Average Error: 0.1 → 0.0
Time: 16.6s
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 r7909905 = x;
        double r7909906 = y;
        double r7909907 = r7909905 + r7909906;
        double r7909908 = r7909907 + r7909906;
        double r7909909 = r7909908 + r7909905;
        double r7909910 = z;
        double r7909911 = r7909909 + r7909910;
        double r7909912 = r7909911 + r7909905;
        return r7909912;
}
↓
double f(double x, double y, double z) {
        double r7909913 = 3.0;
        double r7909914 = x;
        double r7909915 = y;
        double r7909916 = 2.0;
        double r7909917 = z;
        double r7909918 = fma(r7909915, r7909916, r7909917);
        double r7909919 = fma(r7909913, r7909914, r7909918);
        return r7909919;
}

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 2019200 +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))