Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\frac{x}{2} + \mathsf{fma}\left(y, x, z\right)\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\frac{x}{2} + \mathsf{fma}\left(y, x, z\right)
double f(double x, double y, double z) {
        double r276439 = x;
        double r276440 = 2.0;
        double r276441 = r276439 / r276440;
        double r276442 = y;
        double r276443 = r276442 * r276439;
        double r276444 = r276441 + r276443;
        double r276445 = z;
        double r276446 = r276444 + r276445;
        return r276446;
}
↓
double f(double x, double y, double z) {
        double r276447 = x;
        double r276448 = 2.0;
        double r276449 = r276447 / r276448;
        double r276450 = y;
        double r276451 = z;
        double r276452 = fma(r276450, r276447, r276451);
        double r276453 = r276449 + r276452;
        return r276453;
}

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. Using strategy rm

  3. Applied associate-+l+0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019350 +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))