Average Error: 0.1 → 0.0
Time: 916.0ms
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 r154054 = x;
        double r154055 = y;
        double r154056 = r154054 + r154055;
        double r154057 = r154056 + r154055;
        double r154058 = r154057 + r154054;
        double r154059 = z;
        double r154060 = r154058 + r154059;
        double r154061 = r154060 + r154054;
        return r154061;
}
↓
double f(double x, double y, double z) {
        double r154062 = 3.0;
        double r154063 = x;
        double r154064 = 2.0;
        double r154065 = y;
        double r154066 = z;
        double r154067 = fma(r154064, r154065, r154066);
        double r154068 = fma(r154062, r154063, r154067);
        return r154068;
}

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 2020064 +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))