Average Error: 0.0 → 0.0
Time: 1.4s
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 r13761899 = x;
        double r13761900 = 2.0;
        double r13761901 = r13761899 / r13761900;
        double r13761902 = y;
        double r13761903 = r13761902 * r13761899;
        double r13761904 = r13761901 + r13761903;
        double r13761905 = z;
        double r13761906 = r13761904 + r13761905;
        return r13761906;
}
↓
double f(double x, double y, double z) {
        double r13761907 = z;
        double r13761908 = y;
        double r13761909 = x;
        double r13761910 = 2.0;
        double r13761911 = r13761909 / r13761910;
        double r13761912 = fma(r13761908, r13761909, r13761911);
        double r13761913 = r13761907 + r13761912;
        return r13761913;
}

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))