Average Error: 0.1 → 0.0
Time: 4.6s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]
\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)
double f(double x, double y, double z) {
        double r114216 = x;
        double r114217 = y;
        double r114218 = r114216 + r114217;
        double r114219 = r114218 + r114217;
        double r114220 = r114219 + r114216;
        double r114221 = z;
        double r114222 = r114220 + r114221;
        double r114223 = r114222 + r114216;
        return r114223;
}
↓
double f(double x, double y, double z) {
        double r114224 = 3.0;
        double r114225 = x;
        double r114226 = y;
        double r114227 = 2.0;
        double r114228 = z;
        double r114229 = fma(r114226, r114227, r114228);
        double r114230 = fma(r114224, r114225, r114229);
        return r114230;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]

Reproduce

herbie shell --seed 2019326 +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))