Average Error: 0.1 → 0.0
Time: 6.4s
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 r156839 = x;
        double r156840 = y;
        double r156841 = r156839 + r156840;
        double r156842 = r156841 + r156840;
        double r156843 = r156842 + r156839;
        double r156844 = z;
        double r156845 = r156843 + r156844;
        double r156846 = r156845 + r156839;
        return r156846;
}
↓
double f(double x, double y, double z) {
        double r156847 = 3.0;
        double r156848 = x;
        double r156849 = y;
        double r156850 = 2.0;
        double r156851 = z;
        double r156852 = fma(r156849, r156850, r156851);
        double r156853 = fma(r156847, r156848, r156852);
        return r156853;
}

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 2020045 +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))