Average Error: 0.1 → 0.0
Time: 1.2s
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 r217666 = x;
        double r217667 = y;
        double r217668 = r217666 + r217667;
        double r217669 = r217668 + r217667;
        double r217670 = r217669 + r217666;
        double r217671 = z;
        double r217672 = r217670 + r217671;
        double r217673 = r217672 + r217666;
        return r217673;
}
↓
double f(double x, double y, double z) {
        double r217674 = 3.0;
        double r217675 = x;
        double r217676 = 2.0;
        double r217677 = y;
        double r217678 = z;
        double r217679 = fma(r217676, r217677, r217678);
        double r217680 = fma(r217674, r217675, r217679);
        return r217680;
}

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 2020081 +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))