Average Error: 0.1 → 0.0
Time: 997.0ms
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 r171954 = x;
        double r171955 = y;
        double r171956 = r171954 + r171955;
        double r171957 = r171956 + r171955;
        double r171958 = r171957 + r171954;
        double r171959 = z;
        double r171960 = r171958 + r171959;
        double r171961 = r171960 + r171954;
        return r171961;
}
↓
double f(double x, double y, double z) {
        double r171962 = 3.0;
        double r171963 = x;
        double r171964 = 2.0;
        double r171965 = y;
        double r171966 = z;
        double r171967 = fma(r171964, r171965, r171966);
        double r171968 = fma(r171962, r171963, r171967);
        return r171968;
}

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 2019354 +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))