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 r126836 = x;
        double r126837 = y;
        double r126838 = r126836 + r126837;
        double r126839 = r126838 + r126837;
        double r126840 = r126839 + r126836;
        double r126841 = z;
        double r126842 = r126840 + r126841;
        double r126843 = r126842 + r126836;
        return r126843;
}
↓
double f(double x, double y, double z) {
        double r126844 = 3.0;
        double r126845 = x;
        double r126846 = 2.0;
        double r126847 = y;
        double r126848 = z;
        double r126849 = fma(r126846, r126847, r126848);
        double r126850 = fma(r126844, r126845, r126849);
        return r126850;
}

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 2019353 +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))