Average Error: 0.0 → 0.0
Time: 5.6s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y, \frac{x}{2}\right) + z\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y, \frac{x}{2}\right) + z
double f(double x, double y, double z) {
        double r120781 = x;
        double r120782 = 2.0;
        double r120783 = r120781 / r120782;
        double r120784 = y;
        double r120785 = r120784 * r120781;
        double r120786 = r120783 + r120785;
        double r120787 = z;
        double r120788 = r120786 + r120787;
        return r120788;
}
↓
double f(double x, double y, double z) {
        double r120789 = x;
        double r120790 = y;
        double r120791 = 2.0;
        double r120792 = r120789 / r120791;
        double r120793 = fma(r120789, r120790, r120792);
        double r120794 = z;
        double r120795 = r120793 + r120794;
        return r120795;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019347 +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))