Average Error: 0.0 → 0.0
Time: 2.7s
Precision: 64
\[2 \cdot \left(x \cdot x + x \cdot y\right)\]
↓
\[\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2\]
2 \cdot \left(x \cdot x + x \cdot y\right)
↓
\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2
double f(double x, double y) {
        double r527643 = 2.0;
        double r527644 = x;
        double r527645 = r527644 * r527644;
        double r527646 = y;
        double r527647 = r527644 * r527646;
        double r527648 = r527645 + r527647;
        double r527649 = r527643 * r527648;
        return r527649;
}
↓
double f(double x, double y) {
        double r527650 = x;
        double r527651 = y;
        double r527652 = r527650 * r527651;
        double r527653 = fma(r527650, r527650, r527652);
        double r527654 = 2.0;
        double r527655 = r527653 * r527654;
        return r527655;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[\left(x \cdot 2\right) \cdot \left(x + y\right)\]

Derivation

  1. Initial program 0.0

    \[2 \cdot \left(x \cdot x + x \cdot y\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (* (* x 2) (+ x y))

  (* 2 (+ (* x x) (* x y))))