Average Error: 0.1 → 0.0
Time: 4.1s
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 r10913 = x;
        double r10914 = y;
        double r10915 = r10913 + r10914;
        double r10916 = r10915 + r10914;
        double r10917 = r10916 + r10913;
        double r10918 = z;
        double r10919 = r10917 + r10918;
        double r10920 = r10919 + r10913;
        return r10920;
}
↓
double f(double x, double y, double z) {
        double r10921 = 3.0;
        double r10922 = x;
        double r10923 = y;
        double r10924 = 2.0;
        double r10925 = z;
        double r10926 = fma(r10923, r10924, r10925);
        double r10927 = fma(r10921, r10922, r10926);
        return r10927;
}

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 2019315 +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))