Average Error: 0.0 → 0.0
Time: 1.3s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(y, x, \frac{x}{2}\right) + z\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(y, x, \frac{x}{2}\right) + z
double f(double x, double y, double z) {
        double r12640874 = x;
        double r12640875 = 2.0;
        double r12640876 = r12640874 / r12640875;
        double r12640877 = y;
        double r12640878 = r12640877 * r12640874;
        double r12640879 = r12640876 + r12640878;
        double r12640880 = z;
        double r12640881 = r12640879 + r12640880;
        return r12640881;
}
↓
double f(double x, double y, double z) {
        double r12640882 = y;
        double r12640883 = x;
        double r12640884 = 2.0;
        double r12640885 = r12640883 / r12640884;
        double r12640886 = fma(r12640882, r12640883, r12640885);
        double r12640887 = z;
        double r12640888 = r12640886 + r12640887;
        return r12640888;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +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))