Average Error: 0.1 → 0.1
Time: 19.2s
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 r28692 = x;
        double r28693 = y;
        double r28694 = z;
        double r28695 = r28693 * r28694;
        double r28696 = r28695 * r28694;
        double r28697 = r28692 + r28696;
        return r28697;
}
↓
double f(double x, double y, double z) {
        double r28698 = y;
        double r28699 = z;
        double r28700 = r28698 * r28699;
        double r28701 = x;
        double r28702 = fma(r28700, r28699, r28701);
        return r28702;
}

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 2019323 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))