Average Error: 0.1 → 0.0
Time: 997.0ms
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 r129235 = x;
        double r129236 = y;
        double r129237 = r129235 + r129236;
        double r129238 = r129237 + r129236;
        double r129239 = r129238 + r129235;
        double r129240 = z;
        double r129241 = r129239 + r129240;
        double r129242 = r129241 + r129235;
        return r129242;
}
↓
double f(double x, double y, double z) {
        double r129243 = 3.0;
        double r129244 = x;
        double r129245 = 2.0;
        double r129246 = y;
        double r129247 = z;
        double r129248 = fma(r129245, r129246, r129247);
        double r129249 = fma(r129243, r129244, r129248);
        return r129249;
}

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 2019346 +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))