Average Error: 0.1 → 0.0
Time: 1.1s
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 r153240 = x;
        double r153241 = y;
        double r153242 = r153240 + r153241;
        double r153243 = r153242 + r153241;
        double r153244 = r153243 + r153240;
        double r153245 = z;
        double r153246 = r153244 + r153245;
        double r153247 = r153246 + r153240;
        return r153247;
}
↓
double f(double x, double y, double z) {
        double r153248 = 3.0;
        double r153249 = x;
        double r153250 = 2.0;
        double r153251 = y;
        double r153252 = z;
        double r153253 = fma(r153250, r153251, r153252);
        double r153254 = fma(r153248, r153249, r153253);
        return r153254;
}

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 2020060 +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))