Average Error: 0.1 → 0.0
Time: 1.5s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]
\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)
double f(double x, double y, double z) {
        double r19770 = x;
        double r19771 = y;
        double r19772 = r19770 + r19771;
        double r19773 = r19772 + r19771;
        double r19774 = r19773 + r19770;
        double r19775 = z;
        double r19776 = r19774 + r19775;
        double r19777 = r19776 + r19770;
        return r19777;
}
↓
double f(double x, double y, double z) {
        double r19778 = 3.0;
        double r19779 = x;
        double r19780 = y;
        double r19781 = 2.0;
        double r19782 = z;
        double r19783 = fma(r19780, r19781, r19782);
        double r19784 = fma(r19778, r19779, r19783);
        return r19784;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]

Reproduce

herbie shell --seed 2020045 +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))