Average Error: 0.1 → 0.0
Time: 932.0ms
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 r127763 = x;
        double r127764 = y;
        double r127765 = r127763 + r127764;
        double r127766 = r127765 + r127764;
        double r127767 = r127766 + r127763;
        double r127768 = z;
        double r127769 = r127767 + r127768;
        double r127770 = r127769 + r127763;
        return r127770;
}
↓
double f(double x, double y, double z) {
        double r127771 = 3.0;
        double r127772 = x;
        double r127773 = 2.0;
        double r127774 = y;
        double r127775 = z;
        double r127776 = fma(r127773, r127774, r127775);
        double r127777 = fma(r127771, r127772, r127776);
        return r127777;
}

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 2020056 +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))