Average Error: 7.3 → 0.2
Time: 27.0s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(x.re, \left(3 \cdot x.im\right) \cdot x.re, -{x.im}^{3}\right)\]
\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(x.re, \left(3 \cdot x.im\right) \cdot x.re, -{x.im}^{3}\right)
double f(double x_re, double x_im) {
        double r169743 = x_re;
        double r169744 = r169743 * r169743;
        double r169745 = x_im;
        double r169746 = r169745 * r169745;
        double r169747 = r169744 - r169746;
        double r169748 = r169747 * r169745;
        double r169749 = r169743 * r169745;
        double r169750 = r169745 * r169743;
        double r169751 = r169749 + r169750;
        double r169752 = r169751 * r169743;
        double r169753 = r169748 + r169752;
        return r169753;
}
↓
double f(double x_re, double x_im) {
        double r169754 = x_re;
        double r169755 = 3.0;
        double r169756 = x_im;
        double r169757 = r169755 * r169756;
        double r169758 = r169757 * r169754;
        double r169759 = pow(r169756, r169755);
        double r169760 = -r169759;
        double r169761 = fma(r169754, r169758, r169760);
        return r169761;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.3
Target0.3
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 7.3

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :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)))