Average Error: 0.1 → 0.0
Time: 1.3s
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 r164798 = x;
        double r164799 = y;
        double r164800 = r164798 + r164799;
        double r164801 = r164800 + r164799;
        double r164802 = r164801 + r164798;
        double r164803 = z;
        double r164804 = r164802 + r164803;
        double r164805 = r164804 + r164798;
        return r164805;
}
↓
double f(double x, double y, double z) {
        double r164806 = 3.0;
        double r164807 = x;
        double r164808 = 2.0;
        double r164809 = y;
        double r164810 = z;
        double r164811 = fma(r164808, r164809, r164810);
        double r164812 = fma(r164806, r164807, r164811);
        return r164812;
}

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 2020036 +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))