Average Error: 0.0 → 0.0
Time: 1.3s
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 r245260 = x;
        double r245261 = 2.0;
        double r245262 = r245260 / r245261;
        double r245263 = y;
        double r245264 = r245263 * r245260;
        double r245265 = r245262 + r245264;
        double r245266 = z;
        double r245267 = r245265 + r245266;
        return r245267;
}
↓
double f(double x, double y, double z) {
        double r245268 = x;
        double r245269 = y;
        double r245270 = z;
        double r245271 = fma(r245268, r245269, r245270);
        double r245272 = 2.0;
        double r245273 = r245268 / r245272;
        double r245274 = r245271 + r245273;
        return r245274;
}

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 2019353 +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))