Average Error: 0.0 → 0.0
Time: 613.0ms
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 r104263 = x;
        double r104264 = y;
        double r104265 = r104263 * r104264;
        double r104266 = r104265 + r104263;
        double r104267 = r104266 + r104264;
        return r104267;
}
↓
double f(double x, double y) {
        double r104268 = y;
        double r104269 = x;
        double r104270 = fma(r104268, r104269, r104268);
        double r104271 = r104270 + r104269;
        return r104271;
}

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 2020065 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))