Average Error: 0.1 → 0.0
Time: 979.0ms
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 r129486 = x;
        double r129487 = y;
        double r129488 = r129486 + r129487;
        double r129489 = r129488 + r129487;
        double r129490 = r129489 + r129486;
        double r129491 = z;
        double r129492 = r129490 + r129491;
        double r129493 = r129492 + r129486;
        return r129493;
}
↓
double f(double x, double y, double z) {
        double r129494 = 3.0;
        double r129495 = x;
        double r129496 = 2.0;
        double r129497 = y;
        double r129498 = z;
        double r129499 = fma(r129496, r129497, r129498);
        double r129500 = fma(r129494, r129495, r129499);
        return r129500;
}

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 2019347 +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))