Average Error: 0.0 → 0.0
Time: 3.3s
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 r215828 = x;
        double r215829 = 2.0;
        double r215830 = r215828 / r215829;
        double r215831 = y;
        double r215832 = r215831 * r215828;
        double r215833 = r215830 + r215832;
        double r215834 = z;
        double r215835 = r215833 + r215834;
        return r215835;
}
↓
double f(double x, double y, double z) {
        double r215836 = x;
        double r215837 = y;
        double r215838 = z;
        double r215839 = fma(r215836, r215837, r215838);
        double r215840 = 2.0;
        double r215841 = r215836 / r215840;
        double r215842 = r215839 + r215841;
        return r215842;
}

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 2020060 +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))