Average Error: 0.1 → 0.0
Time: 3.6s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]
\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)
double f(double x, double y, double z) {
        double r89784 = x;
        double r89785 = y;
        double r89786 = r89784 + r89785;
        double r89787 = r89786 + r89785;
        double r89788 = r89787 + r89784;
        double r89789 = z;
        double r89790 = r89788 + r89789;
        double r89791 = r89790 + r89784;
        return r89791;
}
↓
double f(double x, double y, double z) {
        double r89792 = 3.0;
        double r89793 = x;
        double r89794 = y;
        double r89795 = 2.0;
        double r89796 = z;
        double r89797 = fma(r89794, r89795, r89796);
        double r89798 = fma(r89792, r89793, r89797);
        return r89798;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  :precision binary64
  (+ (+ (+ (+ (+ x y) y) x) z) x))