Average Error: 0.1 → 0.0
Time: 936.0ms
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 r129963 = x;
        double r129964 = y;
        double r129965 = r129963 + r129964;
        double r129966 = r129965 + r129964;
        double r129967 = r129966 + r129963;
        double r129968 = z;
        double r129969 = r129967 + r129968;
        double r129970 = r129969 + r129963;
        return r129970;
}
↓
double f(double x, double y, double z) {
        double r129971 = 3.0;
        double r129972 = x;
        double r129973 = 2.0;
        double r129974 = y;
        double r129975 = z;
        double r129976 = fma(r129973, r129974, r129975);
        double r129977 = fma(r129971, r129972, r129976);
        return r129977;
}

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 2020062 +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))