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 r219637 = x;
        double r219638 = 2.0;
        double r219639 = r219637 / r219638;
        double r219640 = y;
        double r219641 = r219640 * r219637;
        double r219642 = r219639 + r219641;
        double r219643 = z;
        double r219644 = r219642 + r219643;
        return r219644;
}
↓
double f(double x, double y, double z) {
        double r219645 = z;
        double r219646 = y;
        double r219647 = x;
        double r219648 = 2.0;
        double r219649 = r219647 / r219648;
        double r219650 = fma(r219646, r219647, r219649);
        double r219651 = r219645 + r219650;
        return r219651;
}

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))