Average Error: 0.0 → 0.0
Time: 2.1s
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 r214467 = x;
        double r214468 = 2.0;
        double r214469 = r214467 / r214468;
        double r214470 = y;
        double r214471 = r214470 * r214467;
        double r214472 = r214469 + r214471;
        double r214473 = z;
        double r214474 = r214472 + r214473;
        return r214474;
}
↓
double f(double x, double y, double z) {
        double r214475 = x;
        double r214476 = y;
        double r214477 = z;
        double r214478 = fma(r214475, r214476, r214477);
        double r214479 = 2.0;
        double r214480 = r214475 / r214479;
        double r214481 = r214478 + r214480;
        return r214481;
}

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 2020018 +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))