Average Error: 0.0 → 0.0
Time: 2.4s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, 0.5 + y, z\right)\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, 0.5 + y, z\right)
double f(double x, double y, double z) {
        double r283975 = x;
        double r283976 = 2.0;
        double r283977 = r283975 / r283976;
        double r283978 = y;
        double r283979 = r283978 * r283975;
        double r283980 = r283977 + r283979;
        double r283981 = z;
        double r283982 = r283980 + r283981;
        return r283982;
}
↓
double f(double x, double y, double z) {
        double r283983 = x;
        double r283984 = 0.5;
        double r283985 = y;
        double r283986 = r283984 + r283985;
        double r283987 = z;
        double r283988 = fma(r283983, r283986, r283987);
        return r283988;
}

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. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{0.5 \cdot x + \left(z + x \cdot y\right)}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 + y, z\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, 0.5 + y, z\right)\]

Reproduce

herbie shell --seed 2020047 +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))