Average Error: 0.0 → 0.0
Time: 549.0ms
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y, \frac{x}{2}\right) + z\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y, \frac{x}{2}\right) + z
double f(double x, double y, double z) {
        double r256897 = x;
        double r256898 = 2.0;
        double r256899 = r256897 / r256898;
        double r256900 = y;
        double r256901 = r256900 * r256897;
        double r256902 = r256899 + r256901;
        double r256903 = z;
        double r256904 = r256902 + r256903;
        return r256904;
}
↓
double f(double x, double y, double z) {
        double r256905 = x;
        double r256906 = y;
        double r256907 = 2.0;
        double r256908 = r256905 / r256907;
        double r256909 = fma(r256905, r256906, r256908);
        double r256910 = z;
        double r256911 = r256909 + r256910;
        return r256911;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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))