Average Error: 0.1 → 0.0
Time: 1.1s
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 r149438 = x;
        double r149439 = y;
        double r149440 = r149438 + r149439;
        double r149441 = r149440 + r149439;
        double r149442 = r149441 + r149438;
        double r149443 = z;
        double r149444 = r149442 + r149443;
        double r149445 = r149444 + r149438;
        return r149445;
}
↓
double f(double x, double y, double z) {
        double r149446 = 3.0;
        double r149447 = x;
        double r149448 = 2.0;
        double r149449 = y;
        double r149450 = z;
        double r149451 = fma(r149448, r149449, r149450);
        double r149452 = fma(r149446, r149447, r149451);
        return r149452;
}

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 2020033 +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))