Average Error: 0.0 → 0.0
Time: 546.0ms
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 r179409 = x;
        double r179410 = 2.0;
        double r179411 = r179409 / r179410;
        double r179412 = y;
        double r179413 = r179412 * r179409;
        double r179414 = r179411 + r179413;
        double r179415 = z;
        double r179416 = r179414 + r179415;
        return r179416;
}
↓
double f(double x, double y, double z) {
        double r179417 = x;
        double r179418 = y;
        double r179419 = z;
        double r179420 = fma(r179417, r179418, r179419);
        double r179421 = 2.0;
        double r179422 = r179417 / r179421;
        double r179423 = r179420 + r179422;
        return r179423;
}

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 2020020 +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))