Average Error: 0.0 → 0.0
Time: 3.0s
Precision: 64
\[\left(\frac{x}{2.0} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y + 0.5, z\right)\]
\left(\frac{x}{2.0} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y + 0.5, z\right)
double f(double x, double y, double z) {
        double r10830595 = x;
        double r10830596 = 2.0;
        double r10830597 = r10830595 / r10830596;
        double r10830598 = y;
        double r10830599 = r10830598 * r10830595;
        double r10830600 = r10830597 + r10830599;
        double r10830601 = z;
        double r10830602 = r10830600 + r10830601;
        return r10830602;
}
↓
double f(double x, double y, double z) {
        double r10830603 = x;
        double r10830604 = y;
        double r10830605 = 0.5;
        double r10830606 = r10830604 + r10830605;
        double r10830607 = z;
        double r10830608 = fma(r10830603, r10830606, r10830607);
        return r10830608;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[\left(\frac{x}{2.0} + y \cdot x\right) + z\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \frac{x}{2.0}\right) + z}\]

  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 2019168 +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))