Average Error: 0.0 → 0.0
Time: 540.0ms
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 r224603 = x;
        double r224604 = 2.0;
        double r224605 = r224603 / r224604;
        double r224606 = y;
        double r224607 = r224606 * r224603;
        double r224608 = r224605 + r224607;
        double r224609 = z;
        double r224610 = r224608 + r224609;
        return r224610;
}
↓
double f(double x, double y, double z) {
        double r224611 = x;
        double r224612 = y;
        double r224613 = z;
        double r224614 = fma(r224611, r224612, r224613);
        double r224615 = 2.0;
        double r224616 = r224611 / r224615;
        double r224617 = r224614 + r224616;
        return r224617;
}

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 2020039 +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))