Average Error: 0.4 → 0.3
Time: 24.9s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(\left(\left(x.im \cdot \left(\frac{x.re}{x.im}\right)\right) \cdot \left(x.re - x.im\right)\right)\right)\right)\right), \left(x.re \cdot \left(\frac{x.im}{x.im}\right)\right), x.re\right)\right)\]
\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(\left(\left(x.im \cdot \left(\frac{x.re}{x.im}\right)\right) \cdot \left(x.re - x.im\right)\right)\right)\right)\right), \left(x.re \cdot \left(\frac{x.im}{x.im}\right)\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r2023418 = x_re;
        double r2023419 = r2023418 * r2023418;
        double r2023420 = x_im;
        double r2023421 = r2023420 * r2023420;
        double r2023422 = r2023419 - r2023421;
        double r2023423 = r2023422 * r2023420;
        double r2023424 = r2023418 * r2023420;
        double r2023425 = r2023420 * r2023418;
        double r2023426 = r2023424 + r2023425;
        double r2023427 = r2023426 * r2023418;
        double r2023428 = r2023423 + r2023427;
        return r2023428;
}
↓
double f(double x_re, double x_im) {
        double r2023429 = x_im;
        double r2023430 = x_re;
        double r2023431 = r2023430 + r2023429;
        double r2023432 = r2023429 * r2023431;
        double r2023433 = r2023430 - r2023429;
        double r2023434 = r2023432 * r2023433;
        double r2023435 = /*Error: no posit support in C */;
        double r2023436 = /*Error: no posit support in C */;
        double r2023437 = /*Error: no posit support in C */;
        double r2023438 = r2023429 + r2023429;
        double r2023439 = r2023430 * r2023438;
        double r2023440 = /*Error: no posit support in C */;
        double r2023441 = /*Error: no posit support in C */;
        return r2023441;
}

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

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

  7. Applied distribute-rgt-in0.3

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

  9. Applied introduce-quire0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 
(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)))