Average Error: 0.0 → 0.0
Time: 2.5s
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 r215885 = x;
        double r215886 = 2.0;
        double r215887 = r215885 / r215886;
        double r215888 = y;
        double r215889 = r215888 * r215885;
        double r215890 = r215887 + r215889;
        double r215891 = z;
        double r215892 = r215890 + r215891;
        return r215892;
}
↓
double f(double x, double y, double z) {
        double r215893 = x;
        double r215894 = y;
        double r215895 = 0.5;
        double r215896 = r215894 + r215895;
        double r215897 = z;
        double r215898 = fma(r215893, r215896, r215897);
        return r215898;
}

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 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))