Average Error: 0.1 → 0.0
Time: 956.0ms
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 r117399 = x;
        double r117400 = y;
        double r117401 = r117399 + r117400;
        double r117402 = r117401 + r117400;
        double r117403 = r117402 + r117399;
        double r117404 = z;
        double r117405 = r117403 + r117404;
        double r117406 = r117405 + r117399;
        return r117406;
}
↓
double f(double x, double y, double z) {
        double r117407 = 3.0;
        double r117408 = x;
        double r117409 = 2.0;
        double r117410 = y;
        double r117411 = z;
        double r117412 = fma(r117409, r117410, r117411);
        double r117413 = fma(r117407, r117408, r117412);
        return r117413;
}

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 2020024 +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))