Average Error: 0.1 → 0.0
Time: 1.9s
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 r148342 = x;
        double r148343 = y;
        double r148344 = r148342 + r148343;
        double r148345 = r148344 + r148343;
        double r148346 = r148345 + r148342;
        double r148347 = z;
        double r148348 = r148346 + r148347;
        double r148349 = r148348 + r148342;
        return r148349;
}
↓
double f(double x, double y, double z) {
        double r148350 = 3.0;
        double r148351 = x;
        double r148352 = 2.0;
        double r148353 = y;
        double r148354 = z;
        double r148355 = fma(r148352, r148353, r148354);
        double r148356 = fma(r148350, r148351, r148355);
        return r148356;
}

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 2020035 +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))