Average Error: 0.1 → 0.0
Time: 17.5s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(2, y, \mathsf{fma}\left(3, x, z\right)\right)\]
\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(2, y, \mathsf{fma}\left(3, x, z\right)\right)
double f(double x, double y, double z) {
        double r6443162 = x;
        double r6443163 = y;
        double r6443164 = r6443162 + r6443163;
        double r6443165 = r6443164 + r6443163;
        double r6443166 = r6443165 + r6443162;
        double r6443167 = z;
        double r6443168 = r6443166 + r6443167;
        double r6443169 = r6443168 + r6443162;
        return r6443169;
}
↓
double f(double x, double y, double z) {
        double r6443170 = 2.0;
        double r6443171 = y;
        double r6443172 = 3.0;
        double r6443173 = x;
        double r6443174 = z;
        double r6443175 = fma(r6443172, r6443173, r6443174);
        double r6443176 = fma(r6443170, r6443171, r6443175);
        return r6443176;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  (+ (+ (+ (+ (+ x y) y) x) z) x))