Average Error: 0.1 → 0.0
Time: 1.2s
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 r122970 = x;
        double r122971 = y;
        double r122972 = r122970 + r122971;
        double r122973 = r122972 + r122971;
        double r122974 = r122973 + r122970;
        double r122975 = z;
        double r122976 = r122974 + r122975;
        double r122977 = r122976 + r122970;
        return r122977;
}
↓
double f(double x, double y, double z) {
        double r122978 = 3.0;
        double r122979 = x;
        double r122980 = 2.0;
        double r122981 = y;
        double r122982 = z;
        double r122983 = fma(r122980, r122981, r122982);
        double r122984 = fma(r122978, r122979, r122983);
        return r122984;
}

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 2020100 +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))