Average Error: 0.1 → 0.0
Time: 980.0ms
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 r105344 = x;
        double r105345 = y;
        double r105346 = r105344 + r105345;
        double r105347 = r105346 + r105345;
        double r105348 = r105347 + r105344;
        double r105349 = z;
        double r105350 = r105348 + r105349;
        double r105351 = r105350 + r105344;
        return r105351;
}
↓
double f(double x, double y, double z) {
        double r105352 = 3.0;
        double r105353 = x;
        double r105354 = 2.0;
        double r105355 = y;
        double r105356 = z;
        double r105357 = fma(r105354, r105355, r105356);
        double r105358 = fma(r105352, r105353, r105357);
        return r105358;
}

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 2020047 +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))