Average Error: 0.0 → 0.0
Time: 804.0ms
Precision: 64
\[\left(x \cdot y + x\right) + y\]
↓
\[y + \mathsf{fma}\left(x, y, x\right)\]
\left(x \cdot y + x\right) + y
↓
y + \mathsf{fma}\left(x, y, x\right)
double f(double x, double y) {
        double r86656 = x;
        double r86657 = y;
        double r86658 = r86656 * r86657;
        double r86659 = r86658 + r86656;
        double r86660 = r86659 + r86657;
        return r86660;
}
↓
double f(double x, double y) {
        double r86661 = y;
        double r86662 = x;
        double r86663 = fma(r86662, r86661, r86662);
        double r86664 = r86661 + r86663;
        return r86664;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[\left(x \cdot y + x\right) + y\]

  2. Simplified0.0

    \[\leadsto \color{blue}{y + \mathsf{fma}\left(x, y, x\right)}\]

  3. Final simplification0.0

    \[\leadsto y + \mathsf{fma}\left(x, y, x\right)\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))