Average Error: 0.0 → 0.0
Time: 1.9s
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 r241663 = x;
        double r241664 = 2.0;
        double r241665 = r241663 / r241664;
        double r241666 = y;
        double r241667 = r241666 * r241663;
        double r241668 = r241665 + r241667;
        double r241669 = z;
        double r241670 = r241668 + r241669;
        return r241670;
}
↓
double f(double x, double y, double z) {
        double r241671 = x;
        double r241672 = y;
        double r241673 = z;
        double r241674 = fma(r241671, r241672, r241673);
        double r241675 = 2.0;
        double r241676 = r241671 / r241675;
        double r241677 = r241674 + r241676;
        return r241677;
}

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 2020021 +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))