Average Error: 0.0 → 0.0
Time: 1.9s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, 0.5 + y, z\right)\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, 0.5 + y, z\right)
double f(double x, double y, double z) {
        double r185891 = x;
        double r185892 = 2.0;
        double r185893 = r185891 / r185892;
        double r185894 = y;
        double r185895 = r185894 * r185891;
        double r185896 = r185893 + r185895;
        double r185897 = z;
        double r185898 = r185896 + r185897;
        return r185898;
}
↓
double f(double x, double y, double z) {
        double r185899 = x;
        double r185900 = 0.5;
        double r185901 = y;
        double r185902 = r185900 + r185901;
        double r185903 = z;
        double r185904 = fma(r185899, r185902, r185903);
        return r185904;
}

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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 +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))