Average Error: 7.0 → 0.2
Time: 18.9s
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 r8954598 = x_re;
        double r8954599 = r8954598 * r8954598;
        double r8954600 = x_im;
        double r8954601 = r8954600 * r8954600;
        double r8954602 = r8954599 - r8954601;
        double r8954603 = r8954602 * r8954600;
        double r8954604 = r8954598 * r8954600;
        double r8954605 = r8954600 * r8954598;
        double r8954606 = r8954604 + r8954605;
        double r8954607 = r8954606 * r8954598;
        double r8954608 = r8954603 + r8954607;
        return r8954608;
}
↓
double f(double x_re, double x_im) {
        double r8954609 = x_im;
        double r8954610 = x_re;
        double r8954611 = r8954609 + r8954610;
        double r8954612 = r8954610 - r8954609;
        double r8954613 = r8954612 * r8954609;
        double r8954614 = r8954610 * r8954609;
        double r8954615 = r8954614 + r8954614;
        double r8954616 = r8954615 * r8954610;
        double r8954617 = fma(r8954611, r8954613, r8954616);
        return r8954617;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.0
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 7.0

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

  3. Applied difference-of-squares7.0

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

  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 2019132 +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)))