Average Error: 0.1 → 0.0
Time: 973.0ms
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 r149450 = x;
        double r149451 = y;
        double r149452 = r149450 + r149451;
        double r149453 = r149452 + r149451;
        double r149454 = r149453 + r149450;
        double r149455 = z;
        double r149456 = r149454 + r149455;
        double r149457 = r149456 + r149450;
        return r149457;
}
↓
double f(double x, double y, double z) {
        double r149458 = 3.0;
        double r149459 = x;
        double r149460 = 2.0;
        double r149461 = y;
        double r149462 = z;
        double r149463 = fma(r149460, r149461, r149462);
        double r149464 = fma(r149458, r149459, r149463);
        return r149464;
}

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 2020057 +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))