Average Error: 0.1 → 0.1
Time: 17.3s
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 r21582 = x;
        double r21583 = y;
        double r21584 = z;
        double r21585 = r21583 * r21584;
        double r21586 = r21585 * r21584;
        double r21587 = r21582 + r21586;
        return r21587;
}
↓
double f(double x, double y, double z) {
        double r21588 = y;
        double r21589 = z;
        double r21590 = r21588 * r21589;
        double r21591 = x;
        double r21592 = fma(r21590, r21589, r21591);
        return r21592;
}

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 2019326 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))