Average Error: 0.1 → 0.1
Time: 19.4s
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 r26735 = x;
        double r26736 = y;
        double r26737 = z;
        double r26738 = r26736 * r26737;
        double r26739 = r26738 * r26737;
        double r26740 = r26735 + r26739;
        return r26740;
}
↓
double f(double x, double y, double z) {
        double r26741 = y;
        double r26742 = z;
        double r26743 = r26741 * r26742;
        double r26744 = x;
        double r26745 = fma(r26743, r26742, r26744);
        return r26745;
}

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 2019325 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))