Average Error: 0.1 → 0.0
Time: 981.0ms
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 r144991 = x;
        double r144992 = y;
        double r144993 = r144991 + r144992;
        double r144994 = r144993 + r144992;
        double r144995 = r144994 + r144991;
        double r144996 = z;
        double r144997 = r144995 + r144996;
        double r144998 = r144997 + r144991;
        return r144998;
}
↓
double f(double x, double y, double z) {
        double r144999 = 3.0;
        double r145000 = x;
        double r145001 = 2.0;
        double r145002 = y;
        double r145003 = z;
        double r145004 = fma(r145001, r145002, r145003);
        double r145005 = fma(r144999, r145000, r145004);
        return r145005;
}

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 2019353 +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))