Average Error: 0.1 → 0.0
Time: 917.0ms
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 r154748 = x;
        double r154749 = y;
        double r154750 = r154748 + r154749;
        double r154751 = r154750 + r154749;
        double r154752 = r154751 + r154748;
        double r154753 = z;
        double r154754 = r154752 + r154753;
        double r154755 = r154754 + r154748;
        return r154755;
}
↓
double f(double x, double y, double z) {
        double r154756 = 3.0;
        double r154757 = x;
        double r154758 = 2.0;
        double r154759 = y;
        double r154760 = z;
        double r154761 = fma(r154758, r154759, r154760);
        double r154762 = fma(r154756, r154757, r154761);
        return r154762;
}

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 2020046 +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))