Average Error: 0.0 → 0.0
Time: 1.4s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[z + \mathsf{fma}\left(y, x, \frac{x}{2}\right)\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
z + \mathsf{fma}\left(y, x, \frac{x}{2}\right)
double f(double x, double y, double z) {
        double r8954530 = x;
        double r8954531 = 2.0;
        double r8954532 = r8954530 / r8954531;
        double r8954533 = y;
        double r8954534 = r8954533 * r8954530;
        double r8954535 = r8954532 + r8954534;
        double r8954536 = z;
        double r8954537 = r8954535 + r8954536;
        return r8954537;
}
↓
double f(double x, double y, double z) {
        double r8954538 = z;
        double r8954539 = y;
        double r8954540 = x;
        double r8954541 = 2.0;
        double r8954542 = r8954540 / r8954541;
        double r8954543 = fma(r8954539, r8954540, r8954542);
        double r8954544 = r8954538 + r8954543;
        return r8954544;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +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))