Average Error: 0.0 → 0.0
Time: 5.7s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y, \frac{x}{2}\right) + z\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y, \frac{x}{2}\right) + z
double f(double x, double y, double z) {
        double r238014 = x;
        double r238015 = 2.0;
        double r238016 = r238014 / r238015;
        double r238017 = y;
        double r238018 = r238017 * r238014;
        double r238019 = r238016 + r238018;
        double r238020 = z;
        double r238021 = r238019 + r238020;
        return r238021;
}
↓
double f(double x, double y, double z) {
        double r238022 = x;
        double r238023 = y;
        double r238024 = 2.0;
        double r238025 = r238022 / r238024;
        double r238026 = fma(r238022, r238023, r238025);
        double r238027 = z;
        double r238028 = r238026 + r238027;
        return r238028;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +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))