Average Error: 0.1 → 0.0
Time: 939.0ms
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(3, x, \mathsf{fma}\left(2, y, z\right)\right)\]
\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(3, x, \mathsf{fma}\left(2, y, z\right)\right)
double f(double x, double y, double z) {
        double r122716 = x;
        double r122717 = y;
        double r122718 = r122716 + r122717;
        double r122719 = r122718 + r122717;
        double r122720 = r122719 + r122716;
        double r122721 = z;
        double r122722 = r122720 + r122721;
        double r122723 = r122722 + r122716;
        return r122723;
}
↓
double f(double x, double y, double z) {
        double r122724 = 3.0;
        double r122725 = x;
        double r122726 = 2.0;
        double r122727 = y;
        double r122728 = z;
        double r122729 = fma(r122726, r122727, r122728);
        double r122730 = fma(r122724, r122725, r122729);
        return r122730;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020039 +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))