Average Error: 0.0 → 0.0
Time: 516.0ms
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y, z\right) + \frac{x}{2}\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y, z\right) + \frac{x}{2}
double f(double x, double y, double z) {
        double r232720 = x;
        double r232721 = 2.0;
        double r232722 = r232720 / r232721;
        double r232723 = y;
        double r232724 = r232723 * r232720;
        double r232725 = r232722 + r232724;
        double r232726 = z;
        double r232727 = r232725 + r232726;
        return r232727;
}
↓
double f(double x, double y, double z) {
        double r232728 = x;
        double r232729 = y;
        double r232730 = z;
        double r232731 = fma(r232728, r232729, r232730);
        double r232732 = 2.0;
        double r232733 = r232728 / r232732;
        double r232734 = r232731 + r232733;
        return r232734;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 +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))