Average Error: 0.0 → 0.0
Time: 5.9s
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 r191622 = x;
        double r191623 = 2.0;
        double r191624 = r191622 / r191623;
        double r191625 = y;
        double r191626 = r191625 * r191622;
        double r191627 = r191624 + r191626;
        double r191628 = z;
        double r191629 = r191627 + r191628;
        return r191629;
}
↓
double f(double x, double y, double z) {
        double r191630 = x;
        double r191631 = y;
        double r191632 = 0.5;
        double r191633 = r191631 + r191632;
        double r191634 = z;
        double r191635 = fma(r191630, r191633, r191634);
        return r191635;
}

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 2019323 +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))