Average Error: 0.0 → 0.0
Time: 1.5s
Precision: 64
\[\left(\frac{x}{2.0} + y \cdot x\right) + z\]
↓
\[z + \mathsf{fma}\left(y, x, \frac{x}{2.0}\right)\]
\left(\frac{x}{2.0} + y \cdot x\right) + z
↓
z + \mathsf{fma}\left(y, x, \frac{x}{2.0}\right)
double f(double x, double y, double z) {
        double r8853317 = x;
        double r8853318 = 2.0;
        double r8853319 = r8853317 / r8853318;
        double r8853320 = y;
        double r8853321 = r8853320 * r8853317;
        double r8853322 = r8853319 + r8853321;
        double r8853323 = z;
        double r8853324 = r8853322 + r8853323;
        return r8853324;
}
↓
double f(double x, double y, double z) {
        double r8853325 = z;
        double r8853326 = y;
        double r8853327 = x;
        double r8853328 = 2.0;
        double r8853329 = r8853327 / r8853328;
        double r8853330 = fma(r8853326, r8853327, r8853329);
        double r8853331 = r8853325 + r8853330;
        return r8853331;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[\left(\frac{x}{2.0} + y \cdot x\right) + z\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \frac{x}{2.0}\right) + z}\]

  3. Final simplification0.0

    \[\leadsto z + \mathsf{fma}\left(y, x, \frac{x}{2.0}\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  (+ (+ (/ x 2.0) (* y x)) z))