Average Error: 7.4 → 0.2
Time: 3.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(3, \left(x.re \cdot \left(-x.im\right)\right) \cdot x.im, {x.re}^{3}\right)\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im
↓
\mathsf{fma}\left(3, \left(x.re \cdot \left(-x.im\right)\right) \cdot x.im, {x.re}^{3}\right)
double f(double x_re, double x_im) {
        double r198712 = x_re;
        double r198713 = r198712 * r198712;
        double r198714 = x_im;
        double r198715 = r198714 * r198714;
        double r198716 = r198713 - r198715;
        double r198717 = r198716 * r198712;
        double r198718 = r198712 * r198714;
        double r198719 = r198714 * r198712;
        double r198720 = r198718 + r198719;
        double r198721 = r198720 * r198714;
        double r198722 = r198717 - r198721;
        return r198722;
}
↓
double f(double x_re, double x_im) {
        double r198723 = 3.0;
        double r198724 = x_re;
        double r198725 = x_im;
        double r198726 = -r198725;
        double r198727 = r198724 * r198726;
        double r198728 = r198727 * r198725;
        double r198729 = pow(r198724, r198723);
        double r198730 = fma(r198723, r198728, r198729);
        return r198730;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.4
Target0.3
Herbie0.2
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 7.4

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]

  2. Simplified7.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, x.re \cdot \left(-x.im \cdot x.im\right), {x.re}^{3}\right)}\]
  3. Using strategy rm

  4. Applied distribute-lft-neg-in7.4

    \[\leadsto \mathsf{fma}\left(3, x.re \cdot \color{blue}{\left(\left(-x.im\right) \cdot x.im\right)}, {x.re}^{3}\right)\]

  5. Applied associate-*r*0.2

    \[\leadsto \mathsf{fma}\left(3, \color{blue}{\left(x.re \cdot \left(-x.im\right)\right) \cdot x.im}, {x.re}^{3}\right)\]

  6. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(3, \left(x.re \cdot \left(-x.im\right)\right) \cdot x.im, {x.re}^{3}\right)\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

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