Average Error: 0.0 → 0.0
Time: 5.2s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y + 0.5, z\right)\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y + 0.5, z\right)
double f(double x, double y, double z) {
        double r168541 = x;
        double r168542 = 2.0;
        double r168543 = r168541 / r168542;
        double r168544 = y;
        double r168545 = r168544 * r168541;
        double r168546 = r168543 + r168545;
        double r168547 = z;
        double r168548 = r168546 + r168547;
        return r168548;
}
↓
double f(double x, double y, double z) {
        double r168549 = x;
        double r168550 = y;
        double r168551 = 0.5;
        double r168552 = r168550 + r168551;
        double r168553 = z;
        double r168554 = fma(r168549, r168552, r168553);
        return r168554;
}

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

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +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))