Average Error: 0.0 → 0.0
Time: 10.5s
Precision: 64
\[\left(\frac{x}{2.0} + y \cdot x\right) + z\]
↓
\[z + \mathsf{fma}\left(y, x, \frac{x}{2.0}\right)\]
\left(\frac{x}{2.0} + y \cdot x\right) + z
↓
z + \mathsf{fma}\left(y, x, \frac{x}{2.0}\right)
double f(double x, double y, double z) {
        double r11658974 = x;
        double r11658975 = 2.0;
        double r11658976 = r11658974 / r11658975;
        double r11658977 = y;
        double r11658978 = r11658977 * r11658974;
        double r11658979 = r11658976 + r11658978;
        double r11658980 = z;
        double r11658981 = r11658979 + r11658980;
        return r11658981;
}
↓
double f(double x, double y, double z) {
        double r11658982 = z;
        double r11658983 = y;
        double r11658984 = x;
        double r11658985 = 2.0;
        double r11658986 = r11658984 / r11658985;
        double r11658987 = fma(r11658983, r11658984, r11658986);
        double r11658988 = r11658982 + r11658987;
        return r11658988;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[\left(\frac{x}{2.0} + y \cdot x\right) + z\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \frac{x}{2.0}\right) + z}\]

  3. Final simplification0.0

    \[\leadsto z + \mathsf{fma}\left(y, x, \frac{x}{2.0}\right)\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  (+ (+ (/ x 2.0) (* y x)) z))