Average Error: 0.1 → 0.1
Time: 28.9s
Precision: 64
\[x + \left(y \cdot z\right) \cdot z\]
↓
\[\mathsf{fma}\left(y \cdot z, z, x\right)\]
x + \left(y \cdot z\right) \cdot z
↓
\mathsf{fma}\left(y \cdot z, z, x\right)
double f(double x, double y, double z) {
        double r27882 = x;
        double r27883 = y;
        double r27884 = z;
        double r27885 = r27883 * r27884;
        double r27886 = r27885 * r27884;
        double r27887 = r27882 + r27886;
        return r27887;
}
↓
double f(double x, double y, double z) {
        double r27888 = y;
        double r27889 = z;
        double r27890 = r27888 * r27889;
        double r27891 = x;
        double r27892 = fma(r27890, r27889, r27891);
        return r27892;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[x + \left(y \cdot z\right) \cdot z\]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))