Average Error: 0.0 → 0.0
Time: 3.3s
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 r9917547 = x;
        double r9917548 = 2.0;
        double r9917549 = r9917547 / r9917548;
        double r9917550 = y;
        double r9917551 = r9917550 * r9917547;
        double r9917552 = r9917549 + r9917551;
        double r9917553 = z;
        double r9917554 = r9917552 + r9917553;
        return r9917554;
}
↓
double f(double x, double y, double z) {
        double r9917555 = x;
        double r9917556 = y;
        double r9917557 = 0.5;
        double r9917558 = r9917556 + r9917557;
        double r9917559 = z;
        double r9917560 = fma(r9917555, r9917558, r9917559);
        return r9917560;
}

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

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

  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 2019192 +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))