Average Error: 0.0 → 0.0
Time: 1.2s
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 r137609 = x;
        double r137610 = 2.0;
        double r137611 = r137609 / r137610;
        double r137612 = y;
        double r137613 = r137612 * r137609;
        double r137614 = r137611 + r137613;
        double r137615 = z;
        double r137616 = r137614 + r137615;
        return r137616;
}
↓
double f(double x, double y, double z) {
        double r137617 = x;
        double r137618 = y;
        double r137619 = 2.0;
        double r137620 = r137617 / r137619;
        double r137621 = fma(r137617, r137618, r137620);
        double r137622 = z;
        double r137623 = r137621 + r137622;
        return r137623;
}

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 2019306 +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))