Average Error: 0.1 → 0.0
Time: 1.5s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(3, x, \mathsf{fma}\left(2, y, z\right)\right)\]
\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 r778 = x;
        double r779 = y;
        double r780 = r778 + r779;
        double r781 = r780 + r779;
        double r782 = r781 + r778;
        double r783 = z;
        double r784 = r782 + r783;
        double r785 = r784 + r778;
        return r785;
}
↓
double f(double x, double y, double z) {
        double r786 = 3.0;
        double r787 = x;
        double r788 = 2.0;
        double r789 = y;
        double r790 = z;
        double r791 = fma(r788, r789, r790);
        double r792 = fma(r786, r787, r791);
        return r792;
}

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 2020025 +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))