Average Error: 0.0 → 0.0
Time: 1.7s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y, z\right) + \frac{x}{2}\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y, z\right) + \frac{x}{2}
double f(double x, double y, double z) {
        double r281876 = x;
        double r281877 = 2.0;
        double r281878 = r281876 / r281877;
        double r281879 = y;
        double r281880 = r281879 * r281876;
        double r281881 = r281878 + r281880;
        double r281882 = z;
        double r281883 = r281881 + r281882;
        return r281883;
}
↓
double f(double x, double y, double z) {
        double r281884 = x;
        double r281885 = y;
        double r281886 = z;
        double r281887 = fma(r281884, r281885, r281886);
        double r281888 = 2.0;
        double r281889 = r281884 / r281888;
        double r281890 = r281887 + r281889;
        return r281890;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[\left(\frac{x}{2} + y \cdot x\right) + z\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2) (* y x)) z))