Average Error: 0.0 → 0.0
Time: 529.0ms
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 r256072 = x;
        double r256073 = 2.0;
        double r256074 = r256072 / r256073;
        double r256075 = y;
        double r256076 = r256075 * r256072;
        double r256077 = r256074 + r256076;
        double r256078 = z;
        double r256079 = r256077 + r256078;
        return r256079;
}
↓
double f(double x, double y, double z) {
        double r256080 = x;
        double r256081 = y;
        double r256082 = 2.0;
        double r256083 = r256080 / r256082;
        double r256084 = fma(r256080, r256081, r256083);
        double r256085 = z;
        double r256086 = r256084 + r256085;
        return r256086;
}

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 2020042 +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))