Average Error: 0.0 → 0.0
Time: 3.5s
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 r189733 = x;
        double r189734 = 2.0;
        double r189735 = r189733 / r189734;
        double r189736 = y;
        double r189737 = r189736 * r189733;
        double r189738 = r189735 + r189737;
        double r189739 = z;
        double r189740 = r189738 + r189739;
        return r189740;
}
↓
double f(double x, double y, double z) {
        double r189741 = x;
        double r189742 = y;
        double r189743 = 0.5;
        double r189744 = r189742 + r189743;
        double r189745 = z;
        double r189746 = fma(r189741, r189744, r189745);
        return r189746;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019196 +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))