Average Error: 0.0 → 0.0
Time: 22.7s
Precision: 64
\[\left(x \cdot y + x\right) + y\]
↓
\[y + \mathsf{fma}\left(y, x, x\right)\]
\left(x \cdot y + x\right) + y
↓
y + \mathsf{fma}\left(y, x, x\right)
double f(double x, double y) {
        double r3628453 = x;
        double r3628454 = y;
        double r3628455 = r3628453 * r3628454;
        double r3628456 = r3628455 + r3628453;
        double r3628457 = r3628456 + r3628454;
        return r3628457;
}
↓
double f(double x, double y) {
        double r3628458 = y;
        double r3628459 = x;
        double r3628460 = fma(r3628458, r3628459, r3628459);
        double r3628461 = r3628458 + r3628460;
        return r3628461;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  (+ (+ (* x y) x) y))