Average Error: 0.1 → 0.1
Time: 24.2s
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 r27488 = x;
        double r27489 = y;
        double r27490 = z;
        double r27491 = r27489 * r27490;
        double r27492 = r27491 * r27490;
        double r27493 = r27488 + r27492;
        return r27493;
}
↓
double f(double x, double y, double z) {
        double r27494 = y;
        double r27495 = z;
        double r27496 = r27494 * r27495;
        double r27497 = x;
        double r27498 = fma(r27496, r27495, r27497);
        return r27498;
}

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 2019209 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))