Average Error: 0.1 → 0.1
Time: 19.7s
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 r20237 = x;
        double r20238 = y;
        double r20239 = z;
        double r20240 = r20238 * r20239;
        double r20241 = r20240 * r20239;
        double r20242 = r20237 + r20241;
        return r20242;
}
↓
double f(double x, double y, double z) {
        double r20243 = y;
        double r20244 = z;
        double r20245 = r20243 * r20244;
        double r20246 = x;
        double r20247 = fma(r20245, r20244, r20246);
        return r20247;
}

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 2019304 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))