Average Error: 0.0 → 0.0
Time: 3.4s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y, z\right) + \frac{x}{2}\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y, z\right) + \frac{x}{2}
double f(double x, double y, double z) {
        double r215633 = x;
        double r215634 = 2.0;
        double r215635 = r215633 / r215634;
        double r215636 = y;
        double r215637 = r215636 * r215633;
        double r215638 = r215635 + r215637;
        double r215639 = z;
        double r215640 = r215638 + r215639;
        return r215640;
}
↓
double f(double x, double y, double z) {
        double r215641 = x;
        double r215642 = y;
        double r215643 = z;
        double r215644 = fma(r215641, r215642, r215643);
        double r215645 = 2.0;
        double r215646 = r215641 / r215645;
        double r215647 = r215644 + r215646;
        return r215647;
}

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

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

  3. Final simplification0.0

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

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))