Average Error: 0.0 → 0.0
Time: 628.0ms
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 r242158 = x;
        double r242159 = 2.0;
        double r242160 = r242158 / r242159;
        double r242161 = y;
        double r242162 = r242161 * r242158;
        double r242163 = r242160 + r242162;
        double r242164 = z;
        double r242165 = r242163 + r242164;
        return r242165;
}
↓
double f(double x, double y, double z) {
        double r242166 = x;
        double r242167 = y;
        double r242168 = z;
        double r242169 = fma(r242166, r242167, r242168);
        double r242170 = 2.0;
        double r242171 = r242166 / r242170;
        double r242172 = r242169 + r242171;
        return r242172;
}

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 2020059 +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))