.re) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.im)))