Average Error: 0.0 → 0.0
Time: 3.2s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, 0.5 + y, z\right)\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, 0.5 + y, z\right)
double f(double x, double y, double z) {
        double r195482 = x;
        double r195483 = 2.0;
        double r195484 = r195482 / r195483;
        double r195485 = y;
        double r195486 = r195485 * r195482;
        double r195487 = r195484 + r195486;
        double r195488 = z;
        double r195489 = r195487 + r195488;
        return r195489;
}
↓
double f(double x, double y, double z) {
        double r195490 = x;
        double r195491 = 0.5;
        double r195492 = y;
        double r195493 = r195491 + r195492;
        double r195494 = z;
        double r195495 = fma(r195490, r195493, r195494);
        return r195495;
}

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

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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))