Average Error: 7.7 → 0.2
Time: 21.6s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(x.im, -3 \cdot \left(x.im \cdot x.re\right), {x.re}^{3}\right)\]
\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(x.im, -3 \cdot \left(x.im \cdot x.re\right), {x.re}^{3}\right)
double f(double x_re, double x_im) {
        double r140714 = x_re;
        double r140715 = r140714 * r140714;
        double r140716 = x_im;
        double r140717 = r140716 * r140716;
        double r140718 = r140715 - r140717;
        double r140719 = r140718 * r140714;
        double r140720 = r140714 * r140716;
        double r140721 = r140716 * r140714;
        double r140722 = r140720 + r140721;
        double r140723 = r140722 * r140716;
        double r140724 = r140719 - r140723;
        return r140724;
}
↓
double f(double x_re, double x_im) {
        double r140725 = x_im;
        double r140726 = -3.0;
        double r140727 = x_re;
        double r140728 = r140725 * r140727;
        double r140729 = r140726 * r140728;
        double r140730 = 3.0;
        double r140731 = pow(r140727, r140730);
        double r140732 = fma(r140725, r140729, r140731);
        return r140732;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.7
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 7.7

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

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 +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)))