Average Error: 0.0 → 0.0
Time: 2.5s
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 r202015 = x;
        double r202016 = 2.0;
        double r202017 = r202015 / r202016;
        double r202018 = y;
        double r202019 = r202018 * r202015;
        double r202020 = r202017 + r202019;
        double r202021 = z;
        double r202022 = r202020 + r202021;
        return r202022;
}
↓
double f(double x, double y, double z) {
        double r202023 = x;
        double r202024 = y;
        double r202025 = z;
        double r202026 = fma(r202023, r202024, r202025);
        double r202027 = 2.0;
        double r202028 = r202023 / r202027;
        double r202029 = r202026 + r202028;
        return r202029;
}

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 2020036 +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))