Average Error: 0.1 → 0.0
Time: 1.0s
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 r136679 = x;
        double r136680 = y;
        double r136681 = r136679 + r136680;
        double r136682 = r136681 + r136680;
        double r136683 = r136682 + r136679;
        double r136684 = z;
        double r136685 = r136683 + r136684;
        double r136686 = r136685 + r136679;
        return r136686;
}
↓
double f(double x, double y, double z) {
        double r136687 = 3.0;
        double r136688 = x;
        double r136689 = 2.0;
        double r136690 = y;
        double r136691 = z;
        double r136692 = fma(r136689, r136690, r136691);
        double r136693 = fma(r136687, r136688, r136692);
        return r136693;
}

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 2020001 +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))