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 r228676 = x;
        double r228677 = 2.0;
        double r228678 = r228676 / r228677;
        double r228679 = y;
        double r228680 = r228679 * r228676;
        double r228681 = r228678 + r228680;
        double r228682 = z;
        double r228683 = r228681 + r228682;
        return r228683;
}
↓
double f(double x, double y, double z) {
        double r228684 = x;
        double r228685 = y;
        double r228686 = z;
        double r228687 = fma(r228684, r228685, r228686);
        double r228688 = 2.0;
        double r228689 = r228684 / r228688;
        double r228690 = r228687 + r228689;
        return r228690;
}

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 2019354 +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))