Average Error: 0.1 → 0.0
Time: 8.3s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]
\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 r121681 = x;
        double r121682 = y;
        double r121683 = r121681 + r121682;
        double r121684 = r121683 + r121682;
        double r121685 = r121684 + r121681;
        double r121686 = z;
        double r121687 = r121685 + r121686;
        double r121688 = r121687 + r121681;
        return r121688;
}
↓
double f(double x, double y, double z) {
        double r121689 = 3.0;
        double r121690 = x;
        double r121691 = y;
        double r121692 = 2.0;
        double r121693 = z;
        double r121694 = fma(r121691, r121692, r121693);
        double r121695 = fma(r121689, r121690, r121694);
        return r121695;
}

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 2019323 +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))