Average Error: 0.0 → 0.0
Time: 1.9s
Precision: 64
\[\left(\frac{x}{2.0} + y \cdot x\right) + z\]
↓
\[z + \mathsf{fma}\left(y, x, \frac{x}{2.0}\right)\]
\left(\frac{x}{2.0} + y \cdot x\right) + z
↓
z + \mathsf{fma}\left(y, x, \frac{x}{2.0}\right)
double f(double x, double y, double z) {
        double r11181212 = x;
        double r11181213 = 2.0;
        double r11181214 = r11181212 / r11181213;
        double r11181215 = y;
        double r11181216 = r11181215 * r11181212;
        double r11181217 = r11181214 + r11181216;
        double r11181218 = z;
        double r11181219 = r11181217 + r11181218;
        return r11181219;
}
↓
double f(double x, double y, double z) {
        double r11181220 = z;
        double r11181221 = y;
        double r11181222 = x;
        double r11181223 = 2.0;
        double r11181224 = r11181222 / r11181223;
        double r11181225 = fma(r11181221, r11181222, r11181224);
        double r11181226 = r11181220 + r11181225;
        return r11181226;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[\left(\frac{x}{2.0} + y \cdot x\right) + z\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \frac{x}{2.0}\right) + z}\]

  3. Final simplification0.0

    \[\leadsto z + \mathsf{fma}\left(y, x, \frac{x}{2.0}\right)\]

Reproduce

herbie shell --seed 2019163 +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))