Average Error: 0.0 → 0.0
Time: 3.9s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y + 0.5, z\right)\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y + 0.5, z\right)
double f(double x, double y, double z) {
        double r16948 = x;
        double r16949 = 2.0;
        double r16950 = r16948 / r16949;
        double r16951 = y;
        double r16952 = r16951 * r16948;
        double r16953 = r16950 + r16952;
        double r16954 = z;
        double r16955 = r16953 + r16954;
        return r16955;
}
↓
double f(double x, double y, double z) {
        double r16956 = x;
        double r16957 = y;
        double r16958 = 0.5;
        double r16959 = r16957 + r16958;
        double r16960 = z;
        double r16961 = fma(r16956, r16959, r16960);
        return r16961;
}

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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019315 +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))