Average Error: 0.1 → 0.0
Time: 17.4s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(y, 2, \mathsf{fma}\left(3, x, z\right)\right)\]
\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(y, 2, \mathsf{fma}\left(3, x, z\right)\right)
double f(double x, double y, double z) {
        double r98978 = x;
        double r98979 = y;
        double r98980 = r98978 + r98979;
        double r98981 = r98980 + r98979;
        double r98982 = r98981 + r98978;
        double r98983 = z;
        double r98984 = r98982 + r98983;
        double r98985 = r98984 + r98978;
        return r98985;
}
↓
double f(double x, double y, double z) {
        double r98986 = y;
        double r98987 = 2.0;
        double r98988 = 3.0;
        double r98989 = x;
        double r98990 = z;
        double r98991 = fma(r98988, r98989, r98990);
        double r98992 = fma(r98986, r98987, r98991);
        return r98992;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, 2, \mathsf{fma}\left(3, x, z\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, 2, \mathsf{fma}\left(3, x, z\right)\right)\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  (+ (+ (+ (+ (+ x y) y) x) z) x))