Average Error: 0.0 → 0.0
Time: 3.0s
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 r222631 = x;
        double r222632 = 2.0;
        double r222633 = r222631 / r222632;
        double r222634 = y;
        double r222635 = r222634 * r222631;
        double r222636 = r222633 + r222635;
        double r222637 = z;
        double r222638 = r222636 + r222637;
        return r222638;
}
↓
double f(double x, double y, double z) {
        double r222639 = x;
        double r222640 = y;
        double r222641 = 0.5;
        double r222642 = r222640 + r222641;
        double r222643 = z;
        double r222644 = fma(r222639, r222642, r222643);
        return r222644;
}

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 2019305 +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))