Average Error: 0.0 → 0.0
Time: 5.4s
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 r142158 = x;
        double r142159 = 2.0;
        double r142160 = r142158 / r142159;
        double r142161 = y;
        double r142162 = r142161 * r142158;
        double r142163 = r142160 + r142162;
        double r142164 = z;
        double r142165 = r142163 + r142164;
        return r142165;
}
↓
double f(double x, double y, double z) {
        double r142166 = x;
        double r142167 = y;
        double r142168 = 0.5;
        double r142169 = r142167 + r142168;
        double r142170 = z;
        double r142171 = fma(r142166, r142169, r142170);
        return r142171;
}

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 2019235 +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))