Average Error: 0.0 → 0.0
Time: 1.0s
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 r21949 = x;
        double r21950 = 2.0;
        double r21951 = r21949 / r21950;
        double r21952 = y;
        double r21953 = r21952 * r21949;
        double r21954 = r21951 + r21953;
        double r21955 = z;
        double r21956 = r21954 + r21955;
        return r21956;
}
↓
double f(double x, double y, double z) {
        double r21957 = x;
        double r21958 = y;
        double r21959 = 2.0;
        double r21960 = r21957 / r21959;
        double r21961 = fma(r21957, r21958, r21960);
        double r21962 = z;
        double r21963 = r21961 + r21962;
        return r21963;
}

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 2019310 +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))