Average Error: 0.0 → 0.0
Time: 1.2s
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 r8858843 = x;
        double r8858844 = 2.0;
        double r8858845 = r8858843 / r8858844;
        double r8858846 = y;
        double r8858847 = r8858846 * r8858843;
        double r8858848 = r8858845 + r8858847;
        double r8858849 = z;
        double r8858850 = r8858848 + r8858849;
        return r8858850;
}
↓
double f(double x, double y, double z) {
        double r8858851 = z;
        double r8858852 = y;
        double r8858853 = x;
        double r8858854 = 2.0;
        double r8858855 = r8858853 / r8858854;
        double r8858856 = fma(r8858852, r8858853, r8858855);
        double r8858857 = r8858851 + r8858856;
        return r8858857;
}

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))