Average Error: 0.0 → 0.0
Time: 1.5s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[z + \mathsf{fma}\left(y, x, \frac{x}{2}\right)\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
z + \mathsf{fma}\left(y, x, \frac{x}{2}\right)
double f(double x, double y, double z) {
        double r11857852 = x;
        double r11857853 = 2.0;
        double r11857854 = r11857852 / r11857853;
        double r11857855 = y;
        double r11857856 = r11857855 * r11857852;
        double r11857857 = r11857854 + r11857856;
        double r11857858 = z;
        double r11857859 = r11857857 + r11857858;
        return r11857859;
}
↓
double f(double x, double y, double z) {
        double r11857860 = z;
        double r11857861 = y;
        double r11857862 = x;
        double r11857863 = 2.0;
        double r11857864 = r11857862 / r11857863;
        double r11857865 = fma(r11857861, r11857862, r11857864);
        double r11857866 = r11857860 + r11857865;
        return r11857866;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +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))