Average Error: 6.5 → 0.2
Time: 52.4s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right), \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\right)\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
↓
\mathsf{fma}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right), \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\right)
double f(double x_re, double x_im) {
        double r47627691 = x_re;
        double r47627692 = r47627691 * r47627691;
        double r47627693 = x_im;
        double r47627694 = r47627693 * r47627693;
        double r47627695 = r47627692 - r47627694;
        double r47627696 = r47627695 * r47627693;
        double r47627697 = r47627691 * r47627693;
        double r47627698 = r47627693 * r47627691;
        double r47627699 = r47627697 + r47627698;
        double r47627700 = r47627699 * r47627691;
        double r47627701 = r47627696 + r47627700;
        return r47627701;
}
↓
double f(double x_re, double x_im) {
        double r47627702 = x_im;
        double r47627703 = x_re;
        double r47627704 = r47627702 + r47627703;
        double r47627705 = r47627703 - r47627702;
        double r47627706 = r47627705 * r47627702;
        double r47627707 = r47627703 * r47627702;
        double r47627708 = r47627707 + r47627707;
        double r47627709 = r47627708 * r47627703;
        double r47627710 = fma(r47627704, r47627706, r47627709);
        return r47627710;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.5
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.5

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

  3. Applied difference-of-squares6.5

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

  4. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
  5. Using strategy rm

  6. Applied fma-def0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))