Average Error: 0.0 → 0.0
Time: 1.5s
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 r135512 = x;
        double r135513 = 2.0;
        double r135514 = r135512 / r135513;
        double r135515 = y;
        double r135516 = r135515 * r135512;
        double r135517 = r135514 + r135516;
        double r135518 = z;
        double r135519 = r135517 + r135518;
        return r135519;
}
↓
double f(double x, double y, double z) {
        double r135520 = x;
        double r135521 = y;
        double r135522 = 2.0;
        double r135523 = r135520 / r135522;
        double r135524 = fma(r135520, r135521, r135523);
        double r135525 = z;
        double r135526 = r135524 + r135525;
        return r135526;
}

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 2019304 +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))