Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y, z\right) + \frac{x}{2}\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y, z\right) + \frac{x}{2}
double f(double x, double y, double z) {
        double r207099 = x;
        double r207100 = 2.0;
        double r207101 = r207099 / r207100;
        double r207102 = y;
        double r207103 = r207102 * r207099;
        double r207104 = r207101 + r207103;
        double r207105 = z;
        double r207106 = r207104 + r207105;
        return r207106;
}
↓
double f(double x, double y, double z) {
        double r207107 = x;
        double r207108 = y;
        double r207109 = z;
        double r207110 = fma(r207107, r207108, r207109);
        double r207111 = 2.0;
        double r207112 = r207107 / r207111;
        double r207113 = r207110 + r207112;
        return r207113;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +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))