Average Error: 0.1 → 0.0
Time: 4.2s
Precision: 64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x\]
↓
\[\mathsf{fma}\left(3, x, \mathsf{fma}\left(y, 2, z\right)\right)\]
\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 r14031 = x;
        double r14032 = y;
        double r14033 = r14031 + r14032;
        double r14034 = r14033 + r14032;
        double r14035 = r14034 + r14031;
        double r14036 = z;
        double r14037 = r14035 + r14036;
        double r14038 = r14037 + r14031;
        return r14038;
}
↓
double f(double x, double y, double z) {
        double r14039 = 3.0;
        double r14040 = x;
        double r14041 = y;
        double r14042 = 2.0;
        double r14043 = z;
        double r14044 = fma(r14041, r14042, r14043);
        double r14045 = fma(r14039, r14040, r14044);
        return r14045;
}

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 2019310 +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))