Average Error: 0.1 → 0.1
Time: 31.9s
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 r26527 = x;
        double r26528 = y;
        double r26529 = z;
        double r26530 = r26528 * r26529;
        double r26531 = r26530 * r26529;
        double r26532 = r26527 + r26531;
        return r26532;
}
↓
double f(double x, double y, double z) {
        double r26533 = y;
        double r26534 = z;
        double r26535 = r26533 * r26534;
        double r26536 = x;
        double r26537 = fma(r26535, r26534, r26536);
        return r26537;
}

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 2019199 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  (+ x (* (* y z) z)))