Average Error: 0.0 → 0.0
Time: 813.0ms
Precision: 64
\[\left(x \cdot y + x\right) + y\]
↓
\[\mathsf{fma}\left(y, x, y\right) + x\]
\left(x \cdot y + x\right) + y
↓
\mathsf{fma}\left(y, x, y\right) + x
double f(double x, double y) {
        double r99549 = x;
        double r99550 = y;
        double r99551 = r99549 * r99550;
        double r99552 = r99551 + r99549;
        double r99553 = r99552 + r99550;
        return r99553;
}
↓
double f(double x, double y) {
        double r99554 = y;
        double r99555 = x;
        double r99556 = fma(r99554, r99555, r99554);
        double r99557 = r99556 + r99555;
        return r99557;
}

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. Using strategy rm

  4. Applied fma-udef0.0

    \[\leadsto y + \color{blue}{\left(x \cdot y + x\right)}\]

  5. Applied associate-+r+0.0

    \[\leadsto \color{blue}{\left(y + x \cdot y\right) + x}\]

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))