Average Error: 0.1 → 0.0
Time: 956.0ms
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 r140370 = x;
        double r140371 = y;
        double r140372 = r140370 + r140371;
        double r140373 = r140372 + r140371;
        double r140374 = r140373 + r140370;
        double r140375 = z;
        double r140376 = r140374 + r140375;
        double r140377 = r140376 + r140370;
        return r140377;
}
↓
double f(double x, double y, double z) {
        double r140378 = 3.0;
        double r140379 = x;
        double r140380 = 2.0;
        double r140381 = y;
        double r140382 = z;
        double r140383 = fma(r140380, r140381, r140382);
        double r140384 = fma(r140378, r140379, r140383);
        return r140384;
}

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 2020020 +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))