Average Error: 0.1 → 0.1
Time: 16.7s
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 r27657 = x;
        double r27658 = y;
        double r27659 = z;
        double r27660 = r27658 * r27659;
        double r27661 = r27660 * r27659;
        double r27662 = r27657 + r27661;
        return r27662;
}
↓
double f(double x, double y, double z) {
        double r27663 = y;
        double r27664 = z;
        double r27665 = r27663 * r27664;
        double r27666 = x;
        double r27667 = fma(r27665, r27664, r27666);
        return r27667;
}

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 2019347 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))