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 r154555 = x;
        double r154556 = 2.0;
        double r154557 = r154555 / r154556;
        double r154558 = y;
        double r154559 = r154558 * r154555;
        double r154560 = r154557 + r154559;
        double r154561 = z;
        double r154562 = r154560 + r154561;
        return r154562;
}
↓
double f(double x, double y, double z) {
        double r154563 = x;
        double r154564 = y;
        double r154565 = 2.0;
        double r154566 = r154563 / r154565;
        double r154567 = fma(r154563, r154564, r154566);
        double r154568 = z;
        double r154569 = r154567 + r154568;
        return r154569;
}

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 2019209 +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))