Average Error: 0.0 → 0.0
Time: 3.6s
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 r191634 = x;
        double r191635 = 2.0;
        double r191636 = r191634 / r191635;
        double r191637 = y;
        double r191638 = r191637 * r191634;
        double r191639 = r191636 + r191638;
        double r191640 = z;
        double r191641 = r191639 + r191640;
        return r191641;
}
↓
double f(double x, double y, double z) {
        double r191642 = x;
        double r191643 = y;
        double r191644 = 0.5;
        double r191645 = r191643 + r191644;
        double r191646 = z;
        double r191647 = fma(r191642, r191645, r191646);
        return r191647;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 +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))