Average Error: 0.0 → 0.0
Time: 6.0s
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 r150156 = x;
        double r150157 = 2.0;
        double r150158 = r150156 / r150157;
        double r150159 = y;
        double r150160 = r150159 * r150156;
        double r150161 = r150158 + r150160;
        double r150162 = z;
        double r150163 = r150161 + r150162;
        return r150163;
}
↓
double f(double x, double y, double z) {
        double r150164 = x;
        double r150165 = y;
        double r150166 = 0.5;
        double r150167 = r150165 + r150166;
        double r150168 = z;
        double r150169 = fma(r150164, r150167, r150168);
        return r150169;
}

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 2019303 +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))