Average Error: 0.1 → 0.1
Time: 29.0s
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 r29546 = x;
        double r29547 = y;
        double r29548 = z;
        double r29549 = r29547 * r29548;
        double r29550 = r29549 * r29548;
        double r29551 = r29546 + r29550;
        return r29551;
}
↓
double f(double x, double y, double z) {
        double r29552 = y;
        double r29553 = z;
        double r29554 = r29552 * r29553;
        double r29555 = x;
        double r29556 = fma(r29554, r29553, r29555);
        return r29556;
}

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 2019208 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))