Average Error: 0.1 → 0.0
Time: 983.0ms
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 r151209 = x;
        double r151210 = y;
        double r151211 = r151209 + r151210;
        double r151212 = r151211 + r151210;
        double r151213 = r151212 + r151209;
        double r151214 = z;
        double r151215 = r151213 + r151214;
        double r151216 = r151215 + r151209;
        return r151216;
}
↓
double f(double x, double y, double z) {
        double r151217 = 3.0;
        double r151218 = x;
        double r151219 = 2.0;
        double r151220 = y;
        double r151221 = z;
        double r151222 = fma(r151219, r151220, r151221);
        double r151223 = fma(r151217, r151218, r151222);
        return r151223;
}

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 2020036 +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))