Average Error: 0.1 → 0.0
Time: 961.0ms
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 r121162 = x;
        double r121163 = y;
        double r121164 = r121162 + r121163;
        double r121165 = r121164 + r121163;
        double r121166 = r121165 + r121162;
        double r121167 = z;
        double r121168 = r121166 + r121167;
        double r121169 = r121168 + r121162;
        return r121169;
}
↓
double f(double x, double y, double z) {
        double r121170 = 3.0;
        double r121171 = x;
        double r121172 = 2.0;
        double r121173 = y;
        double r121174 = z;
        double r121175 = fma(r121172, r121173, r121174);
        double r121176 = fma(r121170, r121171, r121175);
        return r121176;
}

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 2020025 +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))