Average Error: 0.1 → 0.0
Time: 987.0ms
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 r202365 = x;
        double r202366 = y;
        double r202367 = r202365 + r202366;
        double r202368 = r202367 + r202366;
        double r202369 = r202368 + r202365;
        double r202370 = z;
        double r202371 = r202369 + r202370;
        double r202372 = r202371 + r202365;
        return r202372;
}
↓
double f(double x, double y, double z) {
        double r202373 = 3.0;
        double r202374 = x;
        double r202375 = 2.0;
        double r202376 = y;
        double r202377 = z;
        double r202378 = fma(r202375, r202376, r202377);
        double r202379 = fma(r202373, r202374, r202378);
        return r202379;
}

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 2020002 +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))