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 r7499077 = x;
        double r7499078 = 2.0;
        double r7499079 = r7499077 / r7499078;
        double r7499080 = y;
        double r7499081 = r7499080 * r7499077;
        double r7499082 = r7499079 + r7499081;
        double r7499083 = z;
        double r7499084 = r7499082 + r7499083;
        return r7499084;
}
↓
double f(double x, double y, double z) {
        double r7499085 = z;
        double r7499086 = y;
        double r7499087 = x;
        double r7499088 = 2.0;
        double r7499089 = r7499087 / r7499088;
        double r7499090 = fma(r7499086, r7499087, r7499089);
        double r7499091 = r7499085 + r7499090;
        return r7499091;
}

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 2019158 +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))