Average Error: 0.0 → 0.0
Time: 6.1s
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 r8485755 = x;
        double r8485756 = 2.0;
        double r8485757 = r8485755 / r8485756;
        double r8485758 = y;
        double r8485759 = r8485758 * r8485755;
        double r8485760 = r8485757 + r8485759;
        double r8485761 = z;
        double r8485762 = r8485760 + r8485761;
        return r8485762;
}
↓
double f(double x, double y, double z) {
        double r8485763 = x;
        double r8485764 = y;
        double r8485765 = 0.5;
        double r8485766 = r8485764 + r8485765;
        double r8485767 = z;
        double r8485768 = fma(r8485763, r8485766, r8485767);
        return r8485768;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \frac{x}{2}\right) + z}\]

  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 2019179 +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))