Average Error: 7.7 → 0.2
Time: 20.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(x.im, -3 \cdot \left(x.im \cdot x.re\right), {x.re}^{3}\right)\]
\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 r121579 = x_re;
        double r121580 = r121579 * r121579;
        double r121581 = x_im;
        double r121582 = r121581 * r121581;
        double r121583 = r121580 - r121582;
        double r121584 = r121583 * r121579;
        double r121585 = r121579 * r121581;
        double r121586 = r121581 * r121579;
        double r121587 = r121585 + r121586;
        double r121588 = r121587 * r121581;
        double r121589 = r121584 - r121588;
        return r121589;
}
↓
double f(double x_re, double x_im) {
        double r121590 = x_im;
        double r121591 = -3.0;
        double r121592 = x_re;
        double r121593 = r121590 * r121592;
        double r121594 = r121591 * r121593;
        double r121595 = 3.0;
        double r121596 = pow(r121592, r121595);
        double r121597 = fma(r121590, r121594, r121596);
        return r121597;
}

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)))