Average Error: 0.0 → 0.0
Time: 1.9s
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 r7640520 = x;
        double r7640521 = 2.0;
        double r7640522 = r7640520 / r7640521;
        double r7640523 = y;
        double r7640524 = r7640523 * r7640520;
        double r7640525 = r7640522 + r7640524;
        double r7640526 = z;
        double r7640527 = r7640525 + r7640526;
        return r7640527;
}
↓
double f(double x, double y, double z) {
        double r7640528 = z;
        double r7640529 = y;
        double r7640530 = x;
        double r7640531 = 2.0;
        double r7640532 = r7640530 / r7640531;
        double r7640533 = fma(r7640529, r7640530, r7640532);
        double r7640534 = r7640528 + r7640533;
        return r7640534;
}

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))