Average Error: 0.0 → 0.0
Time: 1.7s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, y, \frac{x}{2}\right) + z\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, y, \frac{x}{2}\right) + z
double f(double x, double y, double z) {
        double r143875 = x;
        double r143876 = 2.0;
        double r143877 = r143875 / r143876;
        double r143878 = y;
        double r143879 = r143878 * r143875;
        double r143880 = r143877 + r143879;
        double r143881 = z;
        double r143882 = r143880 + r143881;
        return r143882;
}
↓
double f(double x, double y, double z) {
        double r143883 = x;
        double r143884 = y;
        double r143885 = 2.0;
        double r143886 = r143883 / r143885;
        double r143887 = fma(r143883, r143884, r143886);
        double r143888 = z;
        double r143889 = r143887 + r143888;
        return r143889;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019198 +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))