Average Error: 0.0 → 0.0
Time: 6.3s
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 r9617189 = x;
        double r9617190 = 2.0;
        double r9617191 = r9617189 / r9617190;
        double r9617192 = y;
        double r9617193 = r9617192 * r9617189;
        double r9617194 = r9617191 + r9617193;
        double r9617195 = z;
        double r9617196 = r9617194 + r9617195;
        return r9617196;
}
↓
double f(double x, double y, double z) {
        double r9617197 = x;
        double r9617198 = y;
        double r9617199 = 0.5;
        double r9617200 = r9617198 + r9617199;
        double r9617201 = z;
        double r9617202 = fma(r9617197, r9617200, r9617201);
        return r9617202;
}

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 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))