Average Error: 0.1 → 0.0
Time: 2.5s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]
\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)
double f(double x, double y, double z) {
        double r107191 = x;
        double r107192 = y;
        double r107193 = r107191 + r107192;
        double r107194 = r107193 + r107192;
        double r107195 = r107194 + r107191;
        double r107196 = z;
        double r107197 = r107195 + r107196;
        double r107198 = r107197 + r107191;
        return r107198;
}
↓
double f(double x, double y, double z) {
        double r107199 = 3.0;
        double r107200 = x;
        double r107201 = y;
        double r107202 = 2.0;
        double r107203 = z;
        double r107204 = fma(r107201, r107202, r107203);
        double r107205 = fma(r107199, r107200, r107204);
        return r107205;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]

Reproduce

herbie shell --seed 2019351 +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))