Average Error: 0.1 → 0.1
Time: 18.8s
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 r28734 = x;
        double r28735 = y;
        double r28736 = z;
        double r28737 = r28735 * r28736;
        double r28738 = r28737 * r28736;
        double r28739 = r28734 + r28738;
        return r28739;
}
↓
double f(double x, double y, double z) {
        double r28740 = y;
        double r28741 = z;
        double r28742 = r28740 * r28741;
        double r28743 = x;
        double r28744 = fma(r28742, r28741, r28743);
        return r28744;
}

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)))