Average Error: 0.0 → 0.0
Time: 4.8s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y + 0.5, z\right)\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y + 0.5, z\right)
double f(double x, double y, double z) {
        double r127459 = x;
        double r127460 = 2.0;
        double r127461 = r127459 / r127460;
        double r127462 = y;
        double r127463 = r127462 * r127459;
        double r127464 = r127461 + r127463;
        double r127465 = z;
        double r127466 = r127464 + r127465;
        return r127466;
}
↓
double f(double x, double y, double z) {
        double r127467 = x;
        double r127468 = y;
        double r127469 = 0.5;
        double r127470 = r127468 + r127469;
        double r127471 = z;
        double r127472 = fma(r127467, r127470, r127471);
        return r127472;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{x}{2}\right) + z}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{0.5 \cdot x + \left(z + x \cdot y\right)}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + 0.5, z\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, y + 0.5, z\right)\]

Reproduce

herbie shell --seed 2019199 +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))