Average Error: 0.1 → 0.0
Time: 4.3s
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 r129848 = x;
        double r129849 = y;
        double r129850 = r129848 + r129849;
        double r129851 = r129850 + r129849;
        double r129852 = r129851 + r129848;
        double r129853 = z;
        double r129854 = r129852 + r129853;
        double r129855 = r129854 + r129848;
        return r129855;
}
↓
double f(double x, double y, double z) {
        double r129856 = 3.0;
        double r129857 = x;
        double r129858 = y;
        double r129859 = 2.0;
        double r129860 = z;
        double r129861 = fma(r129858, r129859, r129860);
        double r129862 = fma(r129856, r129857, r129861);
        return r129862;
}

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 2019303 +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))