Average Error: 0.0 → 0.0
Time: 4.5s
Precision: 64
\[\left(\frac{x}{2.0} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y + 0.5, z\right)\]
\left(\frac{x}{2.0} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y + 0.5, z\right)
double f(double x, double y, double z) {
        double r9010471 = x;
        double r9010472 = 2.0;
        double r9010473 = r9010471 / r9010472;
        double r9010474 = y;
        double r9010475 = r9010474 * r9010471;
        double r9010476 = r9010473 + r9010475;
        double r9010477 = z;
        double r9010478 = r9010476 + r9010477;
        return r9010478;
}
↓
double f(double x, double y, double z) {
        double r9010479 = x;
        double r9010480 = y;
        double r9010481 = 0.5;
        double r9010482 = r9010480 + r9010481;
        double r9010483 = z;
        double r9010484 = fma(r9010479, r9010482, r9010483);
        return r9010484;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[\left(\frac{x}{2.0} + y \cdot x\right) + z\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \frac{x}{2.0}\right) + z}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{0.5 \cdot x + \left(z + x \cdot y\right)}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + 0.5, z\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, y + 0.5, z\right)\]

Reproduce

herbie shell --seed 2019164 +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))