Average Error: 0.1 → 0.0
Time: 1.0s
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 r125608 = x;
        double r125609 = y;
        double r125610 = r125608 + r125609;
        double r125611 = r125610 + r125609;
        double r125612 = r125611 + r125608;
        double r125613 = z;
        double r125614 = r125612 + r125613;
        double r125615 = r125614 + r125608;
        return r125615;
}
↓
double f(double x, double y, double z) {
        double r125616 = 3.0;
        double r125617 = x;
        double r125618 = 2.0;
        double r125619 = y;
        double r125620 = z;
        double r125621 = fma(r125618, r125619, r125620);
        double r125622 = fma(r125616, r125617, r125621);
        return r125622;
}

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 2020018 +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))