Average Error: 0.1 → 0.0
Time: 1.0s
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 r143723 = x;
        double r143724 = y;
        double r143725 = r143723 + r143724;
        double r143726 = r143725 + r143724;
        double r143727 = r143726 + r143723;
        double r143728 = z;
        double r143729 = r143727 + r143728;
        double r143730 = r143729 + r143723;
        return r143730;
}
↓
double f(double x, double y, double z) {
        double r143731 = 3.0;
        double r143732 = x;
        double r143733 = 2.0;
        double r143734 = y;
        double r143735 = z;
        double r143736 = fma(r143733, r143734, r143735);
        double r143737 = fma(r143731, r143732, r143736);
        return r143737;
}

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 2020034 +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))